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Eberhard Zeidler >matical AppllGQ Mathematical sciences Functional Analysis Applications to Mathematical Physics Springer-Verlag

Eberhard Zeidler Applied Functional Analysis Applications to Mathematical Physics With 56 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest

Eberhard Zeidler Department of Mathematics University of Leipzig August usplatz 10 04109 Leipzig Germany Editors J.E. Marsden L. Sirovich Department of Division of Mathematics Applied Mathematics University of California Brown University Berkeley, CA 94720 Providence, RI 02912 USA USA Mathematics Subject Classification (1991): 34A12, 42A16, 35J05 Library of Congress Cataloging-in-Publication Data Zeidler, Eberhard Applied functional analysis : applications to mathematical physics / Eberhard Zeidler p. cm. - (Applied mathematical sciences ; 108) Includes bibliographical references and index. ISBN 0-387-94442-7 1. Functional analysis. 2. Mathematical physics. I. Title. II. Series: Applied mathematical sciences (Springer-Verlag New York Inc.) ; v. 108. QA1.A647 vol. 108 [QA320] 510 s—dc20 94-43219 [515'.7] Printed on acid-free paper. (c) 1995 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Laura Carlson; manufacturing supervised by Joe Quatela. Photocomposed copy prepared from the author's lATgX files. Printed and bound by R.R. Donnelley &; Sons, Harrisonburg, VA. Printed in the United States of America. 987654321 ISBN 0-387-94442-7 Springer-Verlag New York Berlin H€$|&elbS*£

To My Students Textbooks should be attractive by showing the beauty of the subject. Johann Wolfgang von Goethe (1749-1832) I am not able to learn any mathematics unless I can see some problem I am going to solve with mathematics, and I don't understand how anyone can teach mathematics without having a battery of problems that the student is going to be inspired to want to solve and then see that he or she can use the tools for solving them. Steven Weinberg (Winner of the Nobel Prize in physics in 1979) The more I have learned about physics, the more convinced I am that physics provides, in a sense, the deepest applications of mathematics. The mathematical problems that have been solved, or techniques that have arisen out of physics in the past, have been the lifeblood of mathematics The really deep questions are still in the physical sciences. For the health of mathematics at its research level, I think it is very important to maintain that link as much as possible. Sir Michael Atiyah (Winner of the Fields Medal in 1966)

K David Hilbert Stefan Banach (1862-1943) (1892- 1945) J John von Neumann (1903 1957)

Preface A theory is the more impressive, the simpler are its premises, the more distinct are the things it connects, and the broader is its range of applicability. Albert Einstein There are two different ways of teaching mathematics, namely, (i) the systematic way, and (ii) the application-oriented way. More precisely, by (i), I mean a systematic presentation of the material governed by the desire for mathematical perfection and completeness of the results. In contrast to (i), approach (ii) starts out from the question "What are the most important applications!" and then tries to answer this question as quickly as possible. Here, one walks directly on the main road and does not wander into all the nice and interesting side roads. The present book is based on the second approach. It is addressed to undergraduate and beginning graduate students of mathematics, physics, and engineering who want to learn how functional analysis elegantly solves mathematical problems that are related to our real world and that have played an important role in the history of mathematics. The reader should sense that the theory is being developed, not simply for its own sake, but for the effective solution of concrete problems.

viii Preface This introduction to functional analysis is divided into the following two parts: Part I: Applications to mathematical physics (the present AMS Vol. 108); Part II: Main principles and their applications (AMS Vol. 109). Our presentation of the material is self-contained. As prerequisites we assume only that the reader is familiar with some basic facts from calculus. One of the special features of our introduction to functional analysis is that we try to combine the following topics on a fairly elementary level: (a) linear functional analysis; (b) nonlinear functional analysis; (c) numerical functional analysis; and (d) substantial applications related to the main stream of mathematics and physics. I think that time is ripe for such an approach. Prom a general point of view, functional analysis is based on an assimilation of analysis, geometry, algebra, and topology. The applications to be considered concern the following topics: ordinary differential equations (initial-value problems, boundary-eigenvalue problems, and bifurcation); linear and nonlinear integral equations; variational problems, partial differential equations, and Sobolev spaces; optimization (e.g., Cebysev approximation, control of rockets, game theory, and dual problems); Fourier series and generalized Fourier series; the Fourier transformation; generalized functions (distributions) and the role of the Green function; partial differential equations of mathematical physics (e.g., the Laplace equation, the heat equation, the wave equation, and the Schrodinger equation); time evolution and semigroups; the iV-body problem in celestial mechanics; capillary surfaces; minimal surfaces and harmonic maps; superfluids, superconductors, and phase transition (the Landau-Ginz- burg model); viscous fluids (the Navier-Stokes equations);

Preface ix boundary-value problems and obstacle problems in nonlinear elasticity; quantum mechanics (both the Schrodinger equation approach and the Feynman path integral approach); quantum statistics (both the Hilbert space approach and the C*-algebra approach); quantum field theory (the Fock space); quarks in elementary particle physics; gauge field theory (the Yang-Mills-Dirac equations); string theory. We also study the following fundamental approximation methods: iteration method via fc-contractions; iteration method via monotonicity in ordered Banach spaces; the Ritz method and the method of finite elements; the dual Ritz method (also called the TrefFtz method); the Galerkin method; and cubature formulas. We shall make no attempt to present concepts in the most general way but will rather try to expose their essential core without, on the other hand, trivializing them. In the experience of the author, it is substantially easier for the student to take a mathematical concept and extend it to a more general situation than to struggle through a theorem formulated in its broadest generality and burdened with numerous technicalities in an attempt to divine the basic concept. Here it is the teacher's duty to be helpful. To assist the reader in recognizing the central results, these propositions are denoted as "theorems." A list of the theorems along with a list of the most important definitions can be found at the end of this book. Furthermore, a number of schematic overviews should help the reader to understand the interrelations between the abstract principles and their applications. Functional analysis is a child of the twentieth century. It provides us with a new language that allows us to formulate apparently different topics in a unique way. It seems that functional analysis is deeply rooted in our real world, since it is the appropriate tool for describing quantum phenomena in terms of mathematics. For example, the famous Heisenberg uncertainty principle on position and momentum of particles follows easily from the Schwarz inequality, which represents the most important inequality in Hilbert space theory. In the study of many problems, the following steps are used: (i) translating the given concrete problem into the language of functional analysis;

x Preface (ii) applying abstract functional analytic theorems; (iii) verifying the assumptions in step (ii), which often requires applying very specific analytical tools. The basic idea of functional analysis is to formulate differential and integral equations in terms of operator equations. For example, the operator equation Au = f, ueX, (E) may represent a concise formulation of the following integral equation for the unknown function u: u(x) - / A(x,y)<p(u(y),y)dy = f(x), a<x<b, (Eint) J a provided we introduce the operator A through (Au)(x) := u(x) — I A(x,y)(/)(u(y),y)dy for all x G [a, b]. (Op) J a From the abstract point of view, we assume that u is an element of the "space" X, where X := C[a,b] denotes the set of all continuous functions u: [a, b] -> R. More precisely, the definition of the operator A is to be understood in the following sense. To each function u £ X we assign a new function Au on the interval [a, b) given by (Op). For example, iiu(x) = 1, then the function Au is given by fb ' (Au)(x) = 1 — / A(x,y)(j)(l,y)dy for all x G [a, b], J a The set X is also called a function space. Typically, functional analysis employs the fact that the function spaces possess an additional structure. For example, the space C[a, b] can be equipped with the norm Nl := max \u(x)\. a<x<b We call \\u\\ the length of the vector (function) u. This way, C[a, b] becomes a so-called Banach space. In the special case where </>(u, y) = u, the integral equation (Eint) is said to be linear. Generally, the importance of nonlinear problems stems from the fact that they describe processes in nature with interactions. In contrast to the integral equation (Eint), the operator equation (E) may also correspond to the following boundary-value problem for a differential equation: u"{x) + c(x)u(x) = f(x), a < x <b, (Ediff) u(a) = u(b) = 0 (boundary condition),

Preface xi provided we define the operator A through (Au)(x) := u"(x) + c(x)u(x) for all x G [a, b]. Naturally enough, we now assume that u is an element of the space X, where X denotes the set of all functions u: [a, b] —> R that are twice continuously differentiable on the interval [a, b] and that satisfy the boundary condition u(a) = u(b) = 0. Finally, set u := (1/1,1/2), / := (fi, f2), where 1/1, u2, /1, and /2 are real numbers, i.e., w,/El2. Then, the following system of real equations Ai(ui,u2) = /i, (Esys) A2 (1/1,1/2) = h corresponds to the original operator equation (E), too. Here we define the operator A through An := (Ai(ui,u2), A2(ui, u2)) for all u G X, where we set X := R2. Obviously, u G X implies Au G X. Thus, the operator A: X —> X maps the space X into itself. Furthermore, for example, the abstract minimum problem F(iz) = min!, u G X, (M) corresponds to Euler's classical variational problem I L(x,u(x),u/(x))dx = min!, Ja (Mvar) u(a) = u(b) = 0 (boundary condition), provided we set F(u) := / L(x,u(x),u'(x))dx for all u € X, Ja where X denotes an appropriate space of functions that satisfy the boundary condition u(a) = u(b) = 0. Since F(u) is a real number for each function u G X, the operator F: X —> R from the space X to the space R of real numbers is called a functional In addition, many problems in optimization and control theory can be formulated in terms of the abstract minimum problem (M). Roughly speaking: Functional analysis provides us with existence theorems for both the operator equation (E) and the minimum problem (M) and with convergent approximations methods for (E) and (M).

xii Preface Typically, the spaces X are infinite-dimensional Prom the physical point of view, such spaces describe physical systems with an infinite number of degrees of freedom. Problems of the type minmaxL(u,p) = maxminL(u,p) = L(uo,po) (Minimax) ueA peB peB ueA and, more generally, inf sup L(u,p) = sup inf L(u,p) ueApeB PeBu^A represent basic problems in game theory and duality theory. This will be shown in Chapter 2 of AMS Vol. 109. Functional analysis also establishes a calculus for linear operators. For example, let us consider the abstract differential equation u'{i) = Au(i), t > 0, (D) u(0) = uq (initial condition), where A is a linear operator. Formally, the solution of (D) is given by u(t) = etAu0. It is the goal of the theory of semigroups to give the formal symbol etA a rigorous meaning. Equation (D) describes many time-dependent processes in nature. It turns out that if (D) corresponds to an irreversible process in nature (e.g., diffusion or heat conduction), then the symbol etA only makes sense for time t > 0. Let us briefly discuss the contents of the present AMS Vol. 108 and of AMS Vol. 109. Chapter 1 concerns Banach spaces. For the convenience of the reader, the most important notions of functional analysis are explained in terms of the simple space C[a, b] of continuous functions without using the Lebesgue integral. This way, the first chapter may serve as a quite elementary introduction to functional analysis. The applications to be studied in Chapter 1 concern existence proofs for ordinary differential equations as well as for linear and nonlinear integral equations. Here, we will use the two most important fixed-point theorems due to Banach and Schauder. We also justify the following fundamental principle in mathematics: A priori estimates yield existence. In an abstract functional analytic setting, this principle was established by Leray and Schauder in 1934. Riemann's famous Dirichlet principle stands at the beginning of Chapter 2, which is devoted to Hilbert space theory. We give an elegant functional

Preface xiii analytic justification for the Dirichlet principle based on an existence theorem for quadratic minimum problems in Hilbert spaces. In this connection, the use of the Lebesgue integral is indispensible. Basic facts about this integral are summarized in the appendix. Thus, the book is also accessible to those readers who are not familiar with the Lebesgue integral. In fact, our abstract setting for the Dirichlet principle represents one of several equivalent formulations of the so-called linear orthogonality principle for Hilbert spaces, which will be studied in Section 2.13. In terms of geometry, the linear orthogonality principle tells us that: In Hilbert spaces, there exists a perpendicular from any point to any closed plane. In other words, there exists an orthogonal projection onto closed linear subspaces of Hilbert spaces. If one tries to generalize this fundamental orthogonality principle to nonlinear operators, then one obtains an existence theorem for so-called monotone operator equations. Each Hilbert space is a Banach space. But Hilbert spaces possess a richer structure than Banach spaces, since the concept of orthogonality is available. In Chapter 3 we shall show that complete orthonormal systems in Hilbert spaces are the right tool for solving the convergence problem for Fourier series and more general series expansions of functions. This convergence problem was a famous open problem in the nineteenth century. Hilbert discovered around 1900 that many eigenvalue problems of classical analysis for differential and integral equations can be formulated in terms of a general theory for compact symmetric operators in Hilbert spaces. This approach, which is closely related to Chapter 3, will be studied in Chapter 4. This way, it is possible to understand why the "Fourier method" of physicists works. In terms of physics, this method represents general states as superpositions of so-called eigenstates, which correspond to eigenoscillations of the system under consideration. Functional analysis rigorously establishes the old conjecture by Daniel Bernoulli (1700-1782) that physical systems with an infinite number of degrees of freedom possess an infinite number of eigenoscillations. Around 1935 Friedrichs found out that the partial differential equations of mathematical physics can be understood best by means of the Friedrichs extension of symmetric operators. This extension procedure generates self- adjoint operators, which von Neumann introduced in connection with his mathematical foundations of quantum mechanics in 1932. From the physical point of view, the Friedrichs approach is intimately related to the concept of energy. This will be studied in Chapter 5, where we also show that time-dependent processes in nature can be described mathematically either by semigroups {irreversible processes) or by one-parameter groups {reversible processes). Near 1950 Kato proved that the Schrodinger equation for large classes of

xiv Preface physical systems corresponds to a uniquely determined self-adjoint Hamil- tonian. This way Kato showed that von Neumann's abstract setting for quantum mechanics from 1932 represents the right tool for the mathematical description of the behavior of atoms and molecules. Chapter 5 represents the heart of the present book. It is devoted to the close relations between functional analysis and both classical and modern mathematical physics. For example, in Sections 5.21 through 5.24, which discuss the Dirac calculus and the Feynman path integral in quantum physics, we try to build a bridge between the language and thoughts of physicists and mathematicians. The mathematician should have the following in mind. Until today, it has not been possible to develop a mathematically rigorous quantum field theory for describing the behavior of elementary particles. For about 40 years, however, physicists have worked with dubious mathematical methods that are in fantastic coincidence with experiment (e.g., in quantum electrodynamics) . As a typical example for the difference between the language of physicists and mathematicians, let us consider the "delta function" <5, which the famous physicist Paul Dirac introduced around 1930. In terms of physics, the function 6 = 6(x) describes the mass density of a point of mass m = 1 at x = 0 on the real line. This physical interpretation of 6 leads us immediately to c/ x /0 ifx^O m ^ = \+oo ifz = 0, W as well as /•OO 6(x)dx = total mass = 1 (II) / J — c and /oo f(x)6(x)dx = /(0) • (mass at 0) = /(0). (Ill) -OO Using the substitution x := z — y and g(z) := f(z — y), we also get /oo g(z)6(z - y)dz = g(y) for all y e R. (IV) -OO Set u(x) := 8{x — y). Applying (IV) to the Fourier transformation /oo e~ikxu{x)dx for all k € R -OO and the inverse Fourier transformation /oo eikxv(k)dk for all xGR, -OO

Preface xv we formally obtain that /oo e~ikx6(x - y)dx for all k, y € R (V) -OO and /oo eiHx-y)dk for all x,yeR. (VI) -oo Prom a mathematical point of view, there is no classical function 8 that satisfies (I) and (II). At a first glance, it seems that (I) along with (II) is nonsense. In the introduction to his famous 1932 monograph Foundations of Quantum Mechanics, John von Neumann points that the Dirac calculus lacks a rigorous justification. Therefore, von Neumann did not use this calculus. Around 1950 Laurent Schwartz created the theory of generalized functions (distributions), which allows a rigorous definition of the delta distribution related to Dirac's "delta function." As we will show in Chapters 2 and 3, the theory of generalized functions gives formulas (III) through (VI) a precise meaning. However, physics textbooks do not use the rigorous mathematical approach to generalized functions. Physicists prefer formulas (I) through (VI) because of their mnemotechnical elegance. Experience shows that, generally, the calculi used by physicists possess the advantage of working on their own and leading very quickly to the desired results at least on a heuristic level. Therefore, it is useful to learn both the language of physicists and the language of mathematicians. The present book tries to support this. A mathematician who teaches mathematics to physics students should try to help the students understand the differences and connections between the two different languages of mathematics and physics. In order to avoid confusion, we clearly distinguish between physical motivations and purely mathematical results. The word "proof is always understood in the sense of a rigorous mathematical proof. Let us now briefly discuss the contents of AMS Vol. 109. In Chapter 1 of AMS Vol. 109 we show that the Hahn-Banach theorem allows us to solve interesting convex optimization problems. Here, in terms of geometry, we use the separation of convex sets by hyperplanes. Chapter 2 of AMS Vol. 109 is devoted to variational principles. In particular, we generalize the classical Weierstrass existence theorem for minimum problems via weak convergence. Furthermore, we consider the Ekeland variational principle on the existence of quasi-minimal points. For example, combining this principle with the Palais-Smale condition, we will get the mountain pass theorem on saddle points. Functional analysis explains why the nineteenth-century mathematicians encountered many difficulties in establishing existence theorems for variational problems. The reason for this is the following simple geometric fact: The closed unit ball in an infinite-dimensional Banach space is not compact.

xvi Preface At the end of the 1920s, Banach proved a number of important theorems on linear continuous operators in Banach spaces, which follow from the Baire category theorem, which, in turn, is a consequence of a straightforward generalization of Cantor's nested interval principle to Banach spaces. These so-called principles of linear functional analysis are presented in Chapter 3 of AMS Vol. 109. Applications to linear and nonlinear operator equations are studied in Chapters 4 and 5 in AMS Vol. 109. In particular, in Chapter 4 of AMS Vol. 109 we will use the implicit function theorem in order to study the local behavior of nonlinear operators (diffeomorphisms, submersions, immersions, and subimmersions). This is important for global analysis (i.e., the theory of finite-dimensional and infinite-dimensional manifolds). Chapter 5 of AMS Vol. 109 is devoted to a study of linear and nonlinear Fredholm operators along with bifurcation theory. Many differential and integral operators correspond to Fredholm operators in appropriate function spaces. The theory of Fredholm operators generalizes the classical Fredholm alternative for integral equations formulated first by Fredholm around 1900. In fact, the theory of linear and nonlinear Fredholm operators represents the completely natural generalization of the classical theory for finite systems of real equations to infinite dimensions. Bifurcation theory mathematically models an essential change of the behavior of systems in nature (e.g., the buckling of beams, ecological catastrophes, etc.). The theory of nonlinear Fredholm operators dates back to a 1965 fundamental paper by Smale. The creation of functional analysis by Hilbert around 1900 was strongly influenced by the theory of integral equations. Until the 1930s, partial differential equations were treated by being reduced to integral equations. The more successful modern functional analytic approach to partial differential equations is based on an inspection of the operator equations that correspond directly to the differential equations (cf. (E) and (Ediff)). This approach dates back to von Neumann and Friedrichs in the 1930s. In fact, this point of view works successfully in numerical analysis, too. Note that all the basic equations of physical field theories (elasticity, hydrodynamics, thermodynamics, gas dynamics, electrodynamics, quantum mechanics, quantum field theory, general relativity, gauge field theory, etc.) are partial differential equations. It seems fair to say that the theory of integral equations has reached a certain final shape. In contrast, there are still many deep open questions in the theory of those partial differential equations related to physics. At the end of each chapter, the reader will find problems. Most of them are routine. I hope that such a carefully selected collection of fairly simple problems will help the student to check her or his basic understanding of the material. Some more advanced problems are marked with a star and provided with hints for further reading. For an in-depth presentation of nonlinear functional analysis and its many applications to the natural sciences, the reader is referred to the five-volume treatise Nonlinear Functional Anal-

Preface xvii ysis and Its Applications by the same author. In particular, Vols. 4 and 5 contain a detailed motivation of the basic equations in classical and modern mathematical physics along with both abstract existence proofs and interesting applications to concrete problems in physics, chemistry, biology, and economics,. The representation takes into account that in general no book is read completely from beginning to end. We hope that even a quick skimming of the text will suffice to grasp the essential contents. To this end, we recommend reading the introductions to the individual chapters, the definitions, the "theorems" (without proofs), and the examples (without proofs) as well as the motivations and comments in the text, which point out the meaning of the specific results. The proofs are worked out in great detail. Grasping the individual steps in the proofs as well as their essential ideas is made easier by the careful organization. It is a truism that only a precise study of the proofs enables one to penetrate more deeply into a mathematical theory. Readers have the following two options: (i) Those who want to become acquainted as quickly as possible with the Hilbert space approach to mathematical physics and numerical analysis can immediately begin with Chapter 2 after glancing at the last section of Chapter 1, which summarizes important notions concerning Banach spaces. (ii) Those interested in the main principles of functional analysis and their applications might skip to AMS Vol. 109 after reading Chapter 1. The book is based on lectures I have given for students of mathematics and physics at Leipzig University. The manuscript has been finished during a stay at the "Sonderforschungsbereich 256" of Bonn University and at the Max Planck Institute for Mathematics in Bonn. I would like to thank Professors Stefan Hildebrandt and Priedrich Hirzebruch for the invitations and the kind hospitality. Finally, my special thanks are due to Springer- Verlag for the harmonious collaboration. I hope that the reader of this book enjoys getting a feel for the unity of mathematics by discovering interrelations between apparently completely different subjects. Leipzig Spring 1995 Eberhard Zeidler

Prologue Each progress in mathematics is based on the discovery of stronger tools and easier methods, which at the same time makes it easier to understand earlier methods. By making these stronger tools and easier methods his own, it is possible for the individual researcher to orientate himself in the different branches of mathematics. The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena. David Hilbert, 1900 (Paris lecture)1 In order to understand the great achievement of Hilbert (1862-1943) in the field of analysis, it is necessary to first comment on the state of analysis at the end of the nineteenth century. After Weierstrass (1815-1897) had made sure of the foundations of complex function theory, and it has reached an impressive level, research switched to boundary-value problems, which first arose in physics. The work of Riemann (1826-1866) on complex function theory, however, had shown that boundary-value problems have great importance for pure mathematics as well. Two problems had to be solved: xIn this fundamental lecture, Hilbert formulated his famous 23 open problems, which strongly influenced the development of mathematics in the twentieth century.

xx Prologue (i) the problem of the existence of a potential function for given boundary values; and (ii) the problem of eigenoscillations of elastic bodies, for example, string and membrane. The state of the theory was bad at the end of the nineteenth century. Riemann had believed that, by using the Dirichlet principle, one could deal with these problems in a simple and uniform way. After Weierstrass' substantial criticism of the Dirichlet principle in 1870, special methods had to be developed for these problems. These methods, by C. Neumann, Schwarz, and Poincare, were very elaborate and still have great aesthetic appeal today; but because of their variety they were confusing, although at the end of the nineteenth century, Poincare (1854-1912), in particular, endeavoured with great astuteness to standardize the theory. There was, however, a lack of "simple basic facts" from which one could easily get complete results without sophisticated investigations of limiting processes. Hilbert first looked for these "simple basic facts" in the calculus of variations. He considered so-called regular variational problems which satisfy the Legendre condition. In 1900 he had an immediate and great success; he succeeded in justifying the Dirichlet principle. While Hilbert used variational methods, the Swedish mathematician Fredholm (1866-1927) approached the same goal by developing Poincare's work by using linear integral equations. In the winter semester 1900/01 Holmgren, who had come from Upsala (Sweden) to study under Hilbert in Gottingen, held a lecture in Hilbert's seminar on Predholm's work on linear integral equations which had been published the previous year. This was a decisive day in Hilbert's life, He took up Predholm's new discovering with great zeal, and combined it with his variational methods. In this way he succeeded in creating a uniform theory which solved problems (i) and (ii) above. In 1904 Hilbert's first note on the "Foundations of a General Theory of Linear Integral Equations" was published in the Gottinger Nachrichten. These results were based on lectures which Hilbert held from the summer of 1901 onwards. Fredholm had proved the existence of solutions for linear integral equations of the second kind. His result was sufficient to solve the boundary-value problems of potential theory. But Predholm's theory did not include the eigenoscillations and the expansions of arbitrary functions with respect to eigenfunctions. Only Hilbert solved this problem by using finite- dimensional approximations and a passage to the limit. In this way he obtained a generalization of the classical principal-axis transformation for symmetric matrices to infinite-dimensional matrices. The symmetry of the matrices corresponds to the symmetry of the kernels of integral equations, and it shows that the kernels appearing in oscillation problems are indeed symmetrical.

Prologue xxi Prom our point of view today, Hilbert's paper of 1904 appears clumsy, compared to the elegance of Erhard Schmidt's method published in 1907 which he developed in his dissertation written while a student of Hilbert in Gottingen. But the first step had been made. In the same year, 1904, Hilbert, in his second note, was able to apply his theory to general Sturm-Liouville eigenvalue problems. His third note in 1905 contained a very important result. Of the great problems which had Riemann posed with the complex function theory, there was still one left open; the proof of the existence of differential equations with a prescribed monodromy group. Hilbert solved this problem by reducing it to the determination of two functions which are holomorphic in both the interior and the exterior of a closed curve, and whose real and imaginary parts satisfy appropriate linear combinations on the curve (the Riemann-Hilbert problem). The solution to this problem is a classic example for the axiomatics of limiting processes demanded by Hilbert. No concrete limiting processes are used, but everything results from the existence of the Green function for the interior and the exterior of the closed curve, and from the Fredholm alternative which says that either the homogeneous integral equation has a nontrivial solution or the inhomogeneous integral equation has a solution. Hilbert soon noticed that limits are set to the method of integral equations. In order to overcome these limits he created, in his fourth and fifth notes in 1906, the general theory of quadratic forms of an infinite number of variables. Hilbert believed that with this theory he had provided analysis with a great general basis which corresponds to an axiomatics of limiting processses. The further development of mathematics has proved him to be right. Otto Blumenthal, 1932 The perfection of mathematical beauty is such that whatsoever is most beautiful and regular is also found to be most useful and excellent. D'Arcy W. Thompson, 1917 On Growth and Form

Contents Preface vii Prologue xix Contents of AMS Volume 109 xxvii 1 Banach Spaces and Fixed-Point Theorems 1 1.1 Linear Spaces and Dimension 2 1.2 Normed Spaces and Convergence 7 1.3 Banach Spaces and the Cauchy Convergence Criterion ... 10 1.4 Open and Closed Sets 15 1.5 Operators 16 1.6 The Banach Fixed-Point Theorem and the Iteration Method 18 1.7 Applications to Integral Equations 22 1.8 Applications to Ordinary Differential Equations 24 1.9 Continuity 26 1.10 Convexity 29 1.11 Compactness 33 1.12 Finite-Dimensional Banach Spaces and Equivalent Norms . 42 1.13 The Minkowski Functional and Homeomorphisms 45 1.14 The Brouwer Fixed-Point Theorem 53 1.15 The Schauder Fixed-Point Theorem 61 1.16 Applications to Integral Equations 62 1.17 Applications to Ordinary Differential Equations 63

xxiv Contents 1.18 The Leray-Schauder Principle and a priori Estimates .... 64 1.19 Sub- and Supersolutions, and the Iteration Method in Ordered Banach Spaces 66 1.20 Linear Operators 70 1.21 The Dual Space 74 1.22 Infinite Series in Normed Spaces 76 1.23 Banach Algebras and Operator Functions 76 1.24 Applications to Linear Differential Equations in Banach Spaces 80 1.25 Applications to the Spectrum 82 1.26 Density and Approximation 84 1.27 Summary of Important Notions 88 2 Hilbert Spaces, Orthogonality, and the Dirichlet , Principle 101 2.1 Hilbert Spaces 105 2.2 Standard Examples 109 2.3 Bilinear Forms 120 2.4 The Main Theorem on Quadratic Variational Problems . . 121 2.5 The Functional Analytic Justification of the Dirichlet Principle 125 2.6 The Convergence of the Ritz Method for Quadratic Variational Problems 140 2.7 Applications to Boundary-Value Problems, the Method of Finite Elements, and Elasticity 145 2.8 Generalized Functions and Linear Functionals 156 2.9 Orthogonal Projection 165 2.10 Linear Functionals and the Riesz Theorem 167 2.11 The Duality Map 169 2.12 Duality for Quadratic Variational Problems 169 2.13 The Linear Orthogonality Principle 172 2.14 Nonlinear Monotone Operators 173 2.15 Applications to the Nonlinear Lax-Milgram Theorem and the Nonlinear Orthogonality Principle 174 3 Hilbert Spaces and Generalized Fourier Series 195 3.1 Orthonormal Series 199 3.2 Applications to Classical Fourier Series 203 3.3 The Schmidt Orthogonalization Method 207 3.4 Applications to Polynomials / . . . 208 3.5 Unitary Operators 212 3.6 The Extension Principle 213 3.7 Applications to the Fourier Transformation 214 3.8 The Fourier Transform of Tempered Generalized Functions 219

Contents xxv 4 Eigenvalue Problems for Linear Compact Symmetric Operators 229 4.1 Symmetric Operators 230 4.2 The^ilbert-Schmidt Theory 232 4.3 The Predholm Alternative 237 4.4 Applications to Integral Equations 240 4.5 Applications to Boundary-Eigenvalue Value Problems . . . 245 5 Self-Adjoint Operators, the Friedrichs Extension and the Partial Differential Equations of Mathematical Physics 253 5.1 Extensions and Embeddings 260 5.2 Self-Adjoint Operators 263 5.3 The Energetic Space 273 5.4 The Energetic Extension 279 5.5 The Friedrichs Extension of Symmetric Operators 280 5.6 Applications to Boundary-Eigenvalue Problems for the Laplace Equation 285 5.7 The Poincare Inequality and Rellich's Compactness Theorem 287 5.8 Functions of Self-Adjoint Operators 293 5.9 Semigroups, One-Parameter Groups, and Their Physical Relevance 298 5.10 Applications to the Heat Equation 305 5.11 Applications to the Wave Equation 309 5.12 Applications to the Vibrating String and the Fourier Method 315 5.13 Applications to the Schrodinger Equation 323 5.14 Applications to Quantum Mechanics 327 5.15 Generalized Eigenfunctions 343 5.16 Trace Class Operators 347 5.17 Applications to Quantum Statistics 348 5.18 C*-Algebras and the Algebraic Approach to Quantum Statistics 357 5.19 The Fock Space in Quantum Field Theory and the Pauli Principle 363 5.20 A Look at Scattering Theory 368 5.21 The Language of Physicists in Quantum Physics and the Justification of the Dirac Calculus 373 5.22 The Euclidean Strategy in Quantum Physics 379 5.23 Applications to Feynman's Path Integral 385 5.24 The Importance of the Propagator in Quantum Physics . . 394 5.25 A Look at Solitons and Inverse Scattering Theory 406 Epilogue 425 Appendix 429

xxvi Contents References 443 Hints for Further Reading 455 List of Symbols 459 List of Theorems 465 List of the Most Important Definitions 467 Subject Index 471

Contents of AMS Volume 109 Preface Contents of AMS Volume 108 1 The Hahn-Banach Theorem and Optimization Problems 1.1 The Hahn-Banach Theorem 1.2 Applications to the Separation of Convex Sets 1.3 The Dual Space C[a, b]* 1.4 Applications to the Moment Problem 1.5 Minimum Norm Problems and Duality Theory 1.6 Applications to Cebysev Approximation 1.7 Applications to the Optimal Control of Rockets 2 Variational Principles and Weak Convergence 2.1 The nth Variation 2.2 Necessary and Sufficient Conditions for Local Extrema and the Classical Calculus of Variations 2.3 The Lack of Compactness in Infinite-Dimensional Banach Spaces 2.4 Weak Convergence 2.5 The Generalized Weierstrass Existence Theorem 2.6 Applications to the Calculus of Variations 2.7 Applications to Nonlinear Eigenvalue Problems

XXV111 Contents of AMS Volume 109 2.8 Reflexive Banach Spaces 2.9 Applications to Convex Minimum Problems and Variational Inequalities 2.10 Applications to Obstacle Problems in Elasticity 2.11 Saddle Points 2.12 Applications to Duality Theory 2.13 The von Neumann Minimax Theorem on the Existence of Saddle Points 2.14 Applications to Game Theory 2.15 The Ekeland Principle about Quasi-Minimal Points 2.16 Applications to a General Minimum Principle via the Palais-Smale Condition 2.17 Applications to the Mountain Pass Theorem 2.18 The Galerkin Method and Nonlinear Monotone Operators 2.19 Symmetries and Conservative Laws (The Noether Theorem) 2.20 The Basic Ideas of Gauge Field Theory 2.21 Representations of Lie Algebras 2.22 Applications to Elementary Particles 3 Principles of Linear Functional Analysis 3.1 The Baire Theorem 3.2 Application to the Existence of Nondifferentiable Continuous Functions 3.3 The Uniform Boundedness Theorem 3.4 Applications to Cubature Formulas 3.5 The Open Mapping Theorem 3.6 Product Spaces 3.7 The Closed Graph Theorem 3.8 Applications to Factor Spaces 3.9 Applications to Direct Sums and Projections 3.10 Dual Operators 3.11 The Exactness of the Duality Functor 3.12 Applications to the Closed Range Theorem and to Fredholm Alternatives 4 The Implicit Function Theorem 4.1 ra-Linear Bounded Operators 4.2 The Differential of Operators and the Frechet Derivative 4.3 Applications to Analytic Operators 4.4 Integration 4.5 Applications to the Taylor Theorem 4.6 Iterated Derivatives 4.7 The Chain Rule 4.8 The Implicit Function Theorem

Contents of AMS Volume 109 xxix 4.9 Applications to Differential Equations 4.10 Diffeomorphisms and the Local Inverse Mapping Theorem 4.11 Equivalent Maps and the Linearization Principle 4.12 The Local Normal Form for Nonlinear Double Splitting Maps 4.13 The Surjective Implicit Function Theorem 4.14 Applications to the Lagrange Multiplier Rule 5 Fredholm Operators 5.1 Duality for Linear Compact Operators 5.2 The Riesz-Schauder Theory on Hilbert Spaces 5.3 Applications to Integral Equations 5.4 Linear Fredholm Operators 5.5 The Riesz-Schauder Theory on Banach Spaces 5.6 Applications to the Spectrum of Linear Compact Operators 5.7 The Parametrix 5.8 Applications to the Perturbation of Predholm Operators 5.9 Applications to the Product Index Theorem 5.10 Fredholm Alternatives via Dual Pairs 5.11 Applications to Integral Equations and Boundary-Value Problems 5.12 Bifurcation Theory 5.13 Applications to Nonlinear Integral Equations 5.14 Applications to Nonlinear Boundary-Value Problems 5.15 Nonlinear Fredholm Operators 5.16 Interpolation Inequalities 5.17 Applications to the Navier-Stokes Equations References Subject Index

1 Banach Spaces and Fixed-Point Theorems The role of functional analysis has been decisive exactly in connection with classical problems. Almost all problems are on the applications, where functional analysis enables one to focus on a specific set of concrete analytical tasks and organize material in a clear and transparent form so that you know what the difficulties are. Concrete and functional analysis exist today in an inextricable symbiosis* When someone writes down a system of axioms, no one is going to take them seriously, unless they arise from some intuitive body of concrete subject matter that you would really want to study, and about which you really want to find out something. Felix E. Browder, 1975 In a Banach space, the so-called norm \\u\\ = nonnegative number is assigned to each element u. This generalizes the absolute value \u\ of a real number u. The norm can be used in order to define the convergence lim un = u n—*oo by means of lim \\un — u\\ = 0. n—>oo

1. Banach Spaces and Fixed-Point Theorems UN C[a, b] \ / I Banach space | —► Cauchy convergence criterion normed space ,. , , . II .1 v —► convergence and boundedness linear space ,. „. .. J n n ► dimension and convexity (linear combination au + pv) J FIGURE 1.1. As a standard example for a Banach space we will consider the space C[a,b], which consists of all continuous functions u: [a, b] —> R along with the norm \\u\\ := max |^(x)|, where —oo<a<6<oo. a<x<b Figure 1.1 shows the relations between Banach spaces and other important notions. For example, Figure 1.1 tells us that each Banach space is also a normed space, etc. In this chapter we will prove the two fundamental fixed-point theorems of Banach and Schauder along with applications to integral equations and ordinary differential equations (cf. Figures 1.2 and 1.3). We will show in Chapter 3 of AMS Vol. 109 that the fundamental implicit function theorem is a simple consequence of the Banach fixed-point theorem. The first chapter can be understood without any knowledge of the Lebesgue integral. 1.1 Linear Spaces and Dimension In the following let K := R or K := C, where R and C denote the set of real and complex numbers, respectively. Roughly speaking, in a linear space X over K it is possible to form "linear combinations" au + Pv, where u,v G X and a,/3 G K. In addition, the "usual rules" hold for au + Pv.

1.1 Linear Spaces and Dimension 3 k— contraction A (\\Au - Av\\ < k\\u - v\\, 0 < k < 1) compact operator A l Banach fixed-point theorem (Au = u) Schauder fixed-point theorem {Au = u) I Picard-Lindelof theorem for the ordinary differential equation y? = F(x, u) Peano theorem for the ordinary differential equation y? = F(x, u) FIGURE 1.2. continuous operator convex set # Brower fixed-point theorem in EJ + compactness Schauder fixed-point theorem in Banach spaces \ the Leray-Schauder principle and a priori estimates FIGURE 1.3. Definition 1. A linear space X over K is a set X together with an addition u + v, u, v G X and a scalar multiplication au, a G K, uEl, where all the usual rules are satisfied. More precisely, for all u, v G X and all a G K, u + v and au

4 1. Banach Spaces and Fixed-Point Theorems are defined elements of X such that, for all u,v,w G X and a,/3 G K, the following are true: u + v = v + u, (u + v) +w = u + (v + w), (a + (3)u = au + f3u, a(u + v) = au + av, a(j3u) = (a/3)u, au = u if a = 1. Furthermore, there exists exactly one element 0 in X such that u + 0 = u for all rz G X. (1) Finally, for each given uEl, the equation m + v = 6 (2) has exactly one solution v G X. X is called a real or complex linear space as K = R or K = C, respectively. For simplifying notation, let us write 6 := 0 and v := —u in (1) and (2), respectively. The following proposition shows that this convention makes sense. We also write w — u instead of w + (—u). Proposition 2. Let X be a linear space over K. Then: (i) a6 = 6 for all a G K. (ii) Ou = 0 for all ueX. (iii) (—a)u = —(au) for all a G K and u G X. Proof. Ad (i).1 It follows from au = a(u + 6) = au + a9 and the unique solvability of the equation au = au + v that v — a6 = 6. Ad (ii). Since au = (a + 0)u = au + Ou, we get Ou = 6. Ad (iii). It follows from 9 = Ou = (a + (—a))u = au+ (—a)u xThe Latin term "Ad (i)" stands for "proof of (i)."

1.1 Linear Spaces and Dimension 5 that -(aw) = {—a)u. □ Example 3. Let X := K. Then, X is a linear space over K, where au + (3v with a,(3 G K and u,v E X is to be understood in the classical sense. Example 4. Let X := KN, where JV = 1,2,...; that is, the set X consists of all the TV-tuples x — (f i> • • • > £w) with ffeEK for all k. Define (£i, • • • ,6v) + fai,. • •, ^at) = (£1 + fr,... ,6v + m), a(£i,...,6v) = (afi,...,a&\r), a G K. Then, X becomes a linear space over K. Obviously, 6 = (0,... ,0). Example 5. Let C[a, b] denote the set of all continuous functions u: [a, b] -> R, where —oo < a < b < oo. For u,v e C[a, 6] and a G R, let u + v and au denote the corresponding functions, i.e., (u + v){x) — u{x) + v(x) and (au)(x) = au(x) for all x G [a, 6]. Since the sum and the product of two continuous functions are again continuous, we get u + v£ C[a, b] and au G C[a, 6]. Thus, C[a, b] becomes a real linear space. Definition 6. Let X be a linear space over K. The elements ui,..., u^ of X are called linearly independent iff aiui -\ h a^UM = 0 with a/c G K for all k always implies ai = • • • = a at = 0. Let N = 1,2,.... We write dim X = N

6 1. Banach Spaces and Fixed-Point Theorems iff the maximal number of linearly independent elements in X is equal to N. The number N is called the dimension of X. We write dim X — oo iff, for each N = 1,2,..., there exist N linearly independent elements in X. In this case, X is called an infinite-dimensional space. For X = {0}, we set dim X = 0. The space X is called finite-dimensional iff 0 < dim X < oo. Example 7. Let X := K. Then dim X = 1. Here, X is considered to be a linear space over K. Proof. Let u G X with u ^ 0. Then au = 0 implies a = 0. Hence X contains at least one linearly independent element. Let u and v be two linearly independent elements of X, i.e., au + j3v = 0 with a, /? G K implies a - /? = 0. (3) Hence w^O and v ^ 0. Setting a := — and p := —1, u we obtain a contradiction to (3). Thus, there are no two linearly independent elements in X, i.e., dim X = 1. □ The following result is well known from the course on linear algebra. Example 8. Let X := KN for fixed N = 1,2,..., where X is considered to be a linear space over K. Then, dim X = N. Example 9. Let X := C[a,b]. Then, dim X = oo. Proof. Set Uk(x) := xk for all x G [a, b] and k = 0,1,... . Let ao,..., aw G R. It follows from aoUo + ai^i H h a^^N = 0 in C[a, 6] that a0 + aix + a2#2 H h olnxN = 0 for all x G [a, 6].

1.2 Normed Spaces and Convergence 7 Since a proper polynomial has only a finite number of zeros, we get a\ = • • • = o/v = 0. Consequently, for each N = 1,2,..., the elements uo,..., un in C[a, b] are linearly independent, i.e., dim C[a, b] = oo. □ Definition 10. HA and B are subsets of a linear space over K, then we set A + B olA AxB = {a + b: a G A and b G £}, = {aa:a e A}, a€ = {(a,b):aeA, b G B}. 1.2 Normed Spaces and Convergence Recall that K := R or K := C. Definition 1. Let X be a linear space over K. Then, X is called a normed space over K iff there exists a norm || • || on X, i.e., for all u, v £ X and a G K, the following are true: (i) (ii) (iii) (iv) \u\\ > 0 (i.e., \\u\\ is a nonnegative real number). HI =0iffrz = 0. \au\\ = \a\\\u\\. \u + v\\ < 11xx|| + |H| (triangle inequality). A normed space over K = R or K = C is called a real or complex normed space, respectively. The number \\u — v\\ is called the distance between the two points u and v. In particular, |HI = distance between the point u and the origin v = 0. Since — u — (—l)u, relation (iii) implies || - u\\ = \\u\\ for all u € X. (4) It follows from (iv) that \\(u + v)+ w\\ < \\u + v|| + \\w\\ < \\u\\ + ||v|| + \\w\\. Analogously, by induction, we get N J2uj 3=1 N — /J WujW ^or all ui,... ,^iv ^ -X', N = 1,2,... .

8 1. Banach Spaces and Fixed-Point Theorems Example 2. Let X := R. We set \\u\\ := \u\ for all uGR, where \u\ denotes the absolute value of the real number u. Then, X becomes a real normed space. Example 3. Let X := C. We set HI := \u\ for all u G C, where \u\ denotes the absolute value of the complex number u. Then, X becomes a complex normed space. In these two examples, the triangle inequality (iv) from Definition 1 corresponds to the classical triangle inequality for real and complex numbers. The norm generalizes the absolute value of numbers. Further examples will be considered in the next section. Proposition 4 (Generalized triangle inequality). Let X be a normed space. Then, for all u, v G X, |||«||-H|<||«±t;||<||«|H-|H|. (5) Proof. By the triangle inequality, ||«±t;|| = ||« + (±i;)||<||«|| + ||±i;|| = ||u|| + |H|, and HI = \\(u - v) + v|| < \\u - v\\ + ||v||. Hence ||u||-|M|<||«-i;||. Analogously, H-|M|<||«-u|| = ||«-t;||. This implies |HI-N|<II«-«||. Replacing v with — v and observing that u—(—v) = u+v and ||— v\\ = \\v\\, we also get |H|-||v||| < \\u + v\\. D Definition 5. Let (un) be a sequence in the normed space X, i.e., un G X for all n. We write lim un = u (6) n—>-oo

1.2 Normed Spaces and Convergence 9 iff limn_^oo ||rzn - u\\ = 0. We say that the sequence (un) converges to u. Instead of (6) we also write un —> u as n —> oo. Intuitively, the convergence (6) means that the distance \\un—u\\ between the points un and u goes to zero as n —> oo. Proposition 6. Let X be a normed space over K. Let un,vn, u,v £ X and an, a £ K /or all n = 1,2, T/ien £/ie following are met: (i) T/ie /irm£ poin£ u in (6) 25 uniquely determined. (ii) Jjf un —> u as n —> oo, £/ien £/ie sequence (un) is bounded, i.e., there exists a number r > 0 si/c/i £/m£ ||un|| < r /or all n. (iii) If un ^ u as n —> oo, £/ien ||un|| —> ||u|| as n —> oo. (iv) If un —> u and vn ^ v as n —> oo, £/ien ^n + ^n —> ^ + v as n —> oo. (v) Ifun^u and an —> a as n —> oo, £/ien <^n^n -> cm as n —> oo. Proof. Ad (i). Let un —> u and un —> v as n —> oo. Then ||w-v|| = ||(w-Wn)H-(^n-v)|| < ||u-un|| + ||un-v|| —> 0 as n —► oo. Hence ||u — v|| = 0, i.e., u = v. Ad (ii). Let un —> u as n —> oo. Hence ||un — u|| —> 0 as n —> oo. Thus, the real sequence (\\un — u\\) is bounded, i.e., there is a number R such that ||^n — ^|| < -R for all n. This implies ||^n|| = \\{un —u) + u\\ < \\un - u\\ + \\u\\ < R + ||it|| for all n. Ad (iii). Let un —> u as n —> oo. Then |||^n|| - HI| < ||^n - u\\ -> 0 as n -> oo. Ad (iv). If un —> u and vn —> i? as n —> oo, then || (l/n+Vn) - (W + V)|| = ||(lZn ~ ^) + (vn ~ v)\\ < \\un — u\\ + ||vn — -L^H —^ 0 as n —> oo.

10 1. Banach Spaces and Fixed-Point Theorems Ad (v). If un —> u and an —> a as n —> oo, then ||anun - aw|| = ||(an - a)un + a(un - u)\\ < ||(an - a)rzn|| + \\a(un - u)\\ < \an - a\ • ||itn|| + |a| • \\un - u\\ < \an — a\r + |a| • \\un — u\\ —> 0 as n —> oo. D Definition 7. The sequence (un) in the normed space X is called a Cauchy sequence iff, for each e > 0, there is a number no(e) such that ll^n — ^m|| < £ fc>r all n,m> no(e). Proposition 8. In a normed space, each convergent sequence is Cauchy. Proof. Let un —> u as n —> oo. Hence ||un — u|| —> 0 as n —> oo, i.e., for each e > 0, there is a number no(e) such that \\un — u\\<- for all n > no(e). This implies ||^n - Um\\ = \\(un -U) + (U- Um)\\ < \\un — u\\ + \\u — um\\ < £ for all n,ra > no(e). □ 1.3 Banach Spaces and the Cauchy Convergence Criterion Definition 1. The normed space X is called a Banach space iff each Cauchy sequence is convergent. Therefore, from Proposition 8 in the preceding section, we get the following so-called Cauchy convergence criterion: In a Banach space, a sequence is convergent iff it is Cauchy. Banach spaces are also called complete normed spaces. Example 2. The space X := K is a Banach space over K with the norm \\u\\ := \u\ for all uEK.

1.3 Banach Spaces and the Cauchy Convergence Criterion 11 This follows from the classical Cauchy convergence criterion. Example 3. Let N = 1,2, The space X := KN is a Banach space over K with the norm ||x|| := |x|oo? where Moo := max Ifj|, x= (fi,...,fw). l<j<iV Let xn = (fin,..., fNn). Then lim \xn - x|oo = 0 iff lim ffcn = ffe for all k = 1,..., N. (7) n—►oo n—►oo That is, the convergence xn —> x as n —> oo in X is equivalent to the convergence of the corresponding components. Proof. The inequality If fen - ffc| < \xn - ^|oo = max |fjn - fj| l<j<iV implies statement (7). In fact, if \xn — x\oq —> 0 as n —> oo, then ffcn —> f^ as n —> oo for all fc, and the converse is also true. Let us now prove that | • |oo 1S a norm. Obviously, Moo = 0 ^ f j = 0 for aH j <^> x = 0, and |oMoo = max \a\ |f7| = |a| max |f7| = \a\ |x|oo- Furthermore, the classical triangle inequality |fi+%| < lfjl + 1%1 forfj,% GK implies \x + y\oo= max |f?+ry?|< max |f7|+ max \rjj\ 1 ' i<j<NlJ Jl i<j<NlJl i<j<Nl Jl = Moo + Moo- Finally, we have to show that X is a Banach space with respect to the norm | • l^. To this end, let (xn) be a Cauchy sequence. Then Iffcn - f fern I < \xn ~ ^m|oo < £ for all 71, 771 > n0(s). Thus, the sequence (ffcn) is also Cauchy. The classical Cauchy convergence criterion implies the convergence lim ffcn =ffe, k = l,...,iV.

12 1. Banach Spaces and Fixed-Point Theorems By (7), xn —> x as n —> oo. D Example 4. Let N = 1,2, The space X := R^ is a Banach space with the Euclidean norm \\x\\ := |x|, where Moreover, lim |xn — x\ = 0 iff lim ^n = £& for all k = 1,..., N. (8) n—>-oo n—>-oo Convention 5. If we do not explicitly express the contrary, then the space RN is equipped with the Euclidean norm | • |. Proof. Statement (8) follows from (7) by using the following inequality: |«^n «^|oo _ \^n «£| _ ■** \^n ^loo* Next we want to prove that | • | is a norm. Obviously, \x\ = 0 & £j = 0 for all j o x = 0, and |ax| = \a\ \x\ for all a G R, x G R^. To prove the triangle inequality \x + y\< \x\ + \y\ for all x, y G RN, (9) we will use the classic Schwarz inequality N \ N N E^ ^E^E^ (10) for all real numbers £j, 77^, jf = 1,..., N. Hence N N + y\2 = E& + ^)2 = E 3 + 2^ + ^2 '7l j=l j=l 2 / ^ \ 2 AT /AT \ I M \ N = |x|2 + 2|x|M + M2 = (|x| + M)2.

1.3 Banach Spaces and the Cauchy Convergence Criterion 13 This implies (9). It remains to prove (10). Prom 0 < (a ± b)2 = a2 ± 2ab + b2 we get ±2ab <a2 + b2 for all a, b G R. Choosing a := £j/(X^i£|J2 and & := Wj/ \J2?=iWj) and summing over j, it follows that (Eli, <?) *&•)?) This implies (10). Finally, we have to show that RN is a Banach space with respect to the Euclidean norm | • |. To this end, let (xn) be a Cauchy sequence with respect to the norm | • |. It follows from \xn — ^m|oo < \xn — xm\ < e for all n,m> no{e) that (xn) is also a Cauchy sequence with respect to the norm | • l^. By Example 3, we get the convergence lim £kn = £fc for all fc, n—>-oo and hence (8) implies \xn — x| —> 0 as n —> oo. D Standard Example 6. Let —oo < a < b < oo. Then, X := C[a,b] is a real Banach space with the norm HI := max \u(x)\. a<x<b The convergence wn-^win!asn^oo means \\un — u\\ = max \un{x) — u{x)\ —> 0 as n —> oo, a<x<b i.e., the sequence (un) of continuous functions un:[a,b] —> R converges uniformly on [a, 6] to the continuous function u: [a, 6] —> R. Proof. We first prove that || • || is a norm. Obviously, ||cm|| = max \a\ \u(x)\ = \a\ max \u(x)\ = \a\ \\u\\, a<x<b a<x<b for all a € R, u G X, and i<^ m a< & u{x) = 0 on [a, 6] «» rz = 0 in C[a, b]. \\u\\ = 0 <^> max |^(x)| = 0 a<#<&

14 1. Banach Spaces and Fixed-Point Theorems Moreover, from \u(x) + v(x)\ < \u(x)\ + \v(x)\ we get the triangle inequality \\u + v\\ < H| + ||v||. Finally, we have to show that X = C[a,b] is a Banach space. Let (un) be a Cauchy sequence in X, i.e., \\v>n — v>m\\ — max \un(x) — um(x)\ < e for all n, ra > no(e). (11) a<x<b This implies the pointwise convergence un{x) —> uix) as n —> oo for each x £ [a, 6]. (12) Letting m —> oo in (11), we obtain max \un(x) — u(x)\ < e for all n > no(e). a<x<b Thus, the convergence in (12) is uniform on the interval [a, b]. By a classical result, this implies the continuity of the limit function u: [a, b] —> R. Hence u G X and un —> u in X as n —> oo. D Proposition 7. Le£ (wn) 6e a Cauchy sequence in the normed space X overK, which has a convergent subsequence (unt), that is, un> —> u in X as n —> oo. TAen, £Ae entire sequence converges to u, i.e., un —> u in X as n —> oo. Proof. Let £ > 0 be given. There is an no(e) such that Wun — um\\ < £ for all n, ra > no(e). Since (unf) converges to u, there exists some fixed index m such that ll^m — ^|| < £> where ra > no(s). By the triangle inequality, ll^n — ^|| < ||^n — ^m|| + ll^m ~ ^|| < 2e for all n > no(e). Hence un —> u as n —> oo. □ Corollary 8. Suppose that oo 2lK-+i-«j|| <oo,

1.4 Open and Closed Sets 15 where (un) is a sequence in a normed space X over K. Then, (un) is a Cauchy sequence in X. Proof. By the triangle inequality, for all k = 1,2,..., we get oo \\un - un+k\\ < ^jT \\uj+i - Uj\\ -> 0 as n -> oo. D j=n 1.4 Open and Closed Sets Definition 1. Let X be a normed space. For fixed u0 G X and e > 0, the set U€{u0) := {u e X: \\u - u0\\ < e} is called an e-neighborhood of the point u0. The subset M of X is called open iff, for each point u0 G M, there is some e-neighborhood Ue(u0) such that Ue(uo) C M (cf. Figure 1.4). The subset M of X is called closed iff the set X - M is open. Recall that X — M := {u £ X: u 0 M}. By an open neighborhood U(u) of the point u, we understand an open subset of X containing u. Proposition 2. Let M C X, where X is a normed space. Then, the following are equivalent: (i) M is closed. (ii) It follows from un G M for all n and un —> u as n —> oo that ue M. Proof, (i) => (ii). Let un —> u as n —> oo and un e M for all n. We have to show that u 6 M. If this is not true, then u G X — M. Since the set X — M is open, there is some ^-neighborhood Ue(u) such that U£(u) CX-M. From ||un — u\\ —> 0 as n —> oo we get ||um — u\\ < e for some index m, and hence um e U£(u),

16 1. Banach Spaces and Fixed-Point Theorems \ v •no j \ -^.— " v. (a) open set ■ U£(u0) \ J / / (b) closed set FIGURE 1.4. i.e., um G X — M. This contradicts um € M. (ii) => (i). Suppose that the set M is not closed, i.e., the set X — M is not open. Then, there exists a point ueX-M such that no ^-neighborhood U£(u) is contained in the set X — M. Thus, choosing e = ^, n = 1,2,..., we get a sequence (un) in X — M such that un G CA (u) and un € M for all n. Hence ■■ 1 Nn —v>\\< ► 0 as n —> oo. n By (ii), ue M. This contradicts ue X - M. D Example 3 (Balls). Let X be a normed space. For fixed v e X and fixed r > 0, define B := {ue X: \\u - v\\ < r}. Then, B is closed. The set B is called a closed ball of radius r around the point v. Proof. Let un e B for all n, i.e., \\v>n — v\\ < r for all n. If un —> ia as n —> oo, then ||it — v|| < r, and hence ue B. □ 1.5 Operators Definition 1. Let M and Y be sets. An operator A:M^Y

1.5 Operators 17 associates to each point winMa point v in Y denoted by v = Au. The set M is called the domain of definition of A. We also write D(A) for M. The set A(M) := {v g7:d = Au for some u G M} is called the range of A. We also write R(A) for A(M). The operator A:M —> Y is called surjective iff A(M) = Y. The operator A: M —> Y is called injective iff ^4ia = Ai? implies u = v. The operator ^4: M —> y is called bijective iff ^4 is both surjective and injective. If the operator A: M —> Y is bijective, then there exists the so-called inverse operator A~X:Y ^M defined through A~xv := i£ iff ^4ia = v. This definition makes sense, since for each given v G Y, there exists exactly one u e M such that ^4ia = v. The set ^-!(Ar) :={ueM:AueN} is called the preimage of the set AT. Operators are also called functions. Convention 2. In order to indicate conveniently that the domain of definition M of the operator A: M —> Y is contained in the set X, we frequently write A: M C X -► Y. In particular, if Y = K, then the operator A: M C X —> K is called a Example 3. Let M := [a, 6], Y := [c, d], and X := R, where —oo < a < b < oo and —oo < c < d < oo. The operator pictured in Figures 1.5(a), (b), and (c) is injective, surjective, and bijective, respectively. In addition, A is not injective in Figure 1.5(b). Example 4. Let —oo<a<6<oo, and let the function F: [a, b] x [a, b] x R -> R

18 1. Banach Spaces and Fixed-Point Theorems -H -+- a b (a) injective be continuous. We set a/ (b) surjective FIGURE 1.5. (c) bijective and (Au)(x) := / F(x,y,u(y))dy for all x G [a, 6], -/a fb (Bu)(x) := / F(x,y,u(y))dy for all x G [a, b], J a Then, we obtain the two operators A: C[a, b] -► C[a, b] and B: C[a, b] -► C[a, b]. In fact, it is a well-known classical result that the continuity of the function u: [a, b] -> R implies the continuity of the two functions Au: [a, 6] -> R and £u: [a, 6] -> R, i.e., ia G C[a, 6] implies both ^4ia G (7[a, 6] and £??i G C[a, 6]. The operators A and £? are called integral operators. 1.6 The Banach Fixed-Point Theorem and the Iteration Method The Banach fixed-point theorem represents a fundamental convergence theorem for a broad class of iteration methods. We want to solve the operator equation u = Au, ue M, by means of the following iteration method: un+i = Aun, 71 = 0,1, .... (13) (14)

1.6. The Banach Fixed-Point Theorem and the Iteration Method 19 where uq G M. Each solution of (13) is called a fixed point of the operator A. Theorem l.A (The fixed-point theorem of Banach). We assume that: (a) M is a closed nonempty set in the Banach space X over K, and (b) the operator A: M —> M is k-contractive, i.e., by definition, \\Au - Av\\ < k\\u - v|| for all u,v e M, (15) and fixed k, 0 < k < 1. Then, the following hold true: (i) Existence and uniqueness. The original equation (13) has exactly one solution u, i.e., the operator A has exactly one fixed point u on the set M. (ii) Convergence of the iteration method. For each given Uq G M, the sequence (un) constructed by (14) converges to the unique solution u of equation (13). (iii) Error estimates. For all n = 0,1,... we have the so-called a priori error estimate IK - ti|| < kn{\ - £0_1|K - "oil, (16) and the so-called a posteriori error estimate IK+i - u\\ < fc(l - fc)_1||tin+i - un\\. (17) (iv) Rate of convergence. For all n = 0,1,... we have IK+i -u\\ < k\\un-u\\. This theorem was proved by Banach in 1920. The Banach fixed-point theorem is also called the contraction principle. The a priori estimate (16) makes it possible to use a knowledge of the initial value Uq along with u\ = Auq to determine the maximal number of steps of iteration required to attain a desired level of precision. In contrast to this, the a posteriori estimate (17) allows the use of the computed values un and Mn+i to determine the accuracy of the approximation Mn+i.

20 1. Banach Spaces and Fixed-Point Theorems Experience shows that, as a rule, a posteriori estimates are better than a priori estimates. Proof. Ad (i), (ii). Step 1: We show first that (un) is a Cauchy sequence. Let n = 1,2, Using (15) we get \\Un+l ~ Un\\ = \\AUn - AUn-l\\ < k\\Un - Un-l|| = k\\Aun-i - Aun-2\\ < k2\\un-i - un-2\\ <---<fcn||tii-tio||. Now let n = 0,1,... and m = 1,2, The triangle inequality and the sum formula for the geometric series yield \\un - Un+m\\ = \\(un - Un+i) + (wn+l ~ ^n+2) H h (Mn+m-1 ~ ^n+m)|| < ||?in - tAn+l|| + ||^n+l - ^n+2|| H h ||^n+m-l ~ ^n+m|| < (fcn + fcn+1 + • • • + fcn+m_1)K - tio|| <fcn(l + fc + fc2 + -.-)IK-u0|| = fcn(l-fc)"1||w1 -Uq\\. It follows from 0 < k < 1 that kn —> 0 as n —> oo. Hence the sequence (wn) is Cauchy. Since X is a Banach space, the Cauchy sequence (un) converges, i.e., un —> ia as n —> oo. 5£ep #: We show that the limit point u is a solution of the original equation (13). Prom uq G M and wi = Auq along with A(M) C M, we get wi G M. Similarly, by induction, un e M for all n = 0,1,... . Since the set M is closed, we obtain 1*6 M, and hence Au G M. By (15), \\Aun — Au\\ < k\\un — u\\ —> 0 as n —> oo. Letting n —> oo it follows from izn+i = A?in that u = A?i. 5fep 5: We show the uniqueness of the solution w of (13). It follows from Au = u and Av = v with u,v € M that ||u - v|| = 11 Aw - Av\\ < k\\u - v\\.

1.6. The Banach Fixed-Point Theorem and the Iteration Method 21 Since 0 < k < 1, this implies \\u — v\\ = 0, and hence u = v. Ad (iii). Letting m —> oo it follows from IK - un+m\\ < kn(\ - fc)-1!!^! - w01| that IK - u\\ < kn{\ - k)-1^ - uq\\ for all n = 0,1,... . This is the error estimate (16). Let n = 0,1,... and m = 1,2, To prove the error estimate (17), observe that IK+l - ^n+m+l|| < IK+1 ~ ^n+2|| + IK+2 ~ ^n+3|| H h ||^n+m <(k + k2 + --- + km)\\un-un+1\\. Letting m —> oo we get ||wn+l - u\\ < k{\ - fc)-1|K ~ ^n+l||- Ad (iv). Observe that |K+i - u\\ = \\Aun - Au\\ < k\\un - u\\. D Example 1. Let —oo<a<6<oo. Suppose that we are given the differentiable function A: [a, b] —► [a, b] such that \A'(u)\ < k < 1 for all u G [a, b) and fixed k. Then, Theorem l.A can be applied to the equation u = An, u e [a, b] (18) with M := [a, 6], X := R, and the norm \\u\\ := \u\. In particular, equation (18) has a unique solution u. This solution corresponds to the intersection point between the graph of A and the diagonal in Figure 1.6. Proof. The set M = [a, b] is closed in the real Banach space X = R. By the classical mean value theorem, for each u, v G [a, 6], there exists a point w G [a, b] such that \Au - Av\ = \A'{w){u - v)\ < k\u - v\, i.e., the function A: [a,b] —> [a, b] is fc-contractive. Therefore, the assumptions of Theorem l.A are satisfied. □

22 1. Banach Spaces and Fixed-Point Theorems FIGURE 1.6. 1.7 Applications to Integral Equations We want to solve the integral equation u(x) = \ f F(x, y, u{y))dy + /(x), a < x < 6, (19) J a along with the iteration method un+i{x) = \l F{x,y,un{y))dy + f(x), a <x < 6, ra = 0,1,... , (19*) -/a where u0(x) = 0 and —oc < a < 6 < oo. Proposition 1. Assume the following: (a) TAe function f: [a, 6] —> R is continuous. (b) TAe function F: [a, 6] x [a, b] x R —> R is continuous, and the partial derivative Fu:[a,b] x [a,6] xR^R is also continuous. (c) TAere is a number C such that \Fu(x,y,u)\ < C for all x,y € [a, 6], wGR. (d) Le£ £Ae reaZ number A 6e #wen si/cA that (b — a)|A|£ < 1. (e) Set X := C[a,b] and \\u\\ := maxa<x<6 |w(x)|. TAen, £Ae following hold true: (i) TAe original problem (19) Aas a unique solution u 6 X. (ii) TAe sequence (un) constructed by (19*) converges to u in X.

1.7 Applications to Integral Equations 23 (iii) For all n = 0,1,2,... we get the following error estimates: IIUn-till^fc^l-fcr^ltiiH, \\un+i — u\\ < fc(l — fc)-1||?in+i — Un\\, where k := (b — a)|A|£. Proof. Define the operator (Au)(x) := A / F{x,y,u{y))dy + /(#) for all x G [a, 6]. «/ a Then, the original equation (19) corresponds to the fixed-point problem u = An. /\i u: [a, b] —> R is continuous, then so is the function Aw. [a, b] —> R. This way we get the operator AiX^X. For each x, y G [a, 6] and w,^eR, there exists a w € R such that \F(x,y,u) - F(x,y,v)\ < \Fu(x,y,w)\\u-v\ <C\u-v\, by the classical mean value theorem. This implies \\Au - Av\\ = max \(Au)(x) - (Av)(x)\ a<x<b <\\\{b — a)C max \u{x) — v(x)\. a<x<b Hence \\Au - Av\\ < k\\u - v\\ for all u, v € X. Letting M := X, the assertions follow now from the Banach fixed-point theorem (Theorem l.A in Section 1.6). □ Example 2 (Linear integral equation). Let F(x,y,u):=K(x,y)u, (20) and suppose that the function K : [a, b) x [a, 6] —> R is continuous. Then, the assumptions of Proposition 1 are satisfied with £= max \K(x,y)\. Therefore, all the statements of Proposition 1 are true for the integral equation (19) with (20). In the special case (20), the original problem (19) is called a linear integral equation.

24 1. Banach Spaces and Fixed-Point Theorems FIGURE 1.7. 1.8 Applications to Ordinary Differential Equations We want to solve the following initial-value problem: u' = F(x, u), x0 — ft < x < x0 + ft, U(X0) = Uq, where the point (xo, ^o) G R2 is given. More precisely, we are looking for a solution u = u(x) of (21) such that (21) u: [x0 — ft, x0 + ft] —> R is differentiable, and (x,u(x)) G S for all x G [xq — ft,Xq + ft], (21* with the square S := {(x, u) G R2: \x — Xo| < r and \u — u$\ < r} for fixed r > 0 (see Figure 1.7). We set X := C[xo — ft,Xo + ft] and M := {u G X: \\u — uq\\ < r}. Recall that \\u\\ = maxa<a;<b |^(x)|. Parallel to (21), (21*) let us consider the integral equation u(x) = uq + / F(y, u(y))dy, Xo — ft < x < Xo + ft, u G M, (22) Jxo along with the iteration method px Wn+iW = uo+ F(y, un(y))dy, x0 - ft < x < x0 + ft, n = 0,1,..., where uq(x) = uq. (23) Proposition 1 (The Picard-Lindelof theorem). Assume the following: (a) The function F: S —> R 25 continuous and the partial derivative FU:S ^R 25 a/50 continuous.

1.8 Applications to Ordinary Differential Equations 25 (b) We set M := max \F(x,u)\ and £ := max |Fu(x,u)|, (x,u)es (x,u)es and we choose the real number h in such a way that 0 < h < r, hM < r, and hC < 1. Then, the following hold true: (i) The original problem (21) has a unique solution of the form (21*). (ii) This is also the unique solution of the integral equation (22). (iii) The sequence (un) constructed by (23) converges to u in the Banach space X. (iv) For n = 0,1,... we have the following error estimates: \\u>n — V>\\ < kn(l — fc)_1||ui — M0||, ||un+i — u\\ < k(l — fc)_1||un+i — un\\, where k := h£. Proof. Step 1: The integral equation. Define the operator A through fx (Au)(x) := uq + / JF(y, u{y))dy for all x G [xq — h, Xq + ft]. Jxq Then, the integral equation (22) corresponds to the following fixed-point problem: u = Au, ueM. (22*) If u G M, then the function u: [xq — ft, Xo + ft] —> R is continuous and (x,u(x)) G S for all x G [xq — ft, Xo + ft]. Therefore, the function x h-> F(x,u(x)) is also continuous on the interval [x0 — ft, Xo + ft]. This implies the continuity of the function Au: [xo — ft, x0 + ft] —> M. This way we get the operator A- M ^ X. Let us prove that (a) A{M) C M.

26 1. Banach Spaces and Fixed-Point Theorems (b) \\Au - Av\\ < k\\u - v\\ for all u, v G M. Ad (a). Let u G M. Then I fx \ F{y,u{y))dy \Jxq <|x —x0| max \F{y,u)\ (y,u)eS < hM < r for all x G [x0 - ft,Xo + ft], and hence \\Au-uq\\= max / F(y,u(y))dy xo-h<x<xo+h \JXQ <r, 'Xq i.e., Au e M. Ad (b). By the classical mean value theorem, \F(x,u) — F(x,v)\ = \Fu(x,w)\\u — v\ < C\u — v\ for all (x, u), (x, v) G S. Observe that w depends on u and v, where (x, w) G S. Hence, for all u, v G M, we obtain || Au — Av\\ — max xq— h<x<xo+h f [F(y,u(y)) - F(y,v(y))\dy Jxo < hC max \u{y) — v(y)\ = k\\u — v\\, xo—h<y<xo+h where k := hC. We now apply the Banach fixed-point theorem (Theorem l.A in Section 1.6) to equation (22*). This yields the statements concerning the integral equation (22). Step 2: Equivalence. Let u be a solution of the integral equation (22). Differentiating (22), it follows that the function u is also a solution of the original initial-value problem (21), (21*). Conversely, let u be a solution of (21), (21*). Integration of (21) shows that the function u is also a solution of the integral equation (22). Therefore, the two problems (21), (21*) and (22) are equivalent. □ The following three sections serve as a preparation for the proof of the fundamental Schauder fixed-point theorem, which allows a generalization of the Picard-Lindelof theorem. The basic notions to be considered are: continuity, convexity, and compactness. 1.9 Continuity Definition 1. Let X and Y be normed spaces over K. The operator A'.MCX^Y (24)

1.9 Continuity 27 is called sequentially continuous iff, for each sequence (un) in M, lim un = u with u £ M implies lim Aun = Au. n—►oo n—xx> The operator ^4 in (24) is called continuous iff, for each point u e M and each number e > 0, there is a number 6(e, u) > 0 such that ||v - rz|| < 6(e, u) and wgM imply \\Av - Au\\ < e. (25) In addition, if it is possible to choose the number 6(e, u) > 0 in such a way that it does not depend on the point u £ M, then the operator A in (24) is called uniformly continuous. Example 2. Let X and Y be normed spaces over K. The operator A: M C X —> y is called Lipschitz continuous iff there is a number L > 0 such that \\Av - Au\\ < L\\v - u\\ for all u, v £ M. (26) Each Lipschitz continuous operator is uniformly continuous. In fact, condition (26) implies (25) with 6(e) = f. Proposition 3. We are given the operator A: M C X —> Y, where X and Y are normed spaces over K. Then, the following two statements are equivalent: (i) A is continuous. (ii) A is sequentially continuous. Proof, (i) =4> (ii). Suppose that A is continuous. Let un —> u as n —> oo, where un, u £ M for all n. Then, for each e > 0, there is a number no such that \\un — u\\ < 8(e, u) for all n > no, where the number 8 corresponds to (25). It follows from (25) that \\Aun — Au\\ < e for all n > no- Hence Aun —> Au as n —> oo. (ii) => (i). Suppose that the operator A is not continuous. Then, there exists a number £o > 0 such that condition (25) is violated for each number 8 > 0. In particular, when 8 = ^ there is a point un £ M such that \\un — u\\ < — and \\Aun — Au\\ > £q for all n = 1,2,... .

28 1. Banach Spaces and Fixed-Point Theorems Hence un —> u as n —> oo. By (ii), Aun —> ^4u as n —> oo. This contradicts \\Aun — Au\\ > £q for all n. D Proposition 4 (Composition of continuous operators). Let A.MCX^Y and B:A(M) -► Z be two continuous operators, where X, Y, and Z are normed spaces over K. Set C := B o A, i.e., Cu := B(Aiz) /or a// u G M. Then, the operator C:M ^> Z is continuous. Proof. Let un, u G M for all n. Then, Aun, Au G A(M) for all n. It follows from un —> u as n —> oo that ^4un —> ^4u as n —> oo, and hence B{Aun) —> J3(Am) as n —> oo. D Definition 5. Let M and Y be subsets of normed spaces over K. The operator A:M^Y is called a homeomorphism iff it is continuous and bijective, and the inverse operator A-1: Y —> M is also continuous. The two sets M and Y are called homeomorphic iff there exists a homeomorphism A: M —> Y. Example 6. Let r, a, 6 > 0. The disk D:={(£,77)GR2:£2+y <r2} and the ellipse £:={(£,,) €K2:g + g<l} are homeomorphic. The homeomorphism A: D —> .E is given through along with the inverse operator ^4_1(^,ry) = r(a~1£,b~1r)) (see Figure 1.8). The intuitive meaning of a homeomorphism is a "rubber-sheet transformation. "

1.10 Convexity 29 FIGURE 1.8. l.lp Convexity Definition 1. The set M in a linear space is called convex iff u, v £ M and 0 < a < 1 imply au + (1 — a)v £ M. The function /: M —► R is called convex iff M is convex and /(au + (1 - a)v) < a/(u) + (1 - a)/(v), for all u,v e M and all a, 0 < a < 1. Intuitively, the convexity of the set M means that if the two points u and v belong to M, then the segment joining them also belongs to M (see Figure 1.9(a)). The convexity of the real function /: [a, b] —> R means that the chords always lie above the graph of / (see Figure 1.9(b)). Example 2. Let X be a normed space, and let uq G X, r > 0 be given. Then, the ball B = {u e X: \\u — uo\\ < r} is convex. Proof. If u, v G B and 0 < a < 1, then \\au -f- (1 - a)v - u0|| = \\a(u - u0) + (1 - a)(v - u0)|| <||a(M-wo)|| + ||(l-a)(t;-wo)|| = a||u — uo|| + (1 — a)\\v — uq\\ < ar + (1 — a)r = r. Hence au + (1 — a)i? £ £?. D Example 3. Let X be a normed space with the norm || • ||. Set f(u) := \\u\\. Then, the function /: X —> R is continuous and convex. Proof. It follows from un —> u as n —> oo that ||un|| —> ||u||. Hence / is continuous.

30 1. Banach Spaces and Fixed-Point Theorems H h2 1 1 ► a u v b (a) convex sets (b) convex functions FIGURE 1.9. For u,v e X and 0 < a < 1, ||cm + (1 — a)v\\ < \\oiu\\ + ||(1 — a)v\\ = o^||ix|| + (1 — aO|M|. This proves the convexity of /. D Definition 4. A subset L of the linear space X over K is called a linear subspace of X iff u,v E 1/ and a, /3 E K imply au + (3v E L. By a closed linear subspace L of the normed space X over K we mean a linear subspace that is a closed set. Obviously, each linear subspace is convex. Example 5. Let X := C[a, 6], —oo < a < b < oo, and set L := {wG X:rz(a) = 0}. Then, L is a closed linear subspace of X. Proof. If u, v E L and a, /3 E R, then (aw 4- (3v)(a) — au(a) + /?v(a) = 0, and hence au + (3v E L, i.e., L is a linear subspace of X. If Un E L for all n and un —> u in X as n —> oo, then ^n(^) -*" u{a) as n —> oo, and hence u(a) = 0, i.e., u E L. Thus, the set L is closed. □ Definition 6. Let M be a subset of the linear space X over K. Then: span M := smallest linear subspace of X containing M; co M := smallest convex set of X containing M.

1.10 Convexity 31 Let X be a normed space over K. Then: M := smallest closed set of X containing M; coM:= smallest closed convex set of X containing M; int M := largest open set of X contained in M. Here, we use the following terminology: span M = linear hull of M; M = closure of M; co M = convex hull of M; coM = closed convex hull of M; int M = interior of M. The set _ 8M := Af - int M is called the boundary of M. Finally, the set ext M := int(X — M) is called the exterior of M. The point u is called an interior point, boundary point, or exterior point of M iff u e int M, ue dM, or u e ext M, respectively. Proposition 7. Le£ M be a nonempty subset of the normed space X over K. Tften, the following hold true: (i) u € span M z/f, for some fixed n — 1,2,..., ^ = ai?ii H h an^n, (27) w/iere ui,...,un € M and ai,..., an € K. (ii) wGcoM ijff, /or some fixed n = 1,2,..., ia = ai^i H h an?in, (28) w/iere Mi,..., un € M and 0 < ai,..., an < 1 wi£/i ai H han = 1. (iii) uEMiff, for some sequence (un) in M, un —> u as n —> oo. Proof. Ad (i). Let L be the set of all the linear combinations of the form (27). Then u,v G L and a,(3 € K imply au + (3v e L,

32 1. Banach Spaces and Fixed-Point Theorems i.e., L is a linear subspace of X. In fact, a{aiui + • • • + anun) + /3(/?iVi + • • • + Pmvm) = (aai)ui + • • • + {PPm)vm- Conversely, let £ be a linear subspace of X such that M C £. Then, it follows from wi,..., wn € M that u E C, where ia is given by (27). Hence LCC. Thus, L is the smallest linear subspace of X that contains the set M, i.e., L = span M. Ad (ii). Use a similar argument as in the proof of (i). In this connection, observe that it follows from 0 < ai,...,an, /3i,...,/3m < 1, as well as ai H \- an = 1, fti H h /3m = 1, and a + /3 = 1, that aai H h aan + /3/3i H h /3/3m = a + /3 = 1. Ad (iii). Let C be the set of all the points u € X such that un —> u as n —> ex) for some sequence (wn) in M, i.e., Il^n — u\\ —> 0 as n —> oo and un € M for all n. (29) We want to show that the set C is closed. To this end, let (vn) be a sequence in C such that vn —> v as n —> oo. By the definition of C, for each vn, there is a point wn € M such that Ibn — wn\\ < — for all n = 1, 2,... . n By the triangle inequality, \\wn ~ V\\ = \\{Wn - Vn) + (vn - v)\\ < \\wn - vn\\ + \\vn - v\\ -> 0 as n —► oo. Hence v € C. Thus, the set C is closed. Conversely, let C be a closed subset of X such that M CC. Then CCC. Therefore, C is the smallest closed subset that contains the set M, i.e., C = M. □

open set closed set bounded set norm i 1.1 [convergence compact set I relatively compact set FIGURE 1.10. 1 Compactness compact operator I continuous operator 33 1.11 Compactness We want to study both compact sets and compact operators. The notion of a compact set generalizes the classical Bolzano-Weierstrass convergence theorem. Compact operators allow us to generalize classical results for operator equations in finite-dimensional normed spaces to infinite-dimensional normed spaces via approximation and a limiting process. For example, this method will be used in the proof of the Schauder fixed-point theorem. Compactness plays a key role in functional analysis. Figure 1.10 tells us, for example, that each compact set is closed, and so on. 1.11.1 Compact Sets Definition 1. Let M be a set in a normed space. M is called relatively sequentially compact iff each sequence (un) in M has a convergent subsequence un> —> u as n' —> oo. M is called sequentially compact iff each sequence (un) in M has a convergent subsequence un> —> u as n' —* oo such that u G M. M is called bounded iff there is a number r > 0 such that \\u\\ < r for all ueM. Convention 2. For brevity of terminology, we will use "relatively compact" and "compact" instead of "relatively sequentially compact" and "sequentially compact," respectively. We will show in Problem 1.13 of AMS Vol. 109 that in normed spaces this convention makes sense with respect to the corresponding definitions in general topological spaces. Proposition 3. The set M is compact iff it is relatively compact and closed.

34 1. Banach Spaces and Fixed-Point Theorems Proof. Let M be compact. By definition, this implies that M is also relatively compact. Furthermore, let un —» v as n —> oo with un G M for all n. Since M is compact, there is a subsequence un> —> u as n' -+ oo with u e M. Obviously, u — v. Hence M is closed. Conversely, let M be a relatively compact and closed set. Consider any sequence (un) in M. Then, there is a subsequence such that un> —> u as n' —> oo. Since M is closed, w€M. Thus, M is compact. □ Proposition 4. Eac/i relatively compact set is bounded. Proof. Let the set M be relatively compact and suppose that M is not bounded. Then, there exists a sequence (un) in M such that ||^n|| > n for all n. (30) Since M is relatively compact, there exists a convergent subsequence (un'). Hence (un') is bounded. This contradicts (30). □ Example 5. Let M be a subset of R equipped with the usual norm \\u\\ := \u\. Then, M is relatively compact iff it is bounded. Proof. Let M be bounded, and let (un) be a sequence in M. By the classical Bolzano- Weierstrass theorem, there exists a convergent subsequence un> —> u as n' —> oo. Hence M is relatively compact. Conversely, if the set M is relatively compact, then M is bounded, by Proposition 4. □ Standard Example 6. Let M be a set in KN equipped with the norm IMI •= Moo, where N = 1,2, Then, M is relatively compact iff it is bounded. Proof. By Proposition 4, it is sufficient to prove that each bounded set in KN is relatively compact. Step 1: K = R, N = 1. Observe that R1 = R and use Example 5. Step 2: K = R, N = 2. Suppose that M is bounded. Let (wn) be a sequence in M, i.e., Un '= (Clnj&n)- Then, (wn) is bounded. Since |£ln|, |&n| < |^n|oo for all 71, (31)

1.11 Compactness 35 the sequence (£in) is bounded in R. By Example 5, there is a convergent subsequence fin' -> f i as n' -> oo. Similarly, by (31), the sequence (f2n') is bounded. Thus, there is a convergent subsequence £,2n" —> £2 as n" —> oo. Setting ia := (^1,^2), we get wn// —> u as n" —> 00. Step 3: K = R, AT > 3. Proceed similarly to Step 2. Step ^: IK = C, iV = 1. Suppose that M is a bounded set in C. Consider a sequence (vn) in M, i.e., Vn := fin + «f2n' fin, &n £ R. Since M is bounded, |fin|, |£2n| < |vn| < r for all n and fixed r > 0. As in the proof of Step 2, we get subsequences fin" —► fi and f2n" —> f 2 as n" —► 00. Letting v := f 1 + ^2, this implies vn» -^vas n" —> 00. Step 5: K = C, N > 2. Use the same argument as in Step 2 along with Step 4. □ Standard Example 7 (The Arzela-Ascoli theorem). Let X := C[a,6] with ||u|| := maxa<x<5 \u(x)\ and —00 < a < b < 00. Suppose that we are given a set M in X such that (i) M is bounded, i.e., ||w|| < r for all u G M and fixed r > 0. (ii) M is equicontinuous, i.e., by definition, for each e > 0, there is a <5 > 0 such that \x — y\<6 and ueM imply |^(#) — u(y)\ < e. Then, M is a relatively compact subset of X. Proof. Suppose that we are given a sequence (un) in M, i.e., the functions un: [a, b] —> R, n = 1, 2,... are continuous. Let Q denote the set of rational numbers contained in the interval [a, 6]. This set is countable. Thus, we may write Q = {ri:t = l,2,...}.

36 1. Banach Spaces and Fixed-Point Theorems Step 1: Diagonal sequence. By assumption (i), the sequence (un(ri)) is bounded in R. Thus, there is a subsequence (u<h ) of (un) such that (un (r±)) is convergent, i.e., there is a real number w\ such that u£Hri) —> wi in R as n -> oo. Again by (i), the sequence (uh, (r2)) is bounded in R. Hence there exists a subsequence (uh,') of (un ) such that u^\r2) —> w2 in R as n —> oo. Continuing this construction, for each k — 1,2,..., we obtain a subsequence (uh, ) of (un) such that, as n —> oo, 41}(ri)> 41}(ri)5 41)(ri),--- ^™i> 42)(r2), 42)(r2), 42)(r2),... ^™2, 43)(r3), 43)(r3), 43)(r3),-.. .->^3, In addition, (un ^) is a subsequence of (uh, ') for all k. We now consider the diagonal sequence vn:=u£\ n = l,2,.... Then vn(rj) —> Wj as n —> oo for all jf = 1,2,... . (32) Step 2: Cauchy sequence in [a, 6]. Let £ > 0 be given. We choose the number 6 > 0 as in (ii). Then, there exists a finite number of points %i, • • •, xs G Q such that, for each x G [a, 6], there is some Xj such that \x-Xj\<8. (33) By (32), for each j = l,...,s, the sequence (vn(xj)) is convergent, and hence it is a Cauchy sequence. Thus, there is a number n^e) such that \vn(xj) — vm(xj)\ < £ for all n,m> n0(£), j — 1,... ,5. Finally, for each x G [a, 6], it follows from assumption (ii) and (33) that \vn(x) - Vm(x)\ < \vn(x) - Vn(Xj)\ + \vn(Xj) - Vm(Xj)\ + \vm(Xj) ~ Vm[x)\ <€ + € + £ for all n, m > no(e).

1.11 Compactness 37 This implies \\vn — vm\\ = max \vn(x) — vm(x)\ < 3e for all n,m > no(e), a<x<b i.e., (vn) is a Cauchy subsequence of (un). Since C[a, b] is a Banach space, (vn) represents a convergent subsequence of (un) in C[a, 6]. D Proposition 8 (The Weierstrass theorem). Let f:M^R be a continuous function on the compact nonempty subset M of a normed space. Then, f has a minimum and a maximum on M. Proof. Set a := inf f(u). Then — oo < a < oo. Recall that if the set M:={f(u):ueM} is bounded below, then a is the largest lower bound of M, and hence a > —oo. If M is not bounded below, then a := —oo. By construction of a, there exists a sequence (un) in M such that /(^n) -^ ol as n —> oo. Since the set M is compact, there exists a convergent subsequence uni —> v as n; —> oo. By the continuity of the function /, f(un') -+ /(v) as n; -+ oo. Hence a = f(v). Consequently, a > — oo and f(v)= inf /(«). That is, the function / has a minimum on the set M. Replacing / with —/, we obtain the corresponding result for the maximum of /. □ Proposition 9. Let X and Y be normed spaces over K, and let A:MCX ^Y be a continuous operator on the compact nonempty subset M of X.

38 1. Banach Spaces and Fixed-Point Theorems Then, A is uniformly continuous on M. Proof. Recall that the uniform continuity of A means that, for each e > 0, there is a number 6(e) > 0 such that \\u — v\\ < 6(e) and u,v G M imply ||Am-Av|| < e. (34) Suppose that A is not uniformly continuous. Then, there exist a number, e0 > 0 and two sequences (un) and (vn) in M such that \\un — vn\\ < — and \\Aun — Avn\\ > eo for all n. (35) n Since M is compact, there exists a subsequence of (un), again denoted by (un), such that un —> u as n —> oo and ^ € M. This implies \\vn — u\\ < \\vn — un\\ + ||wn — w|| —> 0 as n —> oo, and hence vn —> ^ as n —> oo. By the continuity of the operator A, Aun — Avn —> 0 as n —> oo. This contradicts condition (35). □ Proposition 10 (Finite e-net). Le£ M be a nonempty set in the Banach space X. Then, the following two statements are equivalent: (i) M is relatively compact. (ii) M has a finite e-net; that is, by definition, for each e > 0, there exists a finite number of points v\,..., v j G M such that min \\u — Vn II < e for all u G M. Proof, (i) => (ii). Let M be relatively compact. Suppose that (ii) is not true. Then, there is a number eo > 0 such that M has no finite £o-net. Choose a fixed point u\ G M. Then, there exists a point U2 G M such that 11^2 — will > £o- Furthermore, there exists a point U3 £ M such that 11^3 — U2W > £0 and \\us — ui\\ > eo- Continuing this construction, we get a sequence (un) in M such that \\v>n ~ v>m\\ > £0 for all n, m = 1,2,... with n ^ m.

1.11 Compactness 39 Consequently, each subsequence of (un) is not Cauchy, i.e., (un) does not contain any convergent subsequence. This is a contradiction to the relative compactness of M. (ii) => (i). Suppose that condition (ii) is satisfied. Let (un) be a sequence in M. Fix e — 1. By (ii), there is some Vj G M such that \\un ~ vj\\ ^ 1 f°r infinitely many indices n. Thus, there is a subsequence (uh ) of (un) such that \\un^ — ^jll ^ 1 f°r a^ n- Hence ll^fc0 - um II ^ Il41} - VJII + IIVj - u$ || < 2 for all n, m. Continuing this construction for e = ^, n = 2,3,..., we obtain the sequences (2) (2) (2) with the following properties. For each k = 1,2,..., (i4 ) is a subsequence of (un ^) and Ik^-^ll^f for all n,m. (36) Consider now the diagonal sequence By (36), 2 Ikn — ^m < — for aU ^> ^ with m>n. n Therefore, (vn) is a Cauchy subsequence of (un). Since X is a Banach space, the sequence (vn) is convergent. This proves the relative compactness of the set M. □ 1.11.2 Compact Operators Definition 11. Let X and Y be normed spaces over K. The operator A:MCX-*Y is called compact iff

40 1. Banach Spaces and Fixed-Point Theorems (i) A is continuous, and (ii) A transforms bounded sets into relatively compact sets. Obviously, property (ii) is equivalent to the following: If (un) is a bounded sequence in M, then there exists a subsequence (unt) of (un) such that the sequence (Aunf) is convergent in Y. Standard Example 12. Let us consider the integral operator (Au)(x) := / F(x,y,u(y))dy for all x G [a, 6], J a where — oo < a < b < oo. Set Q := {(#, y, w) G R3: x, y G [a, 6] and |w| < r for fixed r > 0}. Suppose that the function F: Q —> R is continuous. Set X := C[a, 6] and M := {^GX:||^|| <r}. Then, the operator A M —> X is compact. Proof. By Proposition 9 in Section 1.11.1, the function F is uniformly continuous on the compact set Q. This implies that, for each e > 0, there is a number 6 > 0 such that |F(x,y,ti)-F(z,y,v)|<6: (37) for all (x, y, u), (z, y,v) £ Q with |x — z\ + |w — v\ < 6. We first show that the operator A:M —> X is continuous. In fact, if i£ G M, then the function ?x: [a, b] —> R is continuous, and |^(y)| < r for all y G [a, 6]. Hence the function Au:[a, b] —> R is also continuous. Let u,v G M. Then ||w — v|| = max \u(y) — v(y)\ < 6 a<x<b implies I fb \\Au - Av\\ = max / [F(x, y, u{y)) - F(x, y, v(y))]dy <(b- a)e, (38) by (37). Hence A: M —> X is continuous. We now show that A M —> X is compact Since the set M is bounded it suffices to show that the set A(M) is relatively compact. By the Arzela- Ascoli theorem (Standard Example 7 in Section 1.11.1), it remains to show that

1.11 Compactness 41 (i) A(M) is bounded, and (ii) A(M) is equicontinuous. Ad (i). Set M := max^y^eQ |F(x,y,u)\. Then, for all u G M, ||Au|| = max / F(x,y,u(y))dy a<x<b\Ja <{b-a)M. Ad (ii). Let \x — z\ < 6 and x,z e [a, 6]. Then, by (37), \(Au)(x)-(Au)(z)\< f \F(x,y,u(y))-F(z]y,u(y))\dy J a <(b- a)e for all ueM. D Proposition 13 (Approximation theorem for compact operators). Let A:M CX ^Y be a compact operator, where X and Y are Banach spaces over K, and M is a bounded nonempty subset of X. Then, for every n = 1,2,... there exists a continuous operator An:M^Y such that sup \\Au — Anu\\ < — and dim (span^4n(M)) < oo, as well as An{M) C co A{M). Proof. The set A(M) is relatively compact in Y. Thus, for every n — 1,2,..., there exists a finite ^-net for A(M). That is, there are elements Uj e A(M), j = 1,..., J, such that 1 \\Au-Un\\ < 1<3<J min \\Au — Uj\\ < — for all u G M. Define the Schauder operator y\_i aAu)uj Anu := J~X 3 3 for all u G M, where aj(u) := max{2n_1 — \\Au — Uj||, 0} for all u G M, j = 1,..., J.

42 1. Banach Spaces and Fixed-Point Theorems The function u h-* \\Au — Uj\\ is continuous, by Proposition 6(iii) in Section 1.2. Thus, the function ay.M —> R is also continuous. Moreover, for each i£ G M, the aj(^x) do not all vanish simultaneously. Hence the operator An: M —> Y is continuous. Finally, for each i£ G M, ||;4U - Ati|| = M^J ^ , < \{u)n~ Ejaj(ti) . Ejaj(ti)n l <—4- —-—= n . D 1.12 Finite-Dimensional Banach Spaces and Equivalent Norms Definition 1. Let X be an iV-dimensional linear space over K, where N = 1,2, By a 6aszs {ei,..., e^} of X we understand a set of elements ei,..., ejsi oi X such that, for each u G X, w = aiei H h a^eAr, (39) where the numbers ai,..., a at G K are uniquely determined by u. The numbers ai,..., a at are called the components of i£. In particular, letting u = 0 in (39) it follows from the uniqueness of the components that ai = • • • = a at = 0, i.e., the elements ei,..., ejv of a basis are linearly independent. Proposition 2. Le£ N = 1,2, In eac/i N-dimensional linear space X over K there exists a basis {ei,..., e^}. Proof. Since dim X = iV, there exist N linearly independent elements ei,..., ejsi of X, and N + 1 elements of X are never linearly independent. Thus, for given u G M, there are numbers /30, /?i,..., Pn such that fau + /?iei + • • • + /?Are;v = 0, where fa ^ 0 for some fc. Since fa = 0 implies /3i = • • • = (3n = 0, we get fa 7^ 0. Letting olj; = — S2- we obtain (39). Finally, it follows from (39) and u — ot!xe\-\ (- o!Ne^ that (ai - aiJei H h (qat - oc'N)eN = 0, and hence a^— a^ = 0 for all j. This yields the uniqueness of the components aj. a Definition 3. The two norms || • || and || • ||i on the normed space X are

1.12. Finite-Dimensional Banach Spaces and Equivalent Norms 43 called equivalent iff there are positive numbers a and /3 such that a\\u\\ < \\u\\i < P\\u\\ for all ueX. (40) Proposition 4. Two norms on a finite-dimensional linear space X over K are always equivalent. Proof. If dim X = 0, then X = {0}. In this case, the inequality (40) is satisfied trivially. Let dim X = N for fixed N = 1,2, Suppose that || • || is a norm on X. By (39), two arbitrary elements u and v of X allow the following representations: N N u = Y^ OLdej and v = Y^ Pjej> where a? ,/3j G K for all j. 3=1 3=1 Set a = (ai,..., a at) and define IMIoo := Moo :— max |a7-|. 1<3<N J One checks easily that || • ||oo is a norm on X. We want to show that there exist positive numbers a and b such that a|M| < IMIoo < b\\u\\ for all u G X. (41) Observe first that w- 1 N EaH l n <SK-cjII < &N|oo, where 6 :^ AT i=i Since e^ ^ 0 for all j, we get b > 0. Furthermore, set M:={aGK*:|a|oo = l}. Since | • |oo is a norm on KN, it follows from a^n^ —> a in KN as n —> oo that |« |c» —*• Moo as n —> oo. Thus, the set M is closed and bounded in KN with respect to | • |oo, i.e., M is compact in KN. Define the function /(ai,...,ajv) := N 3=1

44 1. Banach Spaces and Fixed-Point Theorems Then, /: KN —> R is continuous. This follows from \f(au...,aN) -/(/3i,...,0N)\ II N \u2a3e3\ ||i=i - 1 N II E^H |i=i || < N i=i AT < la-^loo^He^l for all a,/? GK^. By the Weierstrass theorem (Proposition 8 in Section 1.11.1), the continuous function /: M —> R on the compact set M has a minimum. Denote the minimal value of / by a. Then /(/?) = \\v\\ > a for all v G X with |H|oo = 1, where v := X)j=i /^jej- Note that |H|oo = 1 implies /3j ^ 0 for some j. Hence a > 0. For given ^ ^ 0, set v := H^H^u. Hence \\u\\ > a\\u\\ for all u £ X. This proves inequality (41). To finish our argument, let || • ||i be a second norm on X. Replacing with || • ||i, from (41) we obtain &ilMli < IMIoo < &1IMI1 f°r all ^ G X (41*) and fixed positive numbers a\ and b\. The desired inequality (40) follows now from (41) and (41*). □ The following consequences of Proposition 4 show that Finite-dimensional normed spaces possess a simple structure. Proposition 5. Let (un) be a sequence in a finite-dimensional normed space X with dim X > 0. Then un —> u in X as n —> oo (42) iff the corresponding components with respect to any fixed basis converge to each other. Proof. Let {ei,..., ejv} be a fixed basis of X. Set AT AT un = 2_]ajn^j and u = \_\ajeji where N = dim X.

1.13. The Minkowski Functional and Homeomorphisms 45 By (41), a\\un — u\\ < max |a7-n — a A < b\\un — u\\. 1<3<N Relation (42) means that \\un—u\\ —> 0 as n —> oo. In turn, this is equivalent to otjn —> olj as n —> oo for all j. (42*) D Corollary 6. .Eac/i finite-dimensional normed space is a Banach space. Proof. Let dim X = 0. Then, X = {0}, and the statement is trivial. Let dim X > 0. Suppose that (un) is a Cauchy sequence. Then \anj — amj\ < b\\un — um\\ < e for all n,m > no(e) and all j. Consequently, each sequence (anj) of the corresponding components is also Cauchy. Hence we obtain (42*), which implies (42). □ Corollary 7. Each finite-dimensional linear subspace L of a normed space is closed. Proof. Let un —> u as n —> oo with un £ L for all n. Then, (un) is Cauchy, and hence u £ L, by Corollary 6. □ Corollary 8. Let M be a subset of a finite-dimensional normed space X. Then (i) M is relatively compact iff it is bounded. (ii) M is compact iff it is bounded and closed. Proof. For dim X = 0, i.e., X = {0}, the statements are trivial. Let N := dim X > 0. By Section 1.11.1, all the statements are true in the special case of the space KN equipped with the norm | • |oo- Now use Proposition 5 along with inequality (41). □ 1.13 The Minkowski Functional and Homeomorphisms The following elementary geometrical considerations will be used in the proof of the Brouwer fixed-point theorem in the next section. Definition 1. Let N = 1,2, The points u0,..., u^ in the linear space X over K are called to be in general position iff Ui - U0, U2 - Uq, . . . ,UN - U0

46 1. Banach Spaces and Fixed-Point Theorems are linearly independent. This definition does not depend on the numbering of the points. For example, if wq, ..., v>n are in general position, then so are ui, uo, U2,..., u^. In fact, it follows from ^0(^0 — ^i) + 0^2(^2 — ui) -\ 1- &n(v>n — uo) = 0 with aj € K for all j that (a0 + ol2 H h aN){u0 - ^1) + ^2(^2 - uq) H h aN(uN - u0) = 0, and hence ao + a2 H h a at = 0, a2 = • • • = a at = 0. This implies aj = 0 for all j. Proposition 2. Le£ iV = 1,2, Suppose that the points uq,.,.,un are in general position, and suppose that u £ span{^o,..., v>n}, (43) then the points uo,..., un, u are also in general position. Proof. Let ai(ui—uo)-\ haAr^Ar— Uo) + a(u —^0) = 0 with a^,a € Kfor all j. By (43), a = 0. Hence ay = 0 for all j. D Definition 3. Let TV = 1,2,..., and let X be a linear space over K. By an N-simplex we understand the set 5:=co{^0,...,^at}, (44) where the points Uo, •.., u^ G X are in general position. By a 0-simplex <S, we understand a single point of X, i.e., S = {u0}. Example 4. 1-simplices are segments, and 2-simplices are triangles (see Figure 1.11). Let N = 0,1, The points uo,...,un in (44) are called the vertices of the simplex <S. Explicitly, (n \ S = < Y^ajUj'.aj > 0 for all j and ao H h a at = 1 > . (44*) (j=o J Using a0 = 1 — (ai H 1- ajv), we also get r at I 5 = < Uq + 2_, Q-j{uj —u0):aij > 0 for all j and ot\ + \- ojjv < 1 > . (44**)

1.13. The Minkowski Functional and Homeomorphisms 47 , So . / . # ' ' ' w HO U\ (a) 1-simplex K2 /^\ * / A / / \y A A /=: \ J= \ U0 ^1 (b) 2-simplex FIGURE 1.11. 1 U0 A"2 / \ §0 / \ y / l \/ • 6 \ \ \ (c) The point N '- N + l ^2Uj is called the barycenter of S (see Figure 1.11(c)). Let N = 1,2,.... The (iV - l)-simplices <S0 :=co{ui,...,uat}, Si := co{u0,^2, • - ,^at}, . •., Sat := co{u0,... ,^at-i} are called the (N — 1)-faces of 5 opposite to the points u0, • •., ^at, respectively (see Figure 1.11(c)). By a k-face of 5 we understand the convex hull of k + 1 distinct vertices of <S, where fc = 0,1,..., N. Definition 5. Let M be a nonempty set in a normed space X. Then, we define the diameter of M through The number diam M := sup \\u — v\\. u,v£M dist(u, M) := inf \\u — w\\ is called the distance of the point u G X from the set M. Instead of dist(u, M), we also write distx(^, M). Standard Example 6 (iV-simplices). Let S = co{u0,...,uN} be an iV-simplex in the normed space X over K, where N = 1,2, Then, the following are true: (i) The set S is convex and compact. (ii) SQL, where L := span{u0,..., ^at}-

48 1. Banach Spaces and Fixed-Point Theorems (iii) The barycenter b is an interior point of S with respect to the linear subspace L of X. (iv) diam S < 2 max \\uj — u$\\. V ' ~ l<j<N J Proof. Ad (i). By (44), S is convex. To show that S is compact, let (vn) be a sequence in S. Then N Vn = y ^QLjnUj, 3=0 where 0 < OL3n < 1 and a0n H V &Nn = 1 for all j, n. By Standard Example 6 in Section 1.11.1, there exist convergent subsequences ajnf —> olj as n' —> oo for all j. Hence 0 < c^ < 1 and ao H h a at = 1 for all j. Letting v := ]T\. otjUj, we get v G S and ?v —> v as nf —> oo. Thus, S is compact. Ad (ii). This is obvious. Ad (iii). Let || • || denote the norm on X. Then, Y := span{^i — uo,..., u^ — v>o} is an iV-dimensional linear subspace of X. We have u G Y iff AT u = S2Pj(uj — uo) with (3j G K for all j. 3 = 1 The norm ll^lloo := max \BA for all u G Y" i<j'<n is equivalent to the norm || • || on F, by Proposition 4 in Section 1.12. Hence \Pj\ < c\\u\\ for all u G Y, j = 1,..., N, and fixed c> 0. (45) Let v G L and ||v — 6|| < r for fixed r > 0. We have to show that this implies v G S provided r is sufficiently small. By (44**), v — Uo£Y and b — u0 G Y. It follows from (45) that the coordinates

1.13. The Minkowski Functional and Homeomorphisms 49 (3j(v — Uq) and (3j(b — uo) of the points v — u$ and b — uo, respectively, satisfy \Pj(v - u0) - (3j(b - u0)\ < c\\(v - uo) - (b - u)\\ < cr, j = 1,..., N, where /3j(b—Uo) = 7]v+T)' .7 = 1> • • •, -W. Consequently, for sufficiently small r > 0, we get (3j(v — uo) > 0 and (5\{u — v0) H h /3n(v> — v0)<1 for all j. By (44**), this implies veS. Ad(iv). Let u,veS. By (44**), m — v = AT ^(aiW ~ ai(v))K' - ^0) j=i AT < max kx7 — Uo \\t^ \oLn(u) — a7-(v) < 2 max kx7 — uo . □ _ i<?'<w J "^' JV y JV y| ~~ i<j<at" 3 " Definition 7. By a barycentric subdivision of the 1-simplex <S = co{^o, ^1}, we understand the collection of the following two 1-simplices: <So -= co{6, uo} and <Si := co{6, ^1}, where b is the barycenter of S (see Figure 1.12(a)). By induction, the barycentric subdivision of an iV-simplex S with bary- center b is the collection of all the iV-simplices co{Mi,...,vat_i}, where i>i,... ,1^-1 are vertices of any (N — l)-simplex obtained by a barycentric subdivision of an (N - l)-face of S. The barycentric subdivision of a 2-simplex is pictured in Figure 1.12(b). Intuitively, a barycentric subdivision corresponds to a triangulation based on barycentric centers. Proposition 8. Let M be a closed, bounded, convex, nonempty subset of a normed space X, where M has an interior point. Then, M is homeomorphic to the closed ball B := {u £ X:\\u\\ <1}. Proof. If X = {0}, then M = {0}, and the statement is trivial. Now let X ^ {0}, and let uo £ M. Replacing u with u — uo, we may assume that u0 = 0.

50 1. Banach Spaces and Fixed-Point Theorems Uq u\ So U0 U2 Uq U\ (a) FIGURE 1.12. (b) A"1™ p(u) u FIGURE 1.13. Step 1: Minkowski functional. For each u G X, we define the Minkowski functional of the set M through p(u) := inf{A: A"1^ G M, A > 0}. The intuitive meaning of p(u) is pictured in Figure 1.13, i.e., the ray through the point u and the origin intersects the boundary dM of the set M at the point p(u)~1u. We want to show that the following are true: (i) a||tt|| < p(u) < b\\u\\ for all u G X and fixed a, b > 0. (ii) p(au) = ap(u) for all a > 0. (iii) p(u + v) < p(u) +p(v) for all u, v G X (triangle inequality), (iv) p: X —> R is continuous. (v) M = {ueX:p(u) < 1}. Ad (i). Since 0 G int M, there is a number r > 0 such that \\u\\ < r implies u G M. Obviously, p(0) = 0. Now let u G X and ^ ^ 0. Then HA-1^!! = r for A := r""1!!^!!. Hence A_1^x G M, i.e., the definition of p(u) makes sense, and p(u) < r-1||^|| for all u G X.

1.13. The Minkowski Functional and Homeomorphisms 51 The set M is bounded, i.e., ||u|| < R for all u G M and fixed R > 0. Consequently, if X~xu G M, then HA"1^ < i?, i.e., A > ^2 x||tx||. This implies p(u) > R^WuW for all ueX. Ad (ii). Let a > 0. Observe that X~xu G M with A > 0 iff (aX^au G M. Ad (iii). Let u,v £ X. For fixed e > 0, choose numbers a and /3 such that p(u) < a < p(u) + e and p(v) < f3 < p(v) + £. Then a~~ V /3_1i; G M. Let 7 := a + /?. Since 7~1a + 7_1/3 = 1 and the set M is convex, the point 7_1(^ + v) = 7~1a(a~1^) + 7~1/3(/3~1^) lives in M. By the definition of p, p(u -\-v)<^ = a-\-/3< p(?x) + p(v) + 2e. Letting e —> 0, we get (iii). Ad (iv). It follows from (iii) that p(?x) = p(v + (u — v)) < p(v) + p(u — v). Replacing u with v and using (i), we obtain that \p{u) —p(v)\ < max{p(u — v),p{v — u)} < b\\u — v\\ for all u, v G X. Thus, p is continuous on X. Ad (v). Let u G M. Since 0 G M and the set M is convex, we get fin G M for all //: 0 < fi < 1. Hence X~xu G M for all A > 1. This implies p(u) < 1. Conversely, let p{u) < 1. If u = 0, then u£ M. Suppose now that ^ 7^ 0. Then, p(u) > 0 by (i), and X~*u G M for all A > p(u) + e. Letting e —> 0, this implies p(^x)-1^ G M, since M is closed. Using 0 G M and p(^)-"1 > 1, the convexity of M implies u G M. Step 2: Homeomorphism A: X —> X. Set I 0 if u = 0. By (i), ||4u|| < b||u|| for all u€X. (46)

52 1. Banach Spaces and Fixed-Point Theorems Thus, A: X —> X is continuous. In fact, let Un —> u as n —> ex). If ^ = 0, then Awn —> 0 as-n —> oo, by (46). Moreover, if w ^ 0 it follows from the continuity of p and || • || along with \\u\\ ^ 0 and Proposition 6(v) in Section 1.2 that Awn —> An as n —> oo. The inverse operator A-1: A" —> X is given through I 0 if v = 0. This follows from v = p(^)||^|| ^ if ^ ^ 0. Again by (i), A 1: X —> X is continuous. Thus, A X —> X is a homeomorphism. Step 5: Obviously, A(M) C B and ^(B) C M, by (v) and (ii), respectively. Hence A(M) — B. Consequently, the restriction A:M —> B represents the desired homeomorphism. □ Proposition 9. Let M be a compact, convex, nonempty set in a finite- dimensional normed space X. Then, M is homeomorphic to some N-simplex S in X with N = 0,1, — Proof. If M consists of a single point, then the statement is true for N = 0. Suppose now that M contains at least two distinct points. Since dim X < oo, the maximal number N of points in the set M being in general position is finite. Let u0,...,uN e M be in general position. Set L := span{i£0, •.., Ujy} and S := co{u0,..., u^}. By Proposition 2 and the maximality of iV, we get Ma, and the convexity of M implies S CM. By Standard Example 6, the simplex <S has an interior point in the normed space L, and hence M also has an interior point in L. By Proposition 8 applied to the space L, the set M is homeomorphic to the ball B :={ueL:\\u\\ < 1}. Again by Proposition 8, the compact convex set S is also homeomorphic to the ball B.

1.14 The Brouwer Fixed-Point Theorem 53 Consequently, there exist homeomorphisms A-.M-+B and C:S^B. Then, the map C~loA.M -^ B ^X S is the desired homeomorphism from the given set M onto the simplex S. □ 1.14 The Brouwer Fixed-Point Theorem Theorem l.B. The continuous operator A:M^M has a fixed point provided M is a compact, convex, nonempty set in a finite- dimensional normed space over K. A variant of this famous theorem was proved by Brouwer in 1912. The Brouwer fixed-point theorem (Theorem l.B) represents one of the most important existence principles in mathematics. It is equivalent to numerous, apparently completely different, propositions. This can be found in Zeidler (1986), Vol. 4, Chapter 77, along with interesting applications to game theory, mathematical economics, and numerical mathematics. By the proof of Example 2 ahead, the Brouwer fixed-point theorem generalizes the classical intermediate-value theorem for continuous functions, which was proved first by Bolzano in 1817. Further important existence principles in mathematics are the following: the Hahn-Banach theorem (Section 1.1 of AMS Vol. 109); the Weierstrass existence theorem for minima (Section 2.5 of AMS Vol. 109); the Baire category theorem (Section 3.1 of AMS Vol. 109). The Brouwer fixed-point theorem implies the Schauder fixed-point theorem (cf. Section 1.15). For example, we will prove in Section 1.18 that the Schauder fixed-point theorem implies the Leray-Schauder principle: a priori estimates yield existence. Corollary 1. The continuous operator B:K^K has a fixed point provided K is a subset of a normed space that is homeo- morphic to a set M as considered in Theorem l.B.

54 1. Banach Spaces and Fixed-Point Theorems a u b FIGURE 1.14. The proof of Theorem l.B will be given in Section 1.14.4. We first show that Corollary 1 is a simple consequence of Theorem l.B. Proof of Corollary 1. Let C: M —> K be a homeomorphism. Then, the operator C~loBoC:M -£* K -^ K — M is continuous. By Theorem l.B, there exists a fixed point u of A := C l 6 BoC, i.e., C-1(B(Cu))=u, ueM. Letting v = Cu, this implies Bv ^= v, v G K, i.e., B has a fixed point. D Example 2. Let M = [a, 6], where — oo<a<6<oo. Then, each continuous function A: [a, b] —► [a, 6] has a fixed point u (see Figure 1.14). This is the simplest special case of the Brouwer fixed-point theorem (Theorem l.B). Let us give a direct proof. To this end, we set B{u) := A(u) — u for all u G [a, 6]. Since A(a), A(b) G [a, 6], we get A(a) > a and A(b) < b. Hence B(a) > 0 and B(b) < 0. By the intermediate-value theorem, the continuous real function B has a zero u G [a, 6], i.e., B(u) = 0. Hence A(w) = u. D 1.14-1 Intuitive Proof of the Brouwer Fixed-Point Theorem Let M be a closed disk in R2, and let A: M —> M be a continuous operator. We want to use a simple intuitive argument in order to prove that A has a fixed point.

1.14 The Brouwer Fixed-Point Theorem 55 FIGURE 1.15. Suppose A: M —> M were a fixed-point free operator, i.e., Au ^ u for all u £ M. Then, we can construct an operator RiM^dM (47) as follows. For each point u £ M follow the directed line segment from the point Au through the point u to its intersection with the boundary dM, and let the intersection point be Ru, as in Figure 1.15. Obviously, the operator R in (47) is a so-called retraction, that is, R is continuous and Ru = u for all u £ dM. Intuitively, such a retraction does not exist. This is the desired contradiction. However, a rigorous proof for the nonexistence of a retraction of the form (47) is highly nontrivial. Such a proof can be found in Zeidler (1986), Vol. 1, p. 51, by means of the mapping degree, which represents an important tool from topology. At this place we want to give a completely different proof, one that uses only elementary facts about simplices. This elegant proof was discovered by Knaster, Kuratowski, and Mazurkiewicz in 1929. Remark 3. Intuitively, all the closed sets M pictured in Figure 1.16 are homeomorphic to a 2-simplex, and hence by Corollary 1 the Brouwer fixed- point theorem applies to these sets. Remark 4 (Counterexamples). We want to show through counterexamples that each of the assumptions of the Brouwer fixed-point theorem is essential. (i) Let M := [0,1]. The function A; M -» M pictured in Figure 1.17(a) has no fixed point. The set M is compact and convex, but A is not continuous.

56 1. Banach Spaces and Fixed-Point Theorems (a) (b) (c) Homeomorphic sets FIGURE 1.16. A - i s -+ y / / y'y- V. 1 ^ (a) (b) FIGURE 1.17. (ii) Let M := R. The continuous function A: M —> M defined through Au := u+1 has no fixed point. The set M is convex, but not compact. (iii) Let M be a closed annulus as pictured in Figure 1.17(b). Then, a proper rotation A: M —> M of the annulus around the center is fixed- point free. Here, the operator A is continuous and M is compact, but M is not convex. For our proof of the Brouwer fixed-point theorem we need the preparations that we consider in Sections 1.14.2 and 1.14.3. 1.14-2 The Sperner Lemma Let S = co{u0,...,uN} be an iV-simplex with N > 1. By a triangulation of S we mean a finite collection <Si,...,Sj (48)

1.14 The Brouwer Fixed-Point Theorem 57 n0 ui / \ S3 S2 FIGURE 1.18. of iV-simplices Sj such that J (a) S — I) Sj, and i=i (b) if j 7^ fc, then the intersection Sj D £& is either empty or a common (N - l)-face (cf. Figure 1.18). Lemma 5. Let one of the numbers 0,1,..., N be associated with each vertex v of the simplices Sj in (48). Suppose that if veco{uio,...,uik}, fc = 0,...,iV, (49) then one of the numbers io,..., ik is associated with v. By definition, Sj is called a Sperner simplex iff all of its vertices carry different numbers, i.e., the vertices of S carry the numbers 0,1,...,N. Then, the number of Sperner simplices is odd. Condition (49) means the following. Each vertex u3 of the original simplex S carries the number j. Moreover, let T be that lowest-dimensional face of the original simplex S that contains the point v. Then, the number associated with v is equal to one of the numbers of the vertices of T (cf. Figure 1.19). Proof. Step 1: Let N — 1 (Figure 1.19(a)). Then, each Sj is a 1-simplex (segment). A 0-face (vertex) of Sj is called distinguished iff it carries the number 0. We have exactly the following two possibilities: (i) Sj has precisely one distinguished (N — l)-face (i.e., Sj is a Sperner simplex); (ii) Sj has precisely two or no distinguished (N — l)-faces (i.e., Sj is not a Sperner simplex). But since the distinguished 0-faces occur twice in the interior and once on the boundary, the total number of distinguished 0-faces is odd. Hence the number of Sperner simplices is odd.

58 1. Banach Spaces and Fixed-Point Theorems no (a) Sperner simplex (b) FIGURE 1.19. Step 2: Let N = 2 (Figure 1.19(b)). Then, Sj is a 2-simplex. A 1-face (segment) of Sj is called distinguished iff it carries the numbers 0,1. Then, conditions (i) and (ii) above are satisfied for N — 2. The distinguished 1-faces occur twice in the interior. By (49), the distinguished 1-faces on the boundary are subsets of co{uo,ui}. It follows from Step 1 that the number of distinguished 1-faces on co{uo,ui} is odd. Thus, the total number of distinguished 1-faces is odd, and hence the number of Sperner 1-simplices is also odd. Step 3: Induction. Let N > 3. Suppose that the lemma is true for N — 1. Then it is also true for N. This follows as in Step 2. In this connection, an (N — l)-face of Sj is called distinguished iff its vertices carry the numbers 0,l,...,iV-l. □ 1.14-3 The Lemma of Knaster, Kuratowski, and Mazurkiewicz Lemma 6. Let S — co{^o,..., u^} be an N-simplex in a finite-dimensional normed space X, where N — 0,1, Suppose that we are given closed sets Co,..., Cn in X such that co{uio,...,uik} C (J d, (50) m=0 for all possible systems of indices {io,..., ik} and all k = 0,..., N. Then, there exists a point v in S such that v G Cj for all j — 0,..., N. Proof. For N — 0, S consists of a single point, and the statement is trivial. Now let N>1. Step 1: Consider a triangulation 5i,..., Sj of S. Let v be any vertex of <Sj, j — 1,..., J, where v e co{i^0,..., uik} for some k = 0,..., N.

1.14 The Brouwer Fixed-Point Theorem 59 By (50), there is a set Ck such that veCk. We associate the number k with the vertex k. It follows from the Sperner lemma (Lemma 5) that there is a Sperner simplex Sj whose vertices carry the numbers 0,..., N. Hence the vertices v$,..., vn of Sj satisfy the condition vk e Ck for all k = 0,..., N. Step 2: We now consider a sequence of triangulations of the simplex S such that the diameters of the simplices of the triangulation go to zero. For example, one can choose a sequence of barycentric subdivisions of S. By Step 1, there are points 4n) eft for all k = 0,..., N and n = 1,2,... such that lim diam co{vq ,..., v^} — 0. (51) n—KX) Since the simplex S is compact, there exists a subsequence, again denoted by (vk), such that Vi —> v as n —* oo and v E S. By (51), vk "^ v as n ~^ °° f°r aU k = 0,..., N. Since the set Ck is closed, this implies veCk for all k = 0,..., N. □ 1.14-4 Proof of the Brouwer Fixed-Point Theorem Step 1: Simplices. Let S be an iV-simplex in a finite-dimensional normed space, and let the operator A.S^S be continuous, where N — 0,1, We want to show that A has a fixed point. For N = 0, the set S consists of a single point and the statement is trivial. For N = 1, the proof has been given in Example 2. Now let N — 2. Then, S = co{^o, wi, ^2}, i-e., S is a triangle. Each point u in <S has the representation i£ = a0(^)^o + oli(u)ui + 0:2(10^2, where 0<a0,ai,a2<l and ao + ai+a^^l. (52)

60 1. Banach Spaces and Fixed-Point Theorems With u-u0 — ai(u)(ui - u0) + oi2{u)(u2 - u0) and ao(u) = 1 — ai(u) — a2(w), it follows from the linear independency of u\ — Wo, U2 — u$ that the barycentric coordinates ao(w), <^i(w), and 012(11) of the points u are uniquely determined by u and depend continuously on u, by Proposition 5 in Section 1.12. We set Cj := {ueSiaj(Au) < o^w)}, j = 0,1,2. Since otj{-) and A are continuous on <S, the set Cj is closed. Furthermore, the crucial condition (50) of the lemma of Knaster, Kuratowski, and Mazurkiewicz is satisfied, i.e., k co{uio,...,uik} C (J Cim, A; = 0,1,2. m=0 In fact, if this is not true, then there exists a point u G co{uio,..., wifc} such that u g Um=o C^-> i-e-> a^m (Aw) > airn (u) for all m — 0,..., k and some k = 0,1,2. (53) This is a contradiction to (52). In fact, if we renumber the vertices, if necessary, condition (53) means that otj(Au) > otj(u) for all j — 0,..., k and some k — 0,1,2. (53*) In addition, since u G S and Aw G <S, it follows from (52) that a0(u) + ai(u) + a2(u) = 1 and a0(Aw) + ai(Aw)+ a2(Aw) = 1. (53**) For k = 2, relation (53*) is impossible, by (53**). If fc = 1 or fc = 0, then w G co{wo,wi} or u G co{wo}, and hence a2(w) = 0 or ai(w) = a2(w) = 0, respectively. Again, (53*) contradicts (53**). The lemma of Knaster, Kuratowski, and Mazurkiewicz (Lemma 6) tells us now that there is a point v G S such that v G Cj for all j = 0,1,2. This implies olj(Av) < <Xj{v) for all j = 0,1,2. According to (53**) with u — v, we get aj(Av) = aj(v) for j = 0,1,2, and hence Av — v. Thus, v is the desired /iced pom£ of A in the case where N = 2.

1.16. Applications to Integral Equations 61 If N > 3, then use the same argument as for N — 2 above. Step 2: Let M be a compact, convex, nonempty subset of a finite-dimensional normed space. By Proposition 9 in Section 1.13, the set M is homeomorphic to some iV-simplex <S. Using Step 1, the same argument as in the proof of Corollary 1 shows that each continuous operator A: M —> M has a fixed point. This finishes the proof of the Brouwer fixed-point theorem (Theorem l.B). □ 1.15 The Schauder Fixed-Point Theorem Theorem l.C. The compact operator A:M^M has a fixed point provided M is a bounded, closed, convex, nonempty subset of a Banach space X over K. This theorem was proved by Schauder in 1930. If dim X < oo, then Theorem l.C coincides with the Brouwer fixed-point theorem (Theorem l.B in Section 1.14). Proof. Let uq e M. Replacing u with u — ^o, if necessary, we may assume that 0 e M. It follows from the approximation theorem for compact operators (Proposition 13 in Section 1.11) that, for every n = 1, 2,..., there exists a finite- dimensional subspace Xn of X and a continuous operator An:M -» Xn such that \\Au-Anu\\<- for all u e M. (54) n Define Mn :- Xn n M. Then, Mn is a bounded, closed, convex subset of Xn with 0 G Mn and An{M) C co A(M) C M, since M is convex. By the Brouwer fixed-point theorem (Theorem l.B in Section 1.14), the operator An: Mn —> Mn has a fixed point un, i.e., Anun = un, un e Mn, for all n = 1,2,... . (55) By (54), \\Aun —un\\ < —, for all n — 1,2,... . (56)

62 1. Banach Spaces and Fixed-Point Theorems Since Mn C M for all n, the sequence (un) is bounded. The compactness of the operator A: M —> M implies that there is a subsequence, again denoted by (un), such that Aun —> v as n —> oo. By (56), ||v — wn|| < ||v — A?xn|| + ||A?xn — wn|| —> 0 as n —> oo. Hence i£n —> v as n —> oo. Since Aun G M for all n and the set M is closed, we get v £ M. Finally, since the operator A: M —> M is continuous, it follows that Av = u, vGM. □ 1.16 Applications to Integral Equations We want to solve the integral equation rb u(x) — A / F(x,y,u(y))dy, a<x<b, (57) J a where "-oo < a < b < oo and A G R. Let Q := {(#, y, w) G R3: x,t/E [a, 6], \u\ < r} for fixed r > 0. Proposition 1. Assume the following: (a) TAe function F:Q —► R is continuous. (b) We de/me .M :— max^^^g |F(x, 2/,w)|. £e£ £Ae real number A 6e #wen such that \X\M < r. (c) We set X := C[o, 6] and M := {u G X: ||u|| < r}. Then, the original integral equation (57) has a solution u G M. This generalizes Proposition 1 in Section 1.7. Proof. Define the operator (Au)(x) := A / F(x, y, u(y))dy for all x G [a, 6]. ./a Then, the integral equation (57) corresponds to the following fixed-point problem: u = Au, ueM. (57*)

1.17 Applications to Ordinary Differential Equations 63 The operator A: M —> M is compact, by Standard Example 12 in Section 1.1L For each ueM, / F(x,y,u{y))dy J a \\Au\\ < |A| max a<x<b < \X\M < r. Hence A(M) C M. Thus, the Schauder fixed-point theorem (Theorem l.C in Section 1.15) tells us that equation (57*) has a solution. □ 1.17 Applications to Ordinary Differential Equations Let us consider the following initial-value problem: v! = F(x, u), x0 — h < x < x0 + h, (58) u(x0) = u0, where the point (#o, ^o) £ M2 is given. Set S := {(#, u) G M2: |x — #o| < ^» |w — ^o| < ?"}• Proposition 1 (The Peano theorem). Assume the following: (a) T/ie function F: S —> M zs continuous. (b) W^e se£ A4 := maxa<a;<6 |F(x, w)|, and we choose a number h in such a way that 0 < h < r and hM < r. Then, the original initial-value problem (58) has a solution. This generalizes the Picard-Lindelof theorem from Section 1.8 based on the Banach fixed-point theorem. We will now use the Schauder fixed-point theorem. Proof. Let us consider the integral equation u(x) =:u0+ F(y, u(y))dy, x0 - h < x < x0 + h. (59) JXq Set X := C[a, b] and M := {u e X: \\u - u0\\ < r}.

64 1. Banach Spaces and Fixed-Point Theorems Define the operator (Au)(x) :=u0 + / F(y,u(y))dy for all x G [x0 - h,x0 + h]. Then, the following are true: (i) A: M —> X is continuous. (ii) A(M) is equicontinuous. (iii) A(M) C M, i.e., in particular, the set A(M) is bounded. Ad (i), (ii). This follows as in the proof of Standard Example 12 in Section 1.11. In this connection, observe that px pz II pz I F(y,u(y))dy- F(y,u(y))dy\ = \ F(y,u(y))dy J Xq J Xq I \J X <\x — z\M, for all x, z e [a, b] and ue M. Ad (iii). If ueM, then I fx \\Au -uo\\= max / F{y,u{y))dy < hM < r. By the Arzela-Ascoli theorem (Standard Example 7 in Section 1.11), it follows from (i), (ii) that the set A(M) is relatively compact in X. Since the set M is bounded, this implies the compactness of the operator A: M -» M. The Schauder fixed-point theorem (Theorem l.C in Section 1.15) tells us that the operator equation Au~u, ue M, has a solution, i.e., the integral equation (59) has a solution u £ M. Differentiating the integral equation (59) with respect to x, we see that u is also a solution of the original problem (58). □ 1.18 The Leray-Schauder Principle and a priori Estimates Let X be a Banach space. We want to solve the equation u = An, ueX, (60) by using properties of the parametrized equation u = tAu, ueX, 0 < t < 1. (61)

1.18 The Leray-Schauder Principle and a priori Estimates 65 For t = 0, equation (61) has the trivial solution u = 0, whereas (61) coincides with (60) if t — 1. The following condition is crucial: (A) A priori estimate. There is a number r > 0 such that if u is a solution of (61), then IMI < r. Observe that we do not assume that equation (61) has a solution. Condition (A) is satisfied trivially if the set A{X) is bounded, i.e., there is a number r > 0 such that \\Au\\ < r for all u £ X. Theorem l.D. Suppose that the compact operator A:X —> X on the Ba- nach space X over K satisfies condition (A). Then, the original equation (60) has a solution. This theorem was proved by Leray and Schauder in 1934. Roughly speaking, Theorem l.C corresponds to the following important principle in mathematics: A priori estimates yield existence. A typical application of this principle to the famous Navier-Stokes equations for viscous fluids will be considered in Section 5.17 of AMS Vol. 109. Further applications to a general class of quasilinear elliptic partial differential equaitons can be found in Zeidler (1986), Vol. 1, Chapter 6. Proof. Set M :~ {u G X: \\u\\ < 2r}. We define an operator ( An if \\Au\\ if \\Au\\ < 2r > 2r. Obviously, \\Bu\\ < 2r for all u 6 X, i.e., B(M) C M. We claim that B:M —> M is compact. In fact, B is continuous. This follows from the continuity of the operator A.X-+X and from 2rAu .„,,,,, Au = Wh lf UA = 2r' by using the (e — <5)-definition of continuity (Definition 1 in Section 1.9). To establish compactness, let (un) be a sequence in the ball M. We consider two cases, namely, there is a subsequence (vn) of (un) such that (a) H^nll < 2r for all n; (b) \\Avn\\ > 2r for all n.

66 1. Banach Spaces and Fixed-Point Theorems In case (a), the boundedness of the set M and the compactness of the operator A imply that there is a subsequence (wn) of (vn) such that Bwn — Awn —> z as n —> oo. In case (b), one can choose a subsequence (wn) of (vn) so that i a and Awn —> z as n —> oo P^n| for suitable a and z, since the sequence f ..^ ■■ j is bounded and the operator A is compact. Hence Bwn —> 2raz as n —> oo. The Schauder fixed-point theorem (Theorem l.C in Section 1.15) applied to the compact operator B:M —> M provides us with a point u G M such that u = Bw. If \\Au\\ < 2r, then Bu = ^4w, and hence ^ = Aw, i.e., w is a solution of the original problem (60). The other case \\Au\\ > 2r is impossible by the a priori estimate (A). In fact, let u~Bu with \\Au\\ > 2r. Then 2r U = BU = tAu With t := -jr-r-rr < 1. (62) ||i4ii|| This forces ||w|| = |t| • \\Au\\ = 2r. On the other hand, equation (62) implies IMI < ^, by (A), which is a contradiction. □ Applications can be found in Problems l.lo and l.lp. 1.19 Sub- and Supersolutions, and the Iteration Method in Ordered Banach Spaces The idea of ordered Banach spaces is to introduce a relation u < v, which generalizes the corresponding relation for real numbers. Definition 1. A subset X+ of a normed space X is called an order cone iff the following are true: (i) X+ is closed, convex, and nonempty, and X+ ^ {0}. (ii) If u e X+ and a > 0, then

1.19. Sub- and Supersolutions 67 (iii) If u G X+ and — u G -X+, then u = 0. Let ^, i; G X. We define ^<v iff i; — u £ X+. By an ordered normed space (resp., ordered Banach space) we understand a normed space (resp., Banach space) together with an order cone. We also define the order interval [u, w] := {v G X: u < v < w}. The order cone X+ is called normal iff there is a number c > 0 such that 0<u<v implies \\u\\ < c\\v\\. Example 2. The Banach space X := R with the norm ||tt|| := |w| is an ordered Banach space with the order cone X+ := R+ := {i£ G X: i£ > 0}. Here, the order relation u < v in X coincides with the corresponding classical relation. Since 0 < u < v implies \u\ < |v|, the order cone X+ is normal. Example 3. The Banach space X :~ RN, N = 1,2,..., with the Euclidean norm | • | is an ordered Banach space with the order cone X+ := R^ - {(ft, • • • ,&v) G R":& > 0 for all j}. Here, (fi, • • • ,€n) < (Vi> • • • > *7at) iff & < % for all j. (63) By (63), 0 < x < y implies 0 < £? < rjj for all j, and hence \x\ < \y\. Thus, the order cone X+ is normal. The order cone R+ in R2 is pictured in Figure 1.20. Standard Example 4. The Banach space X :~ C[a, 6], —oo < a < b < oo, with the usual norm \\u\\ := maxa<a;<5 |m(#)|, is an ordered Banach space with the normal order cone X+ := C+[o, 6] = {u G C[o, b]:u(x) > 0 on [a, 6]}. Here, u <v on X iff ^x(x) < v(x) on [a, 6]. It follows from 0 < u < v in X that 0 < u(x) < v(x) on [a, 6], and hence 11ix11 < \\v\\. Thus, the order cone X+ is normal.

68 1. Banach Spaces and Fixed-Point Theorems FIGURE 1.20. The following proposition shows that the relation u < v has the usual properties. Proposition 5. Let u,v,w,un,vn G X+ for all n, where X+ is an order cone in the Banach space X. Then: (i) u < v and v < w imply u < w. (ii) u < v and v < u imply u = v. (iii) u < v implies u-\-w<v + w and au < av for all a > 0. (iv) un < vn for all n and un —> u and vn —> v as n —> oo imply u < v. (v) // the order cone X+ is normal, then u < v < w implies \\v — u\\ < c\\w — u\\ and \\w — v\\ < c\\w — u\\. Proof. Ad (i). v — u G X+ and w-i> G X+ imply 2~1(v-u) + 2~1(w-v) G X+, since X+ is convex. Hence (v — u) + (w — v) G X+, i.e., w — u G A+. Ad (ii). v — ix £ X+ and — (v — u) G X+ imply v — w = 0. Ad (iii). v — u e X+ implies (v -\-w) — (u + w) G X+ and a(v — u)e X+ for all a > 0. Ad (iv). If vn — un £ X+ for all n and un ^ u and vn -> v as n -> oo, then v — w G -X"+, since X+ is closed. Ad (v). Adding —1£ to i£ < v < w, we get 0<v — u<w — u, and hence ||^ — u\\ < c\\w — u\\. Moreover, u < v < w implies v — u G X+ and (iu — u) — (w — v) G X+. Hence 0 < iu — v <w — u. This yields ||w — v\\ < c\\w — u\\. □ We now want to solve the operator equation u = Au, u0 < u < ^o, u G X, (64) by means of the two iteration methods un+1=Aun and vn+x = Avn, n = 0,1,..., (65)

1.20. Linear Operators 69 where uo,vo € X are given. Theorem I.E. Suppose that the following are met: (a) The operator A: [u0, v0] Q X —> X is compact, where X is an ordered Banach space with normal order cone. (b) The operator A is monotone increasing, i.e., u <v implies Au < Av. (c) uo is a subsolution of (64), i.e., uo < Au$. (d) ^o is a supersolution of (64), i.e., Avo < v0. Then, the iterative sequences (un) and (vn) constructed in (65) converge to a solution u andv of the original equation (64), respectively. In addition, we have the error estimates uo <u\ < " - <un <u <v <vn < vn-i < - - <vo for all n. (66) This theorem corresponds to the following general existence principle in mathematics: The existence of both a subsolution and a supersolution yields the existence of a solution. Proof. We use the same arguments as in the classical case X = R. Step 1: Monotonicity of (un) and (vn). For all n, uo <ui <u2 < "• <un <vn < vn-i < • • • < vi < v0. (66*) In fact, i£0 < Auo implies uo <U\. Since A is monotone increasing, u0 < U\ yields Auo < Au\, i.e., u\ <U2- Moreover, ^o < ^o implies Auo < Avo. By hypothesis, Avo < vo- Hence u\ <v\ < vq. Relation (66*) follows now by induction. Step 2: Convergence of (un). By Proposition 5(v), it follows from (66*) that ll^o — un\\ < c\\vo — uo\\ for all n, i.e., the sequence (un) is bounded. Since the operator A is compact, there exists a subsequence (un>) such that Aun> —> u as n' -^ cxd. Let e > 0 be given. Since un+\ = Aun, there is a number no(e) such that \\un0 -u\\ <e. Letting n' —> oo in (66*), we get Un0 <un<u for all n > no(e).

70 1. Banach Spaces and Fixed-Point Theorems Hence \\u — un\\ < c\\u — uno\\ < e for all n > no(e), i.e., un—±u as n —> oo. Since the operator A is continuous, letting n —> oo in i£n+i = Aun produces u = An. Step 3: Similarly, one proves that vn —> v as n —> oo and v = Av. Letting n —> oo in (66*), we obtain (66). An application to integral equations will be considered in Problem l.ln. 1.20 Linear Operators Definition 1. Let X and Y be linear spaces over K. The operator A:L C X —> Y is called linear iff L is a linear subspace of X and A(cm + /3v) — a An + /3Av for all u,v £ L and a, /3 G K. Recall that R(A) :— {v £Y:v = Au for some u G X}. We also introduce the null space N{A) :={ueX:Au = 0}. The linear operator A:X —> Y is infective iff N(A) = {0}. This follows from Au - Av = A(u - v). In fact, if N(A) = {0}, then Au = Av implies u = v, i.e., ^4 is injective. Conversely, if A is injective, then Aw = 0 implies u = 0, i.e., iV(A) = {0}. Proposition 2. Le£ A: X —> Y 6e a Zmear operator, where X and Y are normed spaces over K. Then the following two conditions are equivalent: (i) A is continuous. (ii) There is a number c > 0 such that \\Au\\ < c\\u\\ for all u G X. For a linear continuous operator A: X —> Y, we define the operator norm through ||4| := sup ||^||. (67) IMI<i By (ii), \\A\\ < oo. Prom (67) we get ||i4u|| < \\A\\ \\u\\ for all u € X, (67*) by letting v := ||u||_1w for u ^ 0. In fact, then ||v|| = 1 and \\Av\\ = ||;4u|| IM!"1 < ||A||.

1.20 Linear Operators 71 Obviously, if X ^ {0}, then \\A\\ = sup \\Av\\. \\v\\ = l Proof, (i) => (ii). Because of the linearity of A, condition (ii) is equivalent to \\Av\\ < c for sllveX with ||v|| < 1. Let A be continuous. If (ii) is not true, then there is a seoueiu i (vn) with ||^n|| < 1 and ||Ayn|| > n for all n = 1,2,... . Setting wn := n~1vn, then wn —> 0 as n —> oo and ||Awn|| > 1 for all n. (68) Since ^4 is linear, A(0) = 0. Moreover, since A is continuous, wn —> 0 as n —> oo implies Awn —> 0 as n —> oo. This contradicts (68). (ii) => (i). For given £ > 0 choose 6 = £C-1. Then ||w — v|| < 6 implies \\Au — Av\\ < e, since ||A(u-v)|| < c||u-v|| < e. D The following proposition tells us that linear continuous operators between finite-dimensional normed spaces correspond to matrices. The two basic formulas are given through (N \ M Y^Znen ) = ^ Vmfm, (69) 71=1 / 771=1 where AT r)m = Y^amn£>n, 771 = 1, . . . , Af. (69*) 71=1 Proposition 3. Let X and Y be finite-dimensional normed spaces over K with dim X = N and dim Y = M, where N,M >1. Let {ei,..., e^} and {/i> • • •, /m} be a basis in X and Y, respectively. Then, the operator A:X —> Y is linear iff there is an (M x N)-matrix (amn) with amn € K for all n = 1,..., JV, m = 1,..., M such that the formulas (69) and (69*) hold. All these operators are continuous.

72 1. Banach Spaces and Fixed-Point Theorems Proof. Suppose that A is linear. Then Aen G Y", and hence there numbers amn in K such that M Aen = ]P amnfm for all n = 1,..., N. m=l Since A is linear, / n \ n I / j snen J == / v sn^n- are \n=l / n=l This yields (69) with (69*). Conversely, define the operator A:X —> Y" through (69) and (69*). Then, A is linear. Recall that |H|oo •= maxn |£n| for u = £iei H h £nen- Then Halloo = niax|r7m| < w4.oo||^x||oo for all uGl, m where AT l<m<M ^—^ n=l Hence H-Alloo := sup ||i4ii||oo < ^oo. (70) N|oo<l Since each norm || • || on a finite-dimensional normed space is equivalent to the norm || • ||oo, we get \\Au\\ < C\\u\\ for all u G X and fixed C > 0, by (40). Hence the operator A: X —> Y is continuous. □ Standard Example 4. Let X := C[a,b] with the norm \\u\\ := max |ti(x)|, a<x<b where —oo<a<6<oo. Suppose that the function K: [a, b] x [a, 6] -» R is continuous. Define the integral operator fb (Au)(x) := / K(x,y)u(y)dy for all x G [a, 6]. ./a Then the operator A: X —> X is linear and continuous with ||A|| < max: |/f(»,|/)|(|r-a).

1.20 Linear Operators 73 In addition, it follows from Standard Example 12 in Section 1.11 that A: X —> X is also compact. Proof. Let u G X. It follows from pb pb \ K(x,y)u(y)dy\< max \K(x,y)\ / \u(y)\dy \Ja a<x,y<b Ja that \\Au\\ = max a<x<b / K(x,y)u(y)dy J a < max \K(x,y)\(b — a)\\u\\. D Proposition 5. Let L(X, Y) denote the space of linear continuous operators A.X^Y, where X is a normed space over K and Y is a Banach space over K. Then L(X, Y) is a Banach space over K with respect to the operator norm Proof. Step 1: L(X,Y) is a linear space, where the linear combination aA + (3B, A,BeL(X,Y), a,/?eK is denned in the usual way through (aA + 0B)u = aAu + (3Bu for all u G X, a, (3 e K. Step 2: The operator norm represents a norm on L(X,Y). In fact, it follows from (67) and (67*) that ||A|| = 0 iff A = 0. Let a e K, and A,B e L(X, Y). Then \\aA\\ = sup ||ai4ifc|| = |a| sup \\Au\\ = \a\ \\A\\. M<i ||u||<i Finally, the triangle inequality follows from \\A + B\\ = sup \\Au + Bu\\ < sup(\\Au\\ + ||J5u||) < sup \\Au\\ + sup ||J5ii|| = \\A\\ + ||JB||, where the supremum is taken over all u G X with \\u\\ < 1. Step 3: Cauchy sequences. Let (An) be a Cauchy sequence in L(X,Y), i.e., \\An — Am\\ < e for all n,m > no(e). Hence \\Anu — Amu\\ < \\An — Am\\ \\u\\ < e\\u\\ for all n,m > no(e). (71)

74 1. Banach Spaces and Fixed-Point Theorems Thus, the sequence (Anu) is Cauchy. Since Y is a Banach space, (Anu) is convergent. Define Au := lim Anu for all u G X. n—>-oo Letting n —> oo, it follows from An(cm + /3v) = aAnM + (3Anv that ^.(a^x + /3v) = olAu + /L4v for all u,v e X, a, /3 G K, i.e., the operator ^4 is linear. By (71), ||Anu|| < ||Anu - Anou|| + ||-Anou|| < e\\u\\ + ||Ai0^ll f°r all n > n0(e). Letting n —> oo, this implies pu|| < (e+||Ano||)|H| forall^GX, i.e., A is continuous. Finally, letting m —> oo in (71), we get ||^4nu — Au\\ < e\\u\\ for all n > no(e) and all u £ X. Hence ||Ai — A\\ < e for all n > n0(e), i.e., An —> ^4 in L(X, Y) as n —> oo. This proves that each Cauchy sequence in L(X,Y) is convergent, i.e., L(X, Y) is a Banach space. □ Proposition 6. Let A:X —> Y and B:Y —> X be linear operators, where X and Y are linear spaces over K. Suppose that AB = I and BA = I. Then A is bijective and A~x = B. Proof. Since A(Bu) = u for all u G Y, the operator A is surjective. Moreover, if Au = Av, then A(u — v) = 0. Hence u — v = BA(u — v) = 0, i.e., A is injective. Consequently, A is bijective. Applying A'1 to AB = /, we get A~XAB = A"1, i.e., £ = A"1. □ 1.21 The Dual Space Definition 1. Let X be a normed space over K. By a linear continuous functional on X we understand a linear continuous operator /:X-K.

1.21 The Dual Space 75 The set of all linear continuous functionals on X is called the dual space X* of X. Obviously, X* = L(X,K). We set (/, u) :- /(«) for all u e X, / e X*. Let / € X*. By (67), the norm of / is given through ll/H := sup |/(„)|. (72) Hl<i Hence |</,«>| = |/(«)| < ll/H ||u|| for all u 6 X, f 6 X*. (72*) Proposition 2. Let X be a normed space over K. Then the dual space X* is a Banach space over K with respect to the norm \\f\\. This follows from Proposition 5 in Section 1.20. Example 3. Let X := C[a, 6], — oo < a < b < oo, and let v G X. Define fb f(u) := / u{x)v{x)dx for all u e X. J a Then, feX*, and ||/|| < (b - a)\\v\\. Proof. For all ueX, \f{u)\<(b — a) max \u(x)\ max \v(x)\ = (b — a)\\u\\ \\v\\. D a<x<b a<x<b A complete description of the dual space C[a, 6]* will be given in Section 2.3 of AMS Vol. 109, along with applications to the famous classical moment problem. Proposition 4. Let X be a finite-dimensional normed space over K with dim X > 1. Let {ei,..., e^} be a basis in X, and let N i=i Then, f G X* iff there exisi numbers an G K, n = 1,..., N, such that N f(U) = 5Z an^n f°r aU UeX' 71=1

76 1. Banach Spaces and Fixed-Point Theorems This is a special case of Proposition 3 in Section 1.20 with Y := K, M = 1, and /i := 1. Important properties of the dual space X* will be studied in Chapters 2 and 3 of AMS Vol. 109, namely, the Hahn-Banach extension theorem and its consequences, the separation of convex sets, reflexive Banach spaces, and variational principles. 1.22 Infinite Series in Normed Spaces Definition 1. Let X be a normed space over K and let u3 G X for all j. We set oo n j=0 3=0 provided this limit exists. This infinite series is called absolutely convergent iff OO Proposition 2. Each absolutely convergent infinite series in a Banach space is convergent. Proof. Set sn := Y^=o ur ^v (73*), for each e > 0, there is an no(e) such that n+k ||sn+fc — sn\\ < ^ \\uj|| < e for all n > no(e) and all k = 1,2,... . Hence the sequence (sn) is Cauchy, i.e., the limit (73) exists. □ 1.23 Banach Algebras and Operator Functions Definition 1. By a Banach algebra B over K we understand a Banach space over K, where an additional multiplication "AB" is denned such that ABeB for all ,4,5 eS. Moreover, for all A,B,CeB and a G K, the following are true: (AB)C = A(BC), A(B + C) = AB + ,4C, (JB + C)A = RA + OA, a(AB) = (aA)B = -A(aB), ||AB|| < ||;4|| ||B||.

1.23 Banach Algebras and Operator Functions 77 In addition, we postulate that there exists an E £ B such that AE = EA for all,4e# and \\E\\ = 1. Standard Example 2. Let X be a Banach space over K with X ^ {0}. Then L(X, X) represents a Banach algebra, where AB corresponds to the usual multiplication of operators denned through (AB)u := A(Bu) for all ueX, and E is equal to the identical operator, i.e., Eu := u for all u G X. Proof. Observe that ||AB|| = sup ||i4(JBii)|| < sup P|| ||£u|| = \\A\ ||JB||, H^ll = sup||^|| = 1, where the supremum is taken over all u G X with ||m|| < 1. □ Proposition 3. Let B be a Banach algebra, and let A, B, An, Bn G B for all n. Then: (i) \\Ak\\ < \\A\\k for all fc = 0,1,2,..., where we set A0 := E. (ii) If An —> A and Bn —> B in B as n —> oo, then AnBn —> AB in B as n —> oo. Proof. Ad (i). Use Pm+1|| = \\AmA\\ < Pm|| ||;4|| for m = 1,2,.... Ad (ii). Since the sequences (An) and (Bn) are bounded, we get \\AnBn - AB\\ = \\{An - A)Bn - A(B - Bn)\\ < \\An - A\\ \\Bn\\ + \\A\\ \\B - Bn\\ -+ 0 as n -+ oo. D Our next goal is the definition of operator functions through oo F(A) := Y, adAj, Aj G K for all j, (74) j=o where oo F(z) = ^a^', zGK, (75) i=o along with oo y^ |aj| |z|J < oo for all z G C with \z\ < r and fixed r > 0. (75*) i=o

78 1. Banach Spaces and Fixed-Point Theorems Recall that K = R or K = C. Proposition 4. Let X be a Banach space over K. Suppose we are given the function F as in (75) with (75*). Then, for each A £ L(X, X) with \\A\\<r, formula (74) defines an operator F(A) £ L(X1X). The following proof shows that this proposition remains valid if we replace L(X,X) with a Banach algebra B. Then, F(A) £ B. Proof. Let ||A|| < r. Then oo oo 2>i^ll<5>i|||.A||''<oo. Thus, the infinite series X]°l0a.7'A? from (74) is absolutely convergent and hence convergent in L(X, X). D Example 5 (The exponential function). Let X be a Banach space over K. Then (i) The infinite series oo = £> 7 converges absolutely for all ^4 £ L(X, X). (ii) For each ,4 £ L(X, X) and all t, s £ K, C*VA = e(t+s)A. (76) Proof. Ad (i). Observe that YlTLo ^ < °° f°r a^ ^ £ C, by a well-known property of the classical exponential function oo .. = ]T -,** for a11 zeC. oo J-0J Ad (ii). As for the classical exponential function, we get n .a n u n ,a v. n r .a t_a j=0J' k=0 ' r=Oj+k=rJ' ' r-0 j=0 y Jr r=0 j=0 ' XJ / r=0

1.23 Banach Algebras and Operator Functions 79 Letting n —> oo and using Proposition 3(ii), we obtain (76). □ Example 6 (The geometric series). Let X be a Banach space over K with X ^ {0}. The classical geometric series oo i=o converges absolutely for all z G C with \z\ < 1. By Proposition 4, for each operator A G L(X,X) with \\A\\ < 1, the infinite series oo 3=0 converges absolutely to an operator B G L(X,X). This series is called the Neumann series. In addition, B = (I-A)-\ Proof. Obviously, (I - A)B = I and B(I - A) = I. RenceB = (I-A)-1. □ Let X and Y be Banach spaces over K with X ^ {0} and Y =^ {0}. Denote by Linv(X, Y) the set of all the operators A G L(X, Y) such that the inverse operator A~l : Y —> X exists and ^4-1 G L(Y, X). Proposition 7. If A G Linv(X, Y) and B G L(X, Y) iwtfi ll^l^p-1!!"1, then A + B e Liny(X,Y). Corollary 8. The set Linv(X, Y) is open in L(X,Y). Proof. Let A G Linv(X, Y). It follows from AA~l = I that A'1 ^ 0, and hence H^"1!! ^ 0. If A G Linv(X, Y) and C G Linv(X, X), then AC G Linv(^, ^) and (ACy1 = C'lA'1. (77) In fact, (C-1^-1)^^) = I = (ACJCC-M-1). Since H-A-1^! < H-A"1!! ||jB|| < 1, it follows from Example 6 that C:=(I + A-1B)eLinv(X,X). By (77), AC = A + B G Linv(X, Y). □

80 1. Banach Spaces and Fixed-Point Theorems 1.24 Applications to Linear Differential Equations in Banach Spaces Definition 1. Let u:U(t0) Cl^I be a function where X is a normed space over K and U(to) is an open neighborhood of the point to £ R- We define the derivative u'(t0) := lim h~l(u(t0 + ft) - u(t0)) h—->0 provided this limit exists. Proposition 2. // the derivative u'(to) exists, then the function u(-) is continuous at the point to. Proof. The identity u(t0 + ft) = u(t0) + hih^iuito + ft) - u(t0))) yields u(t0 + h) —> ti(^o) as ft —► 0. D Let us now consider the following initial-value problem: u'{i) = A?x(£), — oo < £ < oo, (78) u(0) = u0, where Uq € X is given. Proposition 3. Let X be a Banach space over K, and let the operator A e L(X, X) be given. Then the initial-value problem (78) has a unique solution given by u(t) = etAu0 for all t G R. Example 4. Consider the special case where X := RN, N = 1, and A = (djk) is a real (N x iV)-matrix. Then problem (78) corresponds to the following system of linear differential equations: N €'W = 1JL,ajk€k(t), -oo < t < oo, fc=i (78*) 6(o) = 6y, j = i,...,iv.

1.24 Applications to Linear Differential Equations in Banach Spaces 81 By Proposition 3, this system has a unique solution. Proof of Proposition 3. Step 1: Existence. Let ft G R. It follows from ft2 ehA = I + hA + -j- A2 + that A:=\\h-l(ehA-I)-A\\ t^ J=2 ^ I^CMn— WAW3<const|ft|-^0 as ft-> 0. —' 7 =9 J Since i=2 e(t+h)A = e*Ae/iA = e/iAetA for all t, ft G R, we get llft"1^ + ft) - u(*)) - Au(t)\\ = \\(h-\ehA - I) - A)etAu0\ < A||c tA\ \\uo\\ 0 as ft ^ 0. This implies u'(t) = A?x(£) for all t G R. In addition, u(t) = etAu0 = u0 for t = 0. Step 2: For the uniqueness proof, we need the following result, which will be proved in Section 1.1 of AMS Vol. 109 as an easy consequence of the Hahn-Banach theorem: For all v G X, INI = sup ll/»|, ll/ll<i where / G X*. Step 3: Let u = u(t) and v = v(£) be two solutions of the original problem (78). Set w(t) := u(*) - v(t) for all * G R. Then 1t/(£) = Aw(t), — 00 < £ < 00, w(0) = 0. We have to show that (79) implies w(t) = 0 for all t eR. Let w = w(*) be a solution of (79). Choose / G X*. By (79), (/, u/(*)) = (/, j4w(*)> for all t G R. Since /: X —> R is linear and continuous, we get (79) lim /i->0 /i = Km(f,h-1(w(t + h)-w(t))) h—->0 = (f,w'(t)) for all* eR.

82 1. Banach Spaces and Fixed-Point Theorems This implies jt (/, w(t)) = (/, Aw(t)) for all * G R. (80) By Proposition 2, the function t i-» w(t) is continuous on R. Hence, the function t i-» (/, Aw(0)) is also continuous, since A and / are continuous. Integrating (80) and observing that (/, w(0)) = 0, we obtain (/, w(t)) = / (/, Aw(s))ds for all t G R. By (72*), for all t with \t\ < ft, we get l(j>(*)>l< I T11/11 Nl IkWHcfc \Jo <ft||/||||A||maK|KS)||. \s\<h It follows from Step 2 that Hence moil < hm max \\w(s)\\ for ai11: i*i < h> \s\<h max \\w(t)\\ < h\\A\\ max \\w(t)\\. \t\<h \t\<h> This yields w(t) = 0 for all * G R provided ,4 = 0. If A ^ 0, then we choose the number ft := ^OTT- Hence w(j) = 0 for all t: \t\ < ft. Now applying the same argument to the initial-value problems w'(t) = Aw(t), — oo < t < oo, w(±ft) = 0, we get w(t) = 0 for all t G [—2ft, 2ft]. Continuing this, we obtain w(t) = 0 for all teR. D 1.25 Applications to the Spectrum Let us consider the equation Au = \% ueX, Xe C. (81) Definition 1. Let A G L(X, X), where X is a complex Banach space with X ? {0}.

1.25 Applications to the Spectrum 83 The complex number A is called an eigenvalue of the operator A iff equation (81) has a nontrivial solution u ^ 0. The resolvent set p(A) of A is denned to be the set of all the complex numbers A for which the inverse operator (A — A/)-1: X —> X exists and {A-XI)~leL(X,X). In this case, the operator (A — A/)-1 is called a resolvent of A. The spectrum a(A) of A is defined through cr(A) := C — p(A). Proposition 2. The spectrum cr(A) is a compact subset ofC and |A| < \\A\\ forallXea(A). Each eigenvalue X of A belongs to the spectrum of A. The resolvent set p(A) is open in C. Proof. Let A G p(A) and p G C. Hence A - XI G L-mv(X, X). Since IK-A - AZ) - (-A - mZ)|| < |A - m|, it follows from Example 6 in Section 1.23 that A—pi G Linv(X, X) provided |A — >lx| is sufficiently small. Hence the set p(A) is open. If |A| > ||A||, then A G p(A). In fact, since IIa-^ii^ia-1!!^!^!, Example 6 in Section 1.23 tells us that (A"1 A - I)'1 G L(X,X). Hence (A - XI)-1 = X^iX^A - I)-1 G L(X, X). Consequently, the spectrum cr(A) = C — p(A) is closed and bounded, i.e., a(A) is compact. Finally, if A G p(A), then (A - XI)u = 0 implies u = (A - A/)~1(0) = 0, i.e., A is not an eigenvalue of A. □ The operator B: X —> X on the Banach space X is called semi-Fredholm iff the range i?(B) is closed and the null space of B has a finite dimension, i.e., 0 < dim $ < oo. HIV) Definition 3. Let the operator A G L(X,X) be given, where X is a complex Banach space with X ^ {0}. The essential spectrum cre(A) of A consists of all A G C such that the operator A — XI: X —> X is no£ semi-Fredholm.

84 1. Banach Spaces and Fixed-Point Theorems Obviously, ae(A) C a (A). Moreover, ae(A) contains all the eigenvalues A of A that have an infinite multiplicity, i.e., dim N(A — XI) = oo. Example 4. If dim X < oo, then the essential spectrum of A is empty. Proof. Observe that finite-dimensional linear subspaces of Banach spaces are always closed. Hence R(A — XI) is closed for all A G C. □ Example 5. If the operator A is compact, then we shall show in Section 10.6 that the operator A — XI: X —> X is Fredholm (and hence also semi- Fredholm) for all A G C with A ^ 0. Thus, either cre(A) is empty or cr^A) = {0}. The essential spectrum plays a fundamental role in quantum mechanics with respect to scattering processes (cf. Section 5.20). 1.26 Density and Approximation Classical approximation theorems can frequently be formulated in terms of dense sets in normed spaces. Definition 1. Let X be a normed space. A subset M of X is called dense in X iff _ M = X, i.e., for each u £ X and each e > 0, there is a v G M such that \\u — v\\ < e. The space X is called separable iff there is an at most countable, subset M of X. tk^VL Recall that a set M is called countable iff there exists a bijective map A: M —> N, where N denotes the set of natural numbers n = 1,2 The set M is called at most countable iff it is either finite or countable. It is well-known that the set Q of rational numbers is countable. Proposition 2. Let X := C[a, 6], where —oo < a < b < oo. Then the set of all the polynomials p(x) := a0 + a\x H h anxn,... , n = 0,1,... , with real coefficients ai is dense in X. This is the classical Weierstrass approximation theorem. In fact, Proposition 2 tells us that for each continuous function u: [a, b] —*■ R and each e > 0, there is a polynomial p such that ||w — p|| := max \u(x) — p(x)\ < e. a<x<b

1.26 Density and Approximation 85 Corollary 3. The space C[a, 6] is separable. Proof of Corollary 3. For each real number aj and each e > 0, there is (82) a rational number r? such that Letting q(x) := r0 + T\x H h rnxn, it follows from (82) that ||tx — #|| < ||w — p|| + ||p — q\ < e + \J |a? — r?| ( max |x| J < const • e. j=0 \a<x<b J Thus, the set M of all the polynomials q with rational coefficients is dense in C[a, b]. But, the set M is countable, since the set of rational numbers is countable, and the union of a countable number of countable sets is again countable. □ Proof of Proposition 2. Let E := XX=o- ^e on^ consider the case where a = 0 and 6=1. The general case can be reduced to this special case by using the coordinate transformation x = a + (6 — a)y, which transforms [0,1] into [a, 6]. Step 1: Two identities. For bk{x) :=, xk(l — x)n~k, we have E6fc(x) = l, (83a) and E bk(x)(nx - k)2 = nx(l - x) for all x G R and n = 0,1,... . (83b) To prove this, we begin with the binomial theorem {x + y)n = Y,(fyxkyn-k. {x + y)n = Tl['l)xkyn-k. (84) Setting y = 1 — x, we get (83a). Differentiation of (84) with respect to x and multiplication with x (resp., x2) yields nx(x + y)n~l = E ^ } kxkyn~\ n(n - l)x2(x + y)n~2 = E ^ \ k(k - l)xfc2/n-fc.

86 1. Banach Spaces and Fixed-Point Theorems Setting y = 1 — x, we obtain (83b) by summation. In fact, E&fc(x)(n2x2 - 2nxk + k2) = n2x2 - 2n2x2 + [n(n - l)x2 + nx] = nx(l - x). Step 2: The Bernstein polynomials Bn. Let u G C[0,1] and ||tt|| := maxo<a;<i |w(a:)|. We set Bn{x) := Zu (^\ bk(x). By (83a), \u(x) -Bn(x)\ = E f w(a;)-wf- J jbk(x] \u(x) — ^ n bk(x). Let e > 0 be given. Since u is uniformly continuous on [0,1], there is a 8 > 0 such that u(x) — u [ — 1 n <e if I k \x n < 6 and x G [0,1]. k x n Obviously, kx)-ii^|<2|H|<2|H|rx-^ /62 if Hence, by (83), \u(x) - Bn(x)\ < E ( e + 2||ti|| (x - J J /<52 J bk(x) >S. This implies that = £ + 2|M|x(l -x))/82n. \\u — Bn\\ < 2e for all n > n0(e) with suitable n0(e). □ Exaraple 4. The set Q of rational numbers is dense in R. In fact, for each real number x and each e > 0, there is a rational number r such that |x — r\ < e. Example 5. The set Q + iQ = {a + i/3: a, (3 G Q} is dense in C. This follows from Example 4 and from the inequality |a + 2/3-(7 + i6)| < |a-7| + |/3 - <5| for all a, /?,7,8 G R.

1.27. Summary of Important Notions 87 Since the set Q of rational numbers is countable, so is the set Q + iQ. Consequently, R and C are separable. Proposition 6. Each finite-dimensional normed space over K is separable. Proof. Let X = {0}. Then the statement is trivial Now let dim X = N with N > 1. Choose a fixed basis {ei,..., e^} of X. Then, each u G X can be represented in the form N u = 2_\ €jeji where £j G K for all j. j=i Case 1:K = R. Let M be the set of all the u in X with Cj G Q for all j. Let u G X. Then, for given e > 0, there is a w G M such that AT i=i This follows from Example 4. Thus, the set M is dense in X. Since the set Q is countable, so is the set M. Hence X is separable. Case 2: K = C. Let M be the set of all the u in X with ^ G Q + iQ for all j. Then, the countable set M is dense in X, by the same argument as in Case 1 along with Example 5. □ The following proposition is important for the construction of approximation methods such as the Ritz and the Galerkin methods. Proposition 7. Let X be a separable normed space over K. Then there exists a sequence {Xn} of finite-dimensional linear subspaces Xn of X such that oo XiCI2a3C...a and (J Xn = X. 71=1 Proof. First let X = {0}. Then, we set Xn := X for all n. Now let 1 < dim X < oo. Since X is separable, there exists an at most countable dense subset M of X. By the proof of Proposition 6, M is countable, i.e., M = {ui,u2,...}. Set Xn := span{^i,..., un} and K := U^Li Xn- Since M C K, it follows from McZand¥ = I that if = X. □

88 1. Banach Spaces and Fixed-Point Theorems 1.27 Summary of Important Notions We may distinguish between the ants, who read page n before page (n+ 1), and the grasshoppers, who skim and skip until something of interest appears and only then attempt to trace its logical ancestry For the sake of the grasshoppers, herewith is a listing of certain basics. Dan Henry (1981) Let us summarize a number of important notions that have been introduced in this chapter. These notions will be used frequently throughout this book. The most important notion is compactness (compact sets and compact operators). The symbol K stands for either M (the set of real numbers) or C (the set of complex numbers). 1.27.1 Linear Spaces By a linear space X over K we understand a set such that the linear combinations au + fiv are denned for all u,v G X and a, (3 G K. In addition, we postulate that the usual rules for classical vectors in the three-dimensional space of our intuition remain valid. The precise definition can be found in Section 1.1. The points ui,..., um in X are called linearly independent iff, for ai,..., am G K, aiUi H h amum = 0 implies ai = • • • = am = 0. The maximal number of linearly independent points is called the dimension dim X of X. In particular, we write dim X = oo iff there is no finite maximal number of linearly independent points. Let X, 7 be linear spaces over K. A subset M of X is called a linear subspace of X iff u,v G M implies au + (3v G M for all a, /3 G K. The set M is called convex iff u,v e M implies tu + (l- t)v G M for all t G [0,1]. An operator A.MCX^Y

1.27 Summary of Important Notions 89 assigns to each point u G M precisely one point in Y, which is denoted by Au. The set M is called the domain of definition of A. We sometimes write D(A) instead of M. The set A(M) of all the image points of A is called the range of A. We also write R(A) for A(M). The operator A is called surjective (resp., injective) iff R(A) = Y (resp., Au = Av implies u = v). The operator A is called bijective iff it is both injective and surjective. The operator A: M C X —> Y is called linear iff M is a linear subspace of X and A(au + /3v) = aAu + /L4v for all u,v G M and a, /3 G K. Operators of the form A: M C X —> K are also called functionals. The functional i:MCI-^Ris called convex iff M is a convex set and A(foz + (1 + t)v) < tAu + (1 - t)Av for all u,v e M, te [0,1]. 1.27.2 Sets in Normed Spaces It is typical for normed spaces that many notions from topology can be characterized by means of sequences. A linear space X over K is called a normed space over K iff to each u £ X a real number \\u\\ > 0 is assigned such that, for all u, v G X and a G K, the following hold: (a) \\u\\ = 0 iff ^ = 0. (b) ||mx|| = |a| \\u\\. (c) ||m +v|| < \\u\\ + \\v\\. The number \\u — v\\ is called the distance between the two points u and v. A sequence (un) in the normed space X converges to the point u iff \\un — 'ix|| —> 0 as n —> oo. We briefly write ^xn —> ^ as n —> oo. The sequence (un) in X is called a Cauchy sequence iff, for each £ > 0, there is a number Uq{e) such that \\un — v>m\\ < ^ fc>r aU n,m> n0(e). A normed space X over K is called a Banach space over K iff each Cauchy sequence in X is convergent. Let M be a subset of the normed space X. Then, M is called open iff, for each point u G M, there is a number £ > 0 such that the set {v G M: \\v -u\\ < e}

90 1. Banach Spaces and Fixed-Point Theorems is contained in M. The set M is called closed iff the complement X — M is open. This is equivalent to the fact that M is sequentially closed, i.e., un —> u as n —> oo and un G M for all n implies u G M. By definition, the closure M of M is the smallest closed set that contains M, and the interior int M of M is the largest open set contained in M. The set M is called bounded iff there exists a number r > 0 such that ||tt|| < r for all u £ M. The set M is called dense in X iff for each point u G X there exists a sequence (un) in M such that un —> i£ as n —> oo. The set M is called relatively sequentially compact iff each sequence (un) in M has a convergent subsequence. If, in addition, the limits of these convergent subsequences belong to M, then M is called sequentially compact. The set M is called compact iff each family of open sets, that covers M, possesses a finite subfamily that already covers M. The set M is called relatively compact iff the closure M of M is compact. We have the following equivalences:2 M is compact iff M is sequentially compact; M is relatively compact iff M is relatively sequentially compact. By an open neighborhood U(u0) of the point u0 G X, we understand an open subset of X which contains the point Uq. 1.27.3 Operators in Normed Spaces Let A: M C X —> Y be an operator, where X and Y are normed spaces over K. Then, ^4 is called continuous at the point u G M iff, for each e > 0, there is a (5(e) > 0 such that \\v - u\\ < 6(e) and v G M imply ||Av - A?x|| < e. This is equivalent to the fact that A is sequentially continuous at the point u, i.e., for each sequence (un) in M, i£n —> i£ as n —> oo implies ^4^n —> ^4^x as n —> oo. 2This will be proved in Problem 1.12 of AMS Vol. 109. In the following chapters, we only need the concepts of relative sequential compactness and sequential compactness. However, to simplify notation, we will use "compact" and "relatively compact" instead of "sequentially compact" and "relatively sequentially compact." By the equivalences above, this convention cannot cause any misunderst andings.

Problems 91 The operator A: M C X —> Y is called continuous iff it is continuous at each point u G M. If, in addition, A maps bounded sets onto relatively compact sets, then the operator A is called compact. The linear operator A: X —> Y is continuous iff it is bounded, i.e., there exists a number d > 0 such that Pu|| < d\\u\\ for all ugX. The smallest of these numbers d is called the norm \\A\\ of the operator A, \\A\\ := sup HAtill. Nl<i Hence, \\Au\\ < \\A\\ \\u\\ for all u£ X. The space /: X —> K of all linear continuous junctionals on the normed spaces X over K becomes a Banach space X* with respect to the norm ll/H, i.e., we define the linear combination af + fig through (a/ + fig){u) := af(u) + /fy(ti) for all u G X, and we set ||/|| := sup||w||<:L |/(w)|. Here, X* is called the dual space to X. Let X be a normed space over K and let Y be a Banach space over K. Similarly to X*, the space L(X, Y) of all linear continuous operators A\X —> Y becomes a Banach space over K equipped with the operator norm ||A||. Obviously, X* = L(X,K). Let A:X->Xbea linear operator, where X is a normed space over K. The number A G K is called an eigenvalue of ^4 iff the equation An = \u, u G X has a nontrivial solution u ^ 0. Let K = C. The resolvent set p(A) of A consists precisely of all those numbers A G C for which the continuous inverse operator (A-\I)-l:X^X exists. This operator is called a resolvent of A. The complement cr(A) := C — p(A) of p(A) is called the spectrum of the operator A. Each eigenvalue of A is contained in the spectrum of A. However, the spectrum may contain additional points. Problems 1.1. Simple examples. Let X := C[a, 6], where -oo < a < 6 < oo and \\u\\ := maxa<a;<6 |^(#)|- Show that 1.1a. {wGl:/>(^ = 0} is a closed linear subspace of X. This set is not dense in X.

92 1. Banach Spaces and Fixed-Point Theorems 1.1b. {u G X:u(a)2 = u(b)} is a closed subset of X, but not a linear subspace of X. l.lc. {u G X: u(a) > 0} is an open, convex, dense subset of X. l.ld. {u G X:u(a) = 1} is a closed, convex, dense subset of X. l.le. {u G X: \\u\\ < 1} is not a compact subset of X. Hint: Construct a sequence (un) of, say, piecewise linear continuous functions such that un(x) —> 0 as n —> oo for all x G [a, 6], where this convergence is not uniform on [a, 6]. l.lf. {u G X: w(:z) = 0 on [c, d]} is not dense in X provided a < c < d < b. l.lg. {w G -X": w(a) > 0} is the closure of the set {u G X:u(a) > 0} in X. l.lh. If we set 4>{u) := |w(a)|, then 0 is no£ a norm on X. l.li. If we set l^lli := / |w(x)|cte, then || • ||i is a norm on X, but X is not a Banach space with respect to Mli- Hint: Define a discontinuous function w: [a, b] —> R, say, / N fl if a < x < c w \ 0 if c < x < b. <b Construct a sequence (un) in X such that \\un — w\\i —> 0 as n —> oo. Show that (wn) is Cauchy with respect to || • ||i. Suppose that \\un— u\\i —> 0 as n —> oo, where u £ X. Then, ||w - w||i < \\u - Un\\i + \\un - w\\i —► 0 as n —► oo. Hence w(x) = w(x) on [a, 6], contradicting the continuity of the function u. l.lj. The operators A: X —> X and B:X ^> X denned through (Aw)(a;) := w(a) and (Bu)(x) := / u(y)dy J a are linear and continuous with ||A|| = 1 and || JB|| = b — a. 1.1k. If we set /(w) := / yu{y)dy for all w G X,

Problems 93 then /Gl* with Il/H = &£)!. 1.11. Let aGl with |a|(6 — a) < 1. For each given u0 £ X, the iteration method fb un+i(x) = a \ sinun(x)dx + 1, n = 0,1,... , x £ [a, 6], ./a converges uniformly on [a, 6] to the unique solution u £ X of the integral equation fb u(x) = a smu(x)dx + 1, x £[a,b\. J a 1.1m. Let a £ R with |a| < 1. For each given u0 £ R, the iteration method un+i = a sinun + 1, n = 0,1,..., converges to the unique solution wGlof the equation u = a sinu + 1. l.ln. Let K(x, y): [a, b] x [a, 6] —> R be continuous with 0 < K(x, y) <d for all x,y £ [a, 6]. Let 2(6 — a)d < 1 along with u(x) = 0 and ^o(#) = 2. Then, the two iteration methods Un+i(x)= K(x,y)un(y)dy + l, n = 0,l,..., o?G[o,6], ./a vn+i(x)= K(x,y)vn(y)dy + 1 J a converge uniformly on [a, b] to the unique solution u £ X of the integral equation u(x)= K(x,y)u(y)dy + 1, x£[a,b], J a where i£o(#) <v>i(x) < --- < v\{x) < vq(x) for all x £ [a,6]. Hint: Use Theorem l.E on sub- and supersolutions and Example 2 in Section 1.7. l.lo. Let a £ R and / £ X be given. Then, the nonlinear integral equation u(x) = a sinu(x)dx + f(x) J a has a solution u £ X. Hint: Use the Leray-Schauder principle from Section 1.18. l.lp. Let /:R2 —> R be continuous. Then, the system £ = 1027 + sin/(£,r?), rj = cos f&rj)

94 1. Banach Spaces and Fixed-Point Theorems has a solution (£, rj) E M2. Hint: Use the Leray-Schauder principle from Section 1.18. 1.2. Balls. Set B := {u E X: \\u\\ < r} and B0 := {u E X: \\u\\ < r} for fixed r > 0, where X is a normed space over K. Show that B = ~B=T30, ^£ = £0, 9B = {u E X: ||tx|| = r}. 1.3. 7%e spectrum. Let cr(^4) denote the spectrum of the linear operator A: X -» X. Show that 1.3a. <x(A) = {2} provided X := C and Au = 2u. 1.3b. If X = CN, iV > 1, then the spectrum cr(^4) of the matrix operator A X —> X given through AT Vj -^2^jk^ki j = 1,...,JV, fc=l consists precisely of a// £/ie eigenvalues A E C of the matrix (ajfc), i.e., A is a solution of the characteristic equation det(ajfc - X6jk) = 0, where det(-) denotes the determinant of the corresponding (N x iV)-matrix. 1.3c. a(A) = {2^,-2^} and ||A|| = 2 provided -A(f, V) :=(£ + */,£- V) for all (£, 77) E C2, where X := C2 and ||(f,fj)|| := max{|£|, |ij|} on X. 1.3d. If X is a complex, finite-dimensional normed space, then the spectrum cr(A) consists precisely of all the eigenvalues of the operator A. Hint: Use Proposition 3 in Section 1.20. 1.4. The spectral radius. Let A: X —> X be a linear continuous operator on the complex Banach space X. Define the spectral radius r(A) of A through r(A) := sup |A|. \ea(A) Show that 1.4a. r(A) < \\A\\, by Proposition 2 in Section 1.25. 1.4b.*r(i4) = Umn-.ooPn||i. Hint: Cf. Yosida (1980), Chapter 8, Section 2.

Problems 95 1.4c. Volterra integral operator. Let X := C[a, 6]c, where —oo < a < b < oo (cf. Problem 1.6e). Define the operator A:X —> X through (Au)(x) := / K{x,y)u{y)dy for all x G [a, 6], ./a where K: [a, 6] x [a, 6] —> C is continuous. Then, r(A) = 0, and hence a(A) = {0}. Hint: Use Problem 1.4b. Cf. Zeidler (1986), Vol. 1, p. 38. 1.4d. Fredholm integral operator. Let X :— C[a, &]c- Define AX —> X through (Au)(x) := K(x1y)u(y)dy for all x G [a, 6], «/a where iiT is given as in Problem 1.4c. Show that r(A)<(b-a) max \K(x,y)\. x,ye[a,b] Hint: Use Problem 1.4a. 1.5. The Banach space l^. Let K°° denote the space of all sequences (un)n>i, where un G K for all n G N. Moreover, let /^ denote the set of all (tin) G K°° such that IIOOHoo :^SUp|^n| < OO. n>l Define a(un) + P(vn) = (a^n + /3vn) for all a, /? G K. Show that 1.5a. K°° is an infinite-dimensional linear space over K. 1.5b. l^ is an infinite-dimensional Banach space over K with respect to the norm || • ||oo. 1.6. Classical function spaces on [a, 6]. Let —oo < a < b < oo. Show that the following function spaces are Banach spaces. Observe that Holder continuous functions play a fundamental role in the theory of linear and nonlinear elliptic and parabolic partial differential equations3 as well as in classical potential theory.4 3Cf. Zeidler (1986), Vol. 1, Chapter 6, and Gilbarg and Trudinger (1977). 4Cf. Kellogg, Foundations of Potential Theory, Springer-Verlag, 1929.

96 1. Banach Spaces and Fixed-Point Theorems 1.6a. Let B[a, b] denote the set of all bounded functions u: [a, b] —> R and set \\u\\ := sup |ti(x)|. a<x<b 1.6b. For 0 < a < 1, let C°'a[a,6] denote the set of all the so-called Holder continuous functions u: [a, b] —> R, i.e., by definition, l^x(x) - w(y)| < const|x - y\a for all x,y G [a, 6]. (85) Let |u(a:)-u(2/)| #a W := sup L-1r1 p-^-, \x-y\a where the supremum is taken over all x, y G [a, b] with x ^ y, i.e., the so- called Holder constant Ha(u) of ^ is the smallest constant such that (85) holds. In particular, \u(x) — u(y)\ < Ha(u)\x — y\a for all x,y G [a, 6]. Set ||tt|| := max |w(a:)| + Ha(u). a<x<b 1.6c. Let Ck[a,b] with fc = 1,2,... denote the set of all continuous functions u: [a, b] —> R that have continuous derivatives on [a, 6] ttp £o order fc. Set M •= /^ max krJM#) , " " ^a<x<b] V '" j=o - - where u^ denotes the jth derivative. 1.6d. For 0 < a < 1 and k = 1,2,..., let Ck^[a, b] denote the set of all functions u G Ck[a,b] with ^fc) G C°'a[a,6]. Set k \\u\\ : = V max \uW(x)\ + Ha(uw). (a<x<b 1.6e. Let C[a, 6]c denote the set of all complex continuous functions u: [a, b] —> C. Define ||tt|| := max |m(#)|. a<x<b 1.7. Compact embedding. Use the Arzela-Ascoli theorem from Section 1.11.1 in order to prove that the embedding C°'a[a,b]CC[a,b], 0 < a < 1,

Problems 97 is compact, i.e., each bounded set in C°>a[a, b] is relatively compact in C[a,b]. Show also that the embedding C°'a[a,6] C C°^[a,6], 0 < 0 < a < 1, is compact, again by the Arzela-Ascoli theorem. Hint: Cf. Zeidler (1986), Vol. 2A, p. 283. 1.8. Classical function spaces on subsets ofSLN. Let us directly generalize the function spaces from Problem 1.6. Let G be a nonempty bounded open set in RN, N > 1. Show that the following function spaces are Banach spaces. -- 1.8a. Let B(M) denote the set of all bounded functions u: M —> R. Set \\u\\ := sup |w(#)|, x&M where M is an arbitrary nonempty subset of RN. 1.8b. Let C(M) denote the set of all continuous functions u: M —> R. Set ||tt|| := max|^(x)|, 11 " xgm ' v n where M is a nonempty compact subset of R^ (e.g., M = G). 1.8c. For 0 < a < 1, let C0,a(G) denote the set of all Holder continuous functions u: G —> R, i.e., I^x(x) — u(y)\ < const|x — y\a for all x,y £ G. Set ||tt|| := max \u(x)\ + Ha(u), where Ha(u) := sup , _ ' • x,yeG, x^y F 2/1 1.8d. For fc = 1,2,..., let Ck(G) denote the set of all functions u G C(G) which have continuous partial derivatives on G up to order k. Moreover, suppose that all these partial derivatives can be extended continuously to the closure G. Set IMI :== 7^ max|<9^i£(x)|, i«<*xeQ where we sum over the function and all their partial derivatives up to order k.

98 1. Banach Spaces and Fixed-Point Theorems 1.8e. For 0 < aj^ 1 and k = 1,2,..., let Ck^(G) denote the set of all functions u G Ck(G) such that all the partial derivatives of u of order k belong to C°'a(G). We set \\u\\ := J2 max|^ii(x)|+ ^ Ha(d^u). \P\<k X&G |/?|=fc 1.9. Complexification. Let X be a rea/ linear space. Define Xc := {(^,v):ti,i; G X}. Show that 1.9a. Xc forms a complex linear space with respect to the following operations: (u, v) + (w, z) = (u + w,v + 2;), (86) (a + z/?)(w, v) = (cm - /?v, av + /3u), a, /? G R. (87) Instead of (u,v) we also write w + iv. Then, (87) corresponds to the following formal multiplication rule: (a + ip)(u + iv) = au — /3v + i(av + /3u). 1.9b. If X is a real normed space, then Xc becomes a complex normed space equipped with the following norm: ||w + u?||:= max || (cos </>)w + (sin </>)?; ||. 0<(/><27T 1.9c. If X is a real Banach space, then Xc is a complex Banach space. 1.9d. Every linear continuous operator A: X —> X can be extended to a linear continuous operator Ac'- Xc —> Xc by the quite natural definition Ac(u + iv) := Aw + iAv. Then, P||< Pc|| <2||A||. 1.9e. Show that if X := R, then Xc = C along with \\u + iv\\ -= |w + u?| for all complex numbers w + u> with u,vGl. 1.10. Density. Let D bea dense subset of the normed space X over K. Show that (ti*, ti) = 0 for all u G D and fixed w* G X* implies w* = 0.

Problems 99 Solution: Recall that u*(u) = (u*,u). Let v £ X be given. Since D is dense in X, there exists a sequence (un) in D such that un —> v as n —> oo. The functional i£* is continuous and ti*(tin) = 0 for all n. Hence u*(v) = lim ^*(^n) = 0 for all v £ X. n—>oo Therefore, i£* = 0. 1.11. TAe importance of equivalent norms. Let || • || and || • ||i be two equivalent norms on the linear space X over K, and let F be a normed space over K. 1.11a. Show that the following notions are invariant under a passage from the norm || • || to || • ||i: convergent sequence, open set, closed set, bounded set, closure of a set, interior of a set, sequentially compact set, relatively sequentially compact set, dense set in X, continuous (resp.) compact operator A:X —> Y", and Cauchy sequence. In particular, X is a Banach space with respect to || • || iff X is a Banach space with respect to || • ||i. 1.11b. Use Banach's continuous inverse theorem from Section 3.5 of AMS Vol. 109 in order to prove the following. Suppose that there is a constant d > 0 such that \\u\\ < d\\u\\i forall^eX, (88) and suppose that X is a Banach space with respect to both the norms || • || and || • ||i. Then, || • || is equivalent to || • ||i on X. Solution: Let X and Xi denote the linear space X equipped with the norm || • || and || • ||i, respectively. Define the operator A: Xi —> X through Au := u for all u £ Xi. Obviously, A is bijective. By (88), A is continuous. Banach's continuous inverse theorem tells us that the inverse operator A~x: X —> Xi is continuous, too. Hence there is a constant c > 0 such that \\Au\l! <c\\u\\ forallueXL. (89) By (88) and (89), cT^Hi < ||u|| < d|M|i for all u£X.

Hilbert Spaces, Orthogonality, and the Dirichlet Principle When the answers to a mathematical problem cannot be found, then the reason is frequently the fact that we have not recognized the general idea, from which the given problem appears only as a single link in a chain of related problems. David Hilbert, 1900 (Paris lecture) In a famous paper from 1857, Riemann used the Dirichlet principle for the foundation of the theory of complex analytic functions. In 1870 Weier- strass showed that there are variational problems that do not have any solution.1 This way the justification of the Dirichlet principle became an important open problem, which Hilbert solved in 1900. In his Paris lecture, Hilbert formulated 23 open problems. In connection with the twentieth problem, he said The sophisticated methods of Schwarz, C. Neumann, and Poincare essentially solved the boundary-value problems for the Laplace equation. However, these methods cannot be directly extended to more general cases... I am convinced that it will be possible to get these existence proofs by a general basic idea, towards the Dirichlet principle points. Perhaps it will then also be possible to answer the question xThis classical counterexample will be considered in Problem 2.1. A detailed historical discussion of the Dirichlet principle and its influence on modern analysis can be found in Zeidler (1986), Vol. 2A, Sections 18.7 through 18.9.

102 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle of whether or not. every regular variational problem possesses a solution if, with regard to boundary conditions, certain assumptions are fulfilled and if, when necessary, one sensibly generalizes the concept of solution. In this chapter we want to show that the Dirichlet principle can be justified extremely elegantly by using elementary geometric arguments from the theory of Hilbert spaces and by introducing Sobolev spaces based on generalized derivatives. We will also study the convergence of the method of finite elements, which represents one of the most important methods in modern numerical analysis. A physical interpretation in terms of elasticity can be found in Section 2.7. In Hilbert spaces, an inner product (u \ v) is defined, allowing us to introduce the fundamental notion of orthogonality. Figure 2.1 shows the relationship between Hilbert spaces and Banach spaces, and others, that is, each Hilbert space is a Banach space, and so forth. With a view to applications, the most important Hilbert spaces are the Lebesgue spaces L2(G), L%(G) and the related Sobolev spaces Wj (G) o and W\(G). Roughly speaking, the real Lebesgue space L,2(G) (resp., the complex Lebesgue space L%{G)) consists of all functions u:GC J>N (resp., u: G —> C) with / \u{x)\2dx < oo. JG jAT (rN Lebesgue spaces (L2(G), if (O) Sobolev spaces (W21(G),iy21(G» ■► Banach space (the Cauchy criterion is valid) pre-Hilbert space » normed space » linear space (inner product (it | v)) (norm ||ti|| = (u | u) 2) (au + f3v) FIGURE 2.1.

2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle 103 real Lebesgue space Z/2(G) Fourier series and integral equations (Sections 3.2 and 4.4) partial differential equations of mathematical physics (Chapter 5) i i Sobolev spaces W2 (G) and W2 (G) ► Dirichlet principle and calculus of variations (Section 2.4) complex Lebesgue space L% (RN) ► quantum mechanics (Section 5.13) Fourier transformation (Section 3.7) FIGURE 2.2. The corresponding inner product2 (u | v) := / u{x)v{x)dx JG generalizes the classic Euclidean inner product N 3 = 1 on RN and C^, where u = (t*i,..., un) and v — (vi,..., vn). Observe that the integral JG ... is to be understood in the sense of Lebesgue. The theory of Hilbert spaces forces the use of the Lebesgue integral. The deeper reason for this is the fact that in the case of the classical Riemann integral the limiting relation lim / un{x)dx — / u{x)dx n-*°° JG JG 2The bar denotes the conjugate complex number. In the case of the real space Z/2(G), u(x) is real, and hence u\ v) = / u(x)v(x)dx JG

104 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle is only valid under very restrictive assumptions, in contrast to the Lebesgue integral. The Riemann integral leads only to pre-Hilbert spaces for which the fundamental Cauchy criterion is not valid. For the convenience of the reader, basic facts about the Lebesgue integral are summarized in the appendix. Figure 2.2 portrays some applications of Lebesgue spaces and Sobolev spaces to important concrete problems. Moreover, Figure 2.3 displays the logical structure of this chapter. generalized functions (distributions) main theorem on quadratic variational problems existence of a perpendicular orthogonal decomposition generalized derivative Sobolev spaces justification of the Dirichlet principle Riesz theorem A convergence of the Ritz approximation method (finite elements) -► nonlinear Lax-Milgram theorem orthogonality principle nonlinear Lipschitz continuous, strongly monotone operators Banach fixed-point theorem FIGURE 2.3.

2.1 Hilbert Spaces 105 2.1 Hilbert Spaces Recall that K = R or K = C. Definition 1. Let X be a linear space over K. An inner product on X assigns to each pair (u, v) with w,vGla number (u\v)eK such that the following hold for all u, v, w G X and a, (3 G K: (i) (u | u) > 0 and (w | it) = 0 iff u = 0; (ii) (u\av + /3w) = a(u \ v) + (3(u \ w)\ (iii) (u\v) = (y\u). Here, the bar denotes the conjugate complex number. A pre-Hilbert space over K is a linear space X over K together with an inner product. It follows from (ii) and (iii) that (av + (3w | u) = a(v | u) + J3(w | u) for all u,v,w e X, a,/? G K. (1) Let tt, ^ G X. Then, it is called orthogonal to ^ iff (u\v) = 0. (2) The following Schwarz inequality (3) is the most important inequality in pre-Hilbert and Hilbert spaces. In Section 5.14, we shall show that the famous Heisenberg uncertainty relation in quantum mechanics follows from the Schwarz inequality. Proposition 2. Let X be a pre-Hilbert space. Then \(u | v)\ < (u I u)*(v I v)i for all u,v G X. (3) Using the norm || • || introduced in Proposition 3 ahead, we can write the Schwarz inequality in the following form: \(u | v)\ < \\u\\ \\v\\ for all u,v G X. (3*) Proof. Let v =^ 0. Then, we get (3) from 0 < (u — av | u — av) = (u | u) — a{u \ v) — a[(v \ u) — a(v \ v)]

106 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle witha:=gg. D Proposition 3. Each pre-Hilbert X space over K is also a normed space over K with respect to the norm \\u\\ := (u | u)* for all u G X. (4) Proof. We have \\u\\ > 0 for all u G X, and \\u\\ = 0 iff it = 0. Furthermore, ||cm|| = (au | cm)2 = (aa)^(u \ u) = \a\ \\u\\ for all u G X, a G K. Finally, the triangle inequality \\v> + w|| < ||tt|| + H^ll for all u,v E X, follows from the Schwarz inequality (3*). In fact, for all u, v G X, ||w + v\\2 = (u + v \u + v) = (u\u) + (u\v) + (u\v) + (v \v) = ||it||2 + 2Re(it | v) + |H|2 < ||tt||2 + 2||u|| ||t;|| + |M|2 = (\\u\\ + |H|)2, where Re z denotes the real part of the complex number z. □ From Proposition 3 we obtain the following: All the notions and theorems for normed spaces3 remain valid for pre- Hilbert spaces with respect to the norm \\u\\ from (4). In particular, the convergence un —> u as n —> oo in the pre-Hilbert space X is to be understood in the following sense: IWn — u\\ -+ 0 as n —> oo. Proposition 4. Let X be a pre-Hilbert space. Then, the following hold true: (i) The inner product is continuous, that is, un —> u and vn —> v as n —> oo imply (un | vn) —> (u \ v) as n —> oo. 3The basic notions concerning normed spaces and Banach spaces are summarized in Section 1.27.

2.1 Hilbert Spaces 107 (ii) Let M be a dense subset of X. If (u | v) = 0 for fixed u £ X and all v £ M, then u = 0. Proof. Ad (i). Since (vn) is bounded, it follows from the Schwarz inequality (3*) that \{un | Vn) ~ {U | V)\ = \{un - u\vn) + (u\vn - v)\ < \{un - u I vn)\ + \{u I vn - v)\ < \\un — u\\ \\vn\\ + \\u\\ \\vn — v\\ —> 0 as n —> oo. Ad (ii). Since M is dense in X, there is a sequence (vn) in M such that vn —> tt in X as n —> oo. Letting n —> oo, it follows from ^ (u\vn) = 0 for all n 5j^" » that (u\u) = 0. Hence tt = 0. □ Example 5 (The product rule). Let X be a pre-Hilbert space, and let u,v: U(s) C R —> X be two functions defined on an open neighborhood of s e R that are differentiable at the point s. Then the function t h-> (it(£) | v(£)) is differentiable at s, where ^(«(t) | v(t))s = (u'(s) I t;(s)> (u(*) | «'(*)). Proof. Set (j)(t) := (u(t) \ v(i)). Letting h —> 0, the assertion follows from 0(s + h) - <j)(s) fu(s + h) - u(s) \ / v(a + /i) - v(g)' -' \v{s + h)) + [u{s) | D The following definition is basic. Definition 6. By a Hilbert space we mean a pre-Hilbert space that is a Banach space with respect to the norm \\u\\ from (4). In other words, a linear space X over K is a Hilbert space iff the following hold: (i) there exists an inner product in X, and (ii) each Cauchy sequence with respect to the norm \\u\\ from (4) is convergent.

108 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle If K = R or K = C, then X is called a real or complex Hilbert space, respectively. Example 7. Set X := R. Then, X is a real Hilbert space with the inner product (u | v) := uv for all u, v G R. The corresponding norm \\u\\ = (u \ u)* equals \u\. Example 8. Set X := C. Then, X is a complex Hilbert space with the inner product (u | v) := uv for all u, v G C, and the norm \\u\\ = (u \ u)* = \u\. Proposition 9. Each finite-dimensional pre-Hilbert space is a Hilbert space. This follows immediately from the fact that each finite-dimensional normed space is a Banach space. Proposition 10. Let X be a Hilbert space (resp., Banach space) over K, and let L be a linear subspace of X. Then, the closure L of L is also a Hilbert space (resp., Banach space) with respect to the restriction of the inner product (resp., norm) on X to L. Proof. We first prove that L is a linear space over K. In fact, let u,v G L and a, /? G K. Then, there are sequences (un) and (vn) in L such that un —> u and vn —> v in X as n —> oo. Letting n —> oo, it follows from aun + (3vn G L for all n that au + (3v G L. Restrict the inner product (resp., norm) on X to the subset L of X. Then, L is a pre-Hilbert space (resp., normed space). Finally, let (un) be a Cauchy sequence in L. Then, un —> u in X as n —> oo. Since L is closed, u G Z. Hence un -* u in Z as n —> oo. □

2.2 Standard Examples 109 2.2 Standard Examples The reader who wants to pass to important applications of the Hilbert space theory as quickly as possible should only read the examples and propositions of this section without studying the corresponding proofs, which are based on important properties of the Lebesgue integral. Standard Example 1. The space X := KN, N = 1,2,..., is an N- dimensional Hilbert space over K with the inner product N (x | y) := ]jP ZjTfj for all x, y G K, i=i where x = (£i, • •.,£jv) and ?/ = (771,..., tin). The corresponding norm is given through llxll = (x I a)* = [ ]£ |^|2 J for all x G K. This is a real or complex Hilbert space if K = R or K = C, respectively. For X = RN, the norm \\x\\ is identical to the Euclidean norm \x\. Proof. One checks easily that (• | •) represents an inner product. Thus, X is a finite-dimensional pre-Hilbert space, and hence it is a Hilbert space. □ The corresponding space in the case where N = 00 will be studied in Problem 2.2. Example 2. Let —00 < a < b < 00. For all u, v G C[a, &], we define fb (u I v) := / uvdx. ' (5) J a One checks easily that this is an inner product on C[a, b]. The corresponding pre-Hilbert space is denoted by C*[a,b\. Obviously, the norm on C*[a, b], namely, \\u\\ = (u I u)i = I / \u\2dx J for all u G C[a,b] differs from the maximum norm maxa<x<b \u(x)\ introduced in Chapter 1. We shall show in Example 9 that C* [a, b] is not a Hilbert space, but only a dense subset of the Hilbert space L,2(a,b) whose definition is based on the Lebesgue integral. Convention 3. In the following, all integrals are to be understood in the sense of Lebesgue.

110 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle 2.2.1 The Hilbert Space L2 (a, b) Standard Example 4. Suppose that —oo < a < b < oo. Let L2(a,b) denote the set of all measurable functions u:]a,b[ ->R such that fb / |tt|2dx < oo. J a Then (i) L2(a,b) is a real Hilbert space with respect to the following inner product: fb (u\v):= / uvdx for all u, v € L2(a, b). J a (ii) dim L2(a, b) = oo. More presicely, we use the following identification principle: (I) Two functions u and v correspond to the same element in the Hilbert space L2(a,b) iff u{x) = v(x) for almost all x G ]a, b[. Thus, the elements of L2(a,b) are classes of functions characterized by (i). Proof. In the following we will essentially make use of the Fatou lemma for the Lebesgue integral (cf. the appendix). Ad (i). Step 1: The classic Schwarz inequality. We want to show that if u,v € L2(a,&), then the integral J uvdx exists and / J a b uvdx <([ \u\2dx) I f \v\2dx) . (6) To prove this, we start with the simple, classic inequality N<2-1(|^|2 + H2) foraU^eC, (7) which follows from 0 < (|£| - M)2 = |£|2 - 2|£| \v\ + \r)\2. Set fb \ 2 ix) f /»0 / \u\2da i J a

2.2 Standard Examples 111 First let ||u|| = 0 or ||t;|| = 0. Then u(x) = 0 or v(x) = 0 for almost all x G ]a, &[, respectively. Hence J uvdx — 0, i.e., (6) is true. Suppose now that \\u\\ =^ 0 and \\v\\ =^ 0. Replacing u with -rAr and t> with tAt, if necessary, we may assume that ||tt|| = 1 and ||^|| = 1. By (6), \u{x)v{x)\ < 2~l{\u{x)\2 + |v(a;)|2) for all x G ]a, b[. (8) Since the functions u and t> are measurable on ]a, &[, so is the product uv. By (8), the existence of the integrals "2dx rO rO I \u\2dx and / \v\ J a J a b , implies the existence of the integral / |m;|d#, and hence the existence of rb Ja uv dx. Furthermore, it follows from (8) that ft* \ rb I rb rb \ uvdxl < / \uv\dx<2~1 I / \u\2dx + / |v|: Ja Ja \Ja Ja \2dx Ja J This is the desired inequality (6). Step 2: We show that L2(a, b) is a linear space. In fact, for all a, (3 G R and all x G ]a, b[, \au(x) + pv(x)\2 < \a\2\u(x)\2 + |2a/?| |u(a?)v(a:)| + |/3|2|^(x)|2. Let tt,^ G L2(a, &). Then, the integrals Ja |tt|2c?x and Ja |^|2dx exist. By Step 1, this implies the existence of J \uv\dx. Hence the integral rb / \au + (3v\2dx J a Step 3: We prove that (u \ v) := Ja uvdx is an inner product on L2(a, b). exists, i.e., au + (3v G L2(a, b). Step 3: We prove We first show that (u | u) = 0 iff w = 0. In fact, let (it | u) = 0. Then ,6 / |tt|2c?x = 0 implies u(x) = 0 for almost all x G ]a, &[. J a

112 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle By the identification principle (I) given earlier, we obtain that the function u = u{x) corresponds to the zero element u = 0 in L2(a, b). Conversely, let u = 0 be the zero element in L2(a, b). By (I), this element corresponds to the class of all the functions u with u{x) = 0 for almost all x G ]a, b[. Hence (u | u) = 0. Furthermore, it follows from u(x) = u\{x) and v(x) = v\(x) for almost all x G ]a, b[ that u(x)v(x) = ui(x)vi(x) for almost all x G ]a, &[, and pb pb I uvdx = uividx, i.e., (u\v) = {u\ \ v\). J a J a Consequently, the inner product respects the identification principle (I), i.e., it depends only on the corresponding class of functions. Finally, if u,v,w G Z^^ &), then (u | v) = (v \ u) and pb pb pb I u(av + (3w)dx = a uvdx + (3 I uwdx, Ja Ja Ja i.e., (u | av + (3w) = a(u \ v) + /3(u \ w). Consequently, £,2(0,, b) is a pre-Hilbert space. Obviously, the Schwarz inequality \(u I v)\ < \\u\\ \\v\\ for all u,v G L2(a,b) corresponds to the classic Schwarz inequality (6) from Step 1. Step 4- Hilbert space. We want to show that L2(a, b) is a Hilbert space. To this end, we have to show that each Cauchy sequence in L2(a,b) is convergent. By Proposition 7 in Section 1.3, it is sufficient to prove that each Cauchy sequence in L2(a,b) has a convergent subsequence. Let (un) be a Cauchy sequence in L2(a,b), i.e., \Wn — Until < £ f°r aU n,m> no(e). Choosing e = 2_fc, k = 1,2,..., there follows the existence of natural numbers ri\ < n2 < ... such that ll^njb+i ~ unk || < 2~fc for all k = 1,2,... . We set Vk := unk and 771 sm(x) := ^ \vk+i(x) - Vk(x)\. k=l Since the sequence (sm(x)) is monotone increasing, the limit S(x) := lim sm(x)2

2.2 Standard Examples 113 exists for all x G ]a, b[, where 0 < S(x) < oo. Since L2(a, b) is a pre-Hilbert space, the triangle inequality holds. Hence 771 11^7711 < ]T \\vk+i ~ vk\\ < 2"1 + 2"2 + • • • < 1 for all m > 1. This implies / sm(x)2dx < 1 for all m > 1. J a Thus, by the Fatou lemma, the function S is integrable over ]a, &[ with pb pn pb I S(x)dx = / lim sm(x)dx < lim / sm(x)dx < 1. J a J a m^°° rn^ooJa In particular, 5(x) is finite for almost all x G ]a, b[. In the remaining points let us redefine 5 by setting S(x) := 0. Letting s(x) := 5(x)2, we get s(x) = lim sm(x) for almost all x G ]a, &[, (9) m--*oo ^ 1, i.e., s G L We now use the identity and Ja s2dx < 1, i.e., s G 1^2(^5 &)• 7TI—1 vm(x) = ^i(x) + ^ ^+i(z) - ^fc(^)- (10) By (9), 00 s(x) = 2_J \vk+i(x) — ^fc(#)| < 00 for almost all x G ]a, &[. (11) Thus, the finite limit v(x) := lim vm(x) exists for almost all x G ]a, b[. In the remaining points x of the interval ]a, b[ we set ^(x) := 0. As the limit of measurable functions vm, the function v is also measurable on the interval ]a, b[. According to (10) and (11), \v(x)\ < \vi(x)\ + s(x) for almost all x G ]a, b[. (12) Since ^1 G L2(a, &), we get |^i|, s G 1,2(0*, b), and hence |^i| + s G L,2(a,b), by Step 2. It follows from (12) that /»6 pb / MaO|2dx< / (Mx)|+s)2cte J a J a < OO,

114 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle i.e., v G L2(a,b). Finally, we want to show that vn —> v in L2(a, b) as n —> oo. (13) In fact, since (vn) is a Cauchy sequence, for each e > 0 there is an mo(e) such that II Il2 fb / |vn — ^m|2dx < e2 for all n, ra > mo(e). ♦/a Letting ra —> oo, it follows from the Fatou lemma that /•6 \\vn - v\\2 = / |un(aO ~ v(a;)|2da; ■/ b lim |^n(x)-^m(x)|2dx 771—►OO ,6 < lim / \vn{x) — vm(x)\2dx < e2, rn^°° Ja for all n > m0(e). This is (13). Ad (ii). Choose a fixed compact interval [c,d\ with [c,d] C ]a, 6[ and c < d. Define un(x):= {xU [ixe \CA 10 otherwise. Then un G Z>2(a> &) for all n = 0,1,2, It follows as in the proof of Example 9 in Section 1.1 that the functions uo,... ,un are linearly independent for each n. Hence dim L2(a,b) = 00. □ 2.2.2 The Lebesgue Spaces L2(G) and L$(G) Proposition 5 (The space Lf(G)). Let G be a nonempty measurable subset ofRN, N > 1 (e.g., G is open or closed), and let Lf(G) denote the set of all measurable functions u:G-*K such that \u\2dx < 00. (14) L IG Then (i) Lf(G) is a Hilbert space with respect to the following inner product: (u \v):= / uvdx for all u, v G Lf(G). (15) Jg More precisely, two functions u and v correspond to the same element of the Hilbert space Lf(G) iff u(x) = v(x) for almost all x G G.

2.2 Standard Examples 115 For K = R or K = C, Lf(G) is a real or complex Hilbert space, respectively. (ii) If G is open, then dim Lf(G) = oo. In particular, the Schwarz inequality (3) applied to the Hilbert space Lf(G) reads as follows: / uvdxl < / \uv\dx < f / H2cfo] f / |^|2dxj , (16) for all u, v G Lf (G). For brevity of notation, we set L2(G) := I*(G) if K = R. Note also that L2(a, b) = L2(G) with G = ]a, 6[. Proof. Ad (i). Use the same argument as in the proof of Standard Example 4, by replacing the interval ]a, b[ with the set G. Ad (ii). Since G is open, there is a cuboid C := {(£i,... ,&v) G RN: a <£j <b for all j} with C C M, where —oo < a < b < oo. Define ^(x):=\0 ifR"-C, (17) where x = (^i,..., £N). Then, un G L^(G) for all n = 0,1,2,.... It follows as in the proof of Example 9 in Section 1.1 that the functions tto, • •., un are linearly independent for all n. Hence dim Lf (G) = oo. □ 2.2.3 The Space Cq°(G) and Density in the Hilbert Space L2(G) The space Cq?(G) plays a fundamental role in modern analysis. Definition 6. Let G be a nonempty open set in RN, N > 1. Then (a) Ck(G) is the set of all real functions u: G -> R that have continuous partial derivatives of orders4 m = 0,1,..., k. 4As usual, we understand the derivative of order m — 0 to be the function itself.

116 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle JG (a) ueCg°(R) (b) FIGURE 2.4. (b) Ck(G) is the set of all u G Ck(G) for which all partial derivatives of order m = 0,..., k can be extended continuously to the closure G of G. (c) If u G Ck(G) (resp., u G Ck(G)) for all k = 0,1,2,..., then we write u G C°°(G) (resp., u G C°°(G)). (d) Cq°(G) is the set of all functions u G COQ{G) that vanish outside a compact subset C of G that depends on G, i.e., tt(x) = 0 for all x €G -C (see Figure 2.4). £*vApncf r a v , '* r Instead of C°(G) (resp., G°(G)) we write briefly C{G) (resp., (7(G)). That is, C(G) consists of all continuous functions u: G —> R, and G(G) consists of all continuous functions u: G —> R. The set Cq°(G)c consists of all functions tt:G —> C for which both the real and the imaginary parts of u belong to Gq°(G). Similarly, we define Gfc(G)c, and so on. If u G Ck(G), then we say "u is Ck on G." In the one-dimensional special case where G = ]a, ft[, we write briefly Ck{a, ft) := Gfc(G) and Gfc[a, ft] := Ck{G), and so forth. Proposition 7. Le£ G be a nonempty open set in R^, AT > 1. T/ien, £/ie following hold true: (i) T/ie se£ G£°(G) zs dense in L2(G). (ii) T/ie se£ C(G) is dense in L2(G). (iii) The sets C^(G)C and C(G)C are dense in L$(G). Corollary 8. The spaces L2(G) and L%{G) are separable.

2.2 Standard Examples 117 The proofs of Proposition 7 and Corollary 8 will be given in Problems 2.12ff by using an important smoothing technique. Example 9. The pre-Hilbert space C*[a,b] is not a Hilbert space. Proof. Let L := C*[a,b] and X := L2(a,b). By Proposition 7, the linear subspace L is dense in X, i.e., L = X. If L were be a Hilbert space, then L would be closed. Hence L = L = X. But this is impossible, since there are functions with u G X and u $. L. For example, this is true for / n f 1 if a < x < c for fixed c G ]a, b[ /-^ u\x) ,_ I o if c < x < &. ^ ' □ 2.2.4 The Space Cq°(G) and £fte Variational Lemma Variational Lemma 10. Let G be a nonempty open set in RN, N > 1. Then, it follows from u G L,2(G) and f uv dx = 0 for all v G C^{G) (19) £/m£ u(x) = 0 /or almost all x G G. //, m addition, u G C{G), then u{x) = 0 /or all x € G. This lemma plays a fundamental role in the calculus of variations. We shall use it in Section 2.5. Proof. Let X := L2(G). By (19), (u\v)=0 foralUGC0°°(G). Since the set Cq°(G) is dense in X, it follows as in the proof of Proposition 4(ii) in Section 2.1 that (u\u)= f \u\2dx = 0. (20) Jg Hence u{x) = 0 for almost all x G G. If u is continuous on G, then (20) implies u(x) = 0 for all x G G. □ #.#.5 T/ie £pace Cq°(G) and Integration by Parts The classic integration-by-parts formula reads as follows: / u'vdx = uv\a — / uv'dx, (21)

118 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle with the "boundary integral" uv\ = u(b)v(b) — u(a)v(a). In particular, if v(a) = v(b) = 0, then nb nb I u'vdx = - / uv'dx. (21*) J a J a Proposition 11. Let — oo < a < b < oo. Then, the following are met: (i) The integration-by-parts formula (21) holds for all u,v G C1^]. (ii) Formula (21*) Zio/ds /or a// uGC^ajft) and veC%°(a,b). Here, we set C^a.b] := C^G) and C1^) := C1^), where G = ]a,fc[, and so on. Proof. Ad (i). By the fundamental theorem of calculus, I (u'v + uv')dx = / (uv)'dx = w|fl. Ad (ii). Since the function ^ vanishes in a neighborhood of the two boundary points x = a and x = b, we can choose a subinterval [c, d] of ]a, b[ such that t> G Cq°(c, d). Furthermore, u G Cx(a, b) implies u G Cx[c, d]. Hence / (u'v + uv')dx = / (uv)'dx = uv\ = 0, ./a 7c since v(c) = v(d) =0. D The generalization of the integration-by-parts formula (21) to higher dimensions reads as follows: / (dju)vdx= / uvrijdO — I udjvdx, j Jg JdG Jg 1,...,N, (22) where x = (£i,...,6v) and djU := du/d^j. In addition, the outer unit normal vector to the boundary <9G is denoted by n = (ni,..., njq). In the special two-dimensional case (N = 2), the surface integral J^ ... dO is to be understood in the sense of JdG ... ds, where s denotes arclength, and the boundary curve dG is oriented in such a way that the set G lies on the left-hand side of dG (see Figure 2.5(a)).

2.2 Standard Examples 119 H (a) (b) Q FIGURE 2.5. In the special case where v = 0 on <9G, formula (22) passes over to / (djUjv dx = — I udjV dx, j = 1,..., N. Jg Jg (22*) Proposition 12 (Integration by parts). For N = 1,2, ..., the following hold true: (i) Formula (22) holds for all u,veC\G), provided G is a nonempty bounded open set in RN that has a sufficiently smooth boundary. (ii) Formula (22*) holds for all u£C\G) and v€C^{G), provided G is a nonempty open set in R^. The integration-by-parts formula (22) is the key to the modern theory of partial differential equations and to the modern calculus of variations. In this book, we only need the special case (22*). Let us sketch the proof of Proposition 12. Ad (i). The generalization of the fundamental theorem of calculus J a w ax = w\ to higher dimensions is given by the famous Gauss theorem: / djwdx — I JIg Jdc IdG wnjdO.

120 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle Hence / ({dju)v + udjv)dx= I dj(uv)dx = / uvrijdO. JG JG JdG This is (22). Ad (ii). Let v G Cq°(G). Then, the function v vanishes outside a compact subset of G, i.e., v vanishes on a "boundary strip" ofdG. Thus, it is possible to construct a nonempty bounded open subset H of G such that v = 0 outside H and the boundary dH is sufficiently smooth (see Figure 2.5(b)). Hence / ((dju)v + udjv)dx = I dj(uv)dx JG JG = / dj(uv)dx = / uvrijdO = 0, JH JdH since v = 0 on dH. 2.3 Bilinear Forms Definition 1. Let X be a normed space over K. By a bounded bilinear form on X we understand a function a:X xX -+K that has the following properties: (i) Bilinearity. For all u,v,w G X and a, /? G K, a(au + (3v, w) — aa(u, w) + (3a(v, w) and a(w, au + (3v) = aa(w, u) + (3a(w, v). (ii) Boundedness. There is a constant d > 0 such that |a(^, v)\ < d\\u\\ \\v\\ for all u, v G X In addition, a(-, •) is called symmetric iff a(tt, t>) = a(v, u) for all u, v G X Moreover, a(-, •) is called positive iff 0 < a(tt, it) for all u G X Finally, a(-, •) is called strongly positive iff there is a constant c > 0 such that c|M|2 < a(tt, u) for all tt G X

2.4 The Main Theorem on Quadratic Variational Problems 121 Proposition 2. Let a: X x X —> R be a bounded bilinear form on the normed space X over K. Then un —> u and vn —> v as n —> oo zmp/y a(un, vn) —> a(tt, ?;) as n —> oo. Proof. Since the sequence (t>n) is bounded, we get \a(un,vn) -a(u,v)\ = \a(un - u,vn) + a{u,vn - v)\ < d\\un — u\\ \\vn\\ + d\\u\\ \\vn — v|| —> 0 asn-> oo.D 2.4 The Main Theorem on Quadratic Variational Problems We consider the minimum problem 2~1a(u, u) - b{u) = min !, u G X. (23) In the next section we shall show that the famous Dirichlet principle is a special case of the following theorem. Theorem 2.A (Main theorem on quadratic variational problems). Suppose that (a) a: X x X —> R is a symmetric, bounded, strongly positive, bilinear form on the real Hilbert space X. (b) b: X —► R is a linear continuous functional on X. Then the following hold true: (i) The variational problem (23) has a unique solution. (ii) Problem (23) is equivalent to the following so-called variational equation: a(u, v) = b(v) for fixed u G X and all v G X. (23*) Proof of Theorem 2.A. By hypothesis, there are constants c > 0 and d > 0 such that c|M|2 < a{u, u) < d\\u\\2 for all ueX. (24) Step 1: Equivalent equation. We show that (23) is equivalent to (23*). To this end, we set F{u) := 2_1a(it, u) - b{u) for all u G X.

122 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle Moreover, for fixed u, v G X, we set <j)(t) := F[u + tv) for all t G R. Using the symmetry condition a(u,v) = a(v,u), we obtain (j)(t) = 2~~1t2a(v, v) + t[a(u, v) - b(v)} + 2_1a(it, it) - b(u). Note that a(t>,t>) > c\\v\\2 > 0 for all v G X with v ^ 0. Thus, the original problem (23), F(u) = min !, tt G X, has a solution u iff the real quadratic function <j> = </>(£) has a minimum at the point £ = 0 for each fixed v G X, i.e., </>'(0) = 0. (25) Equation (25) is identical to a(u, v) — b(v) = 0 for all v £ X. This is (23*). Step 2: Uniqueness. Let u and w be solutions of the original problem F(u) = min l,ueX. By Step 1, iU'V\ = &\ for all veX, Letting v := u — w, we get c\\u — w\\2 < a(u — w, u — w) = 0. Hence u = w, i.e., the original problem (23) has at most one solution. Step 3: Existence proof. Set a := inf F(u). uex Since F(u) = 2_1a(u,u) - b(u) > 2_1c|M|2 - \\b\\ ||u||, we obtain F(u) —> H-oo if ||tt|| —> +oo. Hence a > — oo. By the definition of a, there is a sequence (un) such that F(un) —> a as n —> oo. Obviously, we have the following key identity: 1^{un,un) + 2a{um,um) = a{un-um,un-um) + a{un + um,un+um). (26)

2.4 The Main Theorem on Quadratic Variational Problems 123 Hence F{un) + F{um) = 4_1o(«n ~ Um, Un - Um) + 2F > 4_1c||ttn - um\\ +2a. Since F{un) + F{um) —> 2a, it follows that (itn) is a Cauchy sequence. Hence un -* u as n —> oo. Since F: X —> R is continuous, F(un) —> F(it) as n —> oo. This implies F(it) = a, i.e., u is a solution of the original problem F(u) = min !, u £ X. D Proposition 1. LetX be apre-Hilbert space overK. Then, for allu, v G X, we have the so-called parallelogram identity 2\\u\\2 + 2|H|2 = \\u + ^||2 + \\u - v\\2. (27) Proof. From (u±v \u±v) = (u\u) ±(u\ v) ± (v \u) + (v \v) we get \\u ± v\\2 = ||tt||2 ±(u\v)±(v\u) + \\v\\2. D Remark 2 (The geometrical meaning of the Dirichlet principle). Figure 2.6(a) shows the geometrical meaning of the parallelogram identity (26). By Figure 2.6(b), it is obvious that (27) generalizes the classic Pythagorean theorem. The proof of Theorem 2.A has been based on the identity (26). If we introduce the energetic inner product (u | v)e := a>(u, v) for all u, v G X, then (26) can be written in the following form: 211^111 + 2111^111 = \\un + Um\\2E + \\Un ~ ^m|||, where \\u\\% = (u \ u)e- This is precisely the parallelogram identity with respect to the energetic inner product. In Section 2.13 we shall prove that Theorem 2.A is equivalent to the perpendicular principle, which says the following: V>n ~r V>r)

124 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle v-u u + v v-u u + v u u (a) 2||u||2+2||^||2 = ||u + ^||2 + ||u-^||2 (b) ||u±*;||2 = |H2+H2 FIGURE 2.6. m u ^— 1 L\ 0 FIGURE 2.7. In a Hilbert space, there exists a perpendicular from each point u to each given closed linear subspace L. This principle is pictured in Figure 2.7. Since the Dirichlet principle follows from Theorem 2.A, we can say the following: The functional analytic justification of the Dirichlet principle is based on the idea of orthogonality. There are ideas in mathematics that remain eternally young and that lose nothing in their intellectual freshness, even after thousands of years. Mathematicians of the Pythagorean school in ancient Greece attributed the Pythagorean theorem to the master of their school, Pythagoras of Samos (circa 560 B.C.-480 B.C.). It is said that Pythagoras sacrified one hundred oxen to the gods in gratitude. In fact, this theorem was already known in Babylon at the time of King Hammurabi (circa 1728 B.C.-1686 B.C.). Presumably, however, it was a mathematician of the Pythagorean school who first proved the Pythagorean theorem. This theorem appears as Proposition 47 in Book I of Euclid's Elements (300 B.C.). The theory of Hilbert spaces is the abstract and very efficient formulation of the idea of orthogonality. It seems that this idea has deep roots in our real world, since the Hilbert space theory is the right mathematical tool for describing quantum physics. This will be discussed in Section 5.14.

2.5. The Functional Analytic Justification of the Dirichlet Principle 125 2.5 The Functional Analytic Justification of the Dirichlet Principle We want to study the following variational problem: F(u) := 2"1 / J2(dju)2dx - [ fudx = min !, u = g on OG. (28) Jg~[ jg This problem is also called the Dirichlet problem. Here, we assume that (H) G is a nonempty bounded open set in R^, N = 1, 2, In addition, we set x = (£i,..., £;v) and djU := du/d£j. The Dirichlet principle says_Jhat problem (28) has a solution u. After the necessary preparations, the final existence theorem will be proved at the end of this section. 2.5.1 The Classic Euler-Lagrange Equation Along with (28) let us consider the following boundary-value problem for the Poisson equation: -Au = f on G u = g on oG. v ' In addition, let us also study the so-called generalized problem to (28a): / J2 djudjVdx= f fv dx for all v e C^{G). (28b) Jgj=1 Jg Here, the Laplacian is defined through If / = 0, then (28a) is called the first boundary-value problem for the Laplace equation. Proposition 1. Assume (H). Let the continuous functions g: dG —> R and f: G —> R be given. Suppose that u G C2(G). Then, the following hold true: (i) If the function u is a solution of the original variational problem (28), i.e., more precisely, u is a solution of F(w)=mm\, weC2(G), (28*) w = g on dG, then u is a solution of the boundary-value problem (28a).

126 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle (ii) The function u is a solution of the boundary-value problem (28a) iff it is a solution of the generalized boundary-value problem (28b). Equation (28a) is called the Euler-Lagrange equation to the variational problem (28). The following arguments are typical for the calculus of variations. Proof. Ad (i). Step 1: Admissible functions. Let u be a solution of (28*). Then, for each fixed v G Cq°(G) and <Gi, the function w := u + tv is admissible for the variational problem (28*), i.e., w = g ondG and w G C2(G). Step 2: Reduction to a minimum problem for real functions. For fixed v G C$>{G), we set <j){t) := F(u + tv) for all t G R. Explicitly, (j)(t) = 2"1 / Y](djU + tdjvfdx - J f(u + tv)dx. Jgj^[ Jg Since u is a solution of (28*), the function 0: R —> R has a minimum at the point t = 0. Hence *'(0) = 0. Explicitly, r N <£'(0)= / Y,dJudJvdx- / fvdx = 0 for all v € Cg°(G). (29a) Thus, it is a solution of the generalized problem (28b). Step 3: The variational lemma. Applying integration by parts to (29a), we get j that is, / T^idjdjtfivdx - / fvdx = 0 for all v G Cg°(G), (29b) Jg j=1 Jg - [ (Au + f)vdx = 0 for all v G C%°(G). Jg

By the variational lemma from Section 2.2.4, this implies Au + / = 0 on G, (29c) i.e., u is a solution to the boundary-value problem (28a). Ad (iii). By Step 3, equation (29a) implies (29c). Conversely, integration by parts tells us that (29c) implies (29a). □ Remark 2 (Lack of classic solutions). By Proposition 1, each sufficiently smooth solution u to the Dirichlet problem (28) is also a solution to the boundary-value problem (28a). However, the point is that There exist reasonable situations where the Dirichlet problem (28) lacks smooth solutions. In order to understand this typical difficulty of the calculus of variations, let us consider the following two simple minimum problems: f(u) = mini, ue[a,b], (30) and f{u) = min !, ue [a, b] n Q, (30*) where —oo < a < b < oo, and Q denotes the set of rational numbers. Suppose that the function /: [a, b] —> R is continuous. Then (a) problem (30) has always a solution, but (b) there are reasonable functions / for which problem (30*) has no solution. In fact, statement (a) follows from the classical Weierstrass theorem. Suppose now that all the solutions u of (30) are irrational numbers. Then, problem (30*) has no solution. Consequently, mathematicians who do not know irrational numbers cannot prove the Weierstrass existence theorem for minimum problems. A similar situation is encountered with respect to the Dirichlet problem. Roughly speaking, the search for smooth solutions to the Dirichlet problem (28) corresponds to problem (30*). In order to get a situation comparable with the solvable problem (30), we have to add ideal elements to the class of smooth solutions. These ideal elements correspond to functions uew}{G) in the Sobolev space WjJ (G), which will be introduced ahead. Such functions only possess generalized derivatives of first order. Summarizing: The introduction of Sobolev spaces corresponds to the introduction of real numbers by completion of the set of rational numbers via irrational numbers. Our program is now the following:

128 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle (i) We define generalized derivatives via integration by parts. (ii) We define Sobolev spaces. (iii) We prove the inequality of Poincare-Priedrichs. (iv) We apply the main theorem on quadratic variational problems (Theorem 2.A) to the Dirichlet problem (28) with respect to a suitable o linear closed subspace W\{G) of the Sobolev space W2 (G). Here, the Poincare-Priedrichs inequality ensures the fundamental strong positivity of the quadratic main part of the Dirichlet problem. This way we obtain both generalized solutions of the Dirichlet problem (28) and generalized solutions of the classical boundary-value problem (28a) for the Poisson equation. The modern theory of variational problems and partial differential equations is governed by the notion of generalized solutions. In this connection, one uses the following general strategy: (a) One proves the existence of generalized solutions by using the methods of functional analysis. (/?) One uses sophisticated analytical methods in order to prove that generalized solutions are also classic solutions, provided the situation is sufficiently regular (e.g., the boundary dG and the functions / and g in (28) are sufficiently smooth). Step (/?) represents the subject of the so-called regularity theory. An elementary introduction to regularity theory can be found in Zeidler (1986), Vol. 2A. 2.5.2 Generalized Derivatives The point of departure for the definition of generalized derivatives is the classic integration-by-parts formula: / udjvdx = - [ {dju)vdx for all v G Cg°(G), (31) Jg Jg where u G C1(G). The simple trick is to set w = djU. This way we obtain the key formula [ udjvdx = - [ wvdx for all v G Cg°(G). (31*) Jg Jg

2.5. The Functional Analytic Justification of the Dirichlet Principle 129 £■ The point is that this formula remains valid for certain nonsmooth functions u and w. Definition 3. Let G be a nonempty open set in RN, N > 1. Let u,w G L2(G), and suppose that (31*) holds. Then, the function w is called a generalized derivative of the function u on the set G of type dj. As in the classic case, we write w = djU. Proposition 4. The generalized derivative w = djU is uniquely determined up to the values of w on a set of N-dimensional measure zero. Proof. Suppose that (31*) holds for w,w £ L2(G). This yields (w - w)vdx = 0 for all v G C£°(G), IG and the variational lemma from Section 2.2.4 implies that w(x) = w(x) for almost all x G G. □ Example 5. Consider the function u: ] — 1,1[ —> R with u{x) := \x\ for all x G ] — 1,1[. Set r-1 if —1 < a; < 0 w(x) :— < c if x = 0 I 1 if 0 < x < 1, where c is a fixed, but otherwise arbitrary, real number. Then, the function w represents the generalized derivative of the function u on the interval ]-l,l[. We write u' — w on ] — 1,1[. Note that w is the classic derivative of u on both the subintervals ] — 1,0[ and ]0,1[, but the classic derivative of u does not exist at the point x — 0. Proof. For all v G Cq°(—1,1), integration by parts yields /l pO pi uv'dx — I uv'dx + / uv'dx -l J-i Jo /o pi u'vdx— / u'vdx+ uv\_l+uv\0 = — / wvdx— / wvdx + u(0)v(0) — u(—l)v(—1) + u(l)v(l)-u(0)v(0) = — / wvdx,

130 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle since v(±l) = 0. More precisely, for small e > 0, note that integration by parts yields / uv'dx = — / u'vdx + uv\Jv since u and v are C1 on ] — 1, — e[. Letting e —> +0, we get uv'dx = / u'vdx + uv\_v since uv is continuous on [—1,0]. Similarly, applying e —> +0 to J ..., we get the corresponding formula /o1-• = ...• □ 2.5.3 The Sobolev Space W$ (G) Definition 6. Let G be a nonempty open set in R^, N > 1. The Sobolev space W2 (G) consists precisely of all the functions u € L2(G) that have generalized derivatives dju E L2{G) for all j = 1,..., N. Furthermore, for all u,v £ W2(G), we set r I \ (v> I ^)i,2 •= / j uv + jT^djudjV J dx, uv ■ and ||w||i,2 •= (u | tt)i 2- Proposition 7. T/ie space VK^G) together with the inner product (• | •)i,2 becomes a real Hilbert space, provided we identify two functions whose values differ only on a set of N-dimensional measure zero. Proof. Let u G Wj (G). From (u | u)\^ = 0 we get JG u2dx = 0, and hence u{x) = 0 for almost all x G G, i.e., it is the zero element. Hence (• | -)i?2 is an inner product on Wj (G). Thus, WjJ (G) is a pre-Hilbert space. In order to prove that Wj (G) is a Hilbert space, let (un) be a Cauchy sequence in Wj(G), i.e., Il^n -^m||i,2 < ^ for all n,m> rio(e). Hence (un) and (djUn) are Cauchy sequences in L2(G). Since Z/2(G) is a Hilbert space, there are functions tt,^ G L2(G) such that, as n —> 00, un -* u in L,2(G) and djU^ Wj in Z>2(G) for all j. (32)

2.5. The Functional Analytic Justification of the Dirichlet Principle 131 Letting n —> oo, from / undjvdx = — (djUn)vdx for all v G Cq°(G) Jg Jg IG we obtain // udjvdx = - / Wjvdx for all v € C£°(G), (33) using the continuity of the inner product on the Hilbert space L2(G) (see Proposition 4 in Section 2.1). Equation (33) tells us that the function u has the generalized derivatives Wj = djU on G for all j. Since djU G L2(G) for all j, we get u G Wj(G). Finally, it follows from (32) that \Wn — ^||i,2 —► 0 as n —> oo, i.e., un —► u in Wj(G) as n —> oo. Hence VK2X(G) is a Hilbert space. D 2.5.4 The Sobolev Space W\{G) o Definition 8. Let W2(G) denote the closure of the set Cq°(G) in the Hilbert space W%(G). o We will later discuss that it makes sense to say that the functions u £W 1(G) satisfy the boundary condition u = 0 on dG in some generalized sense. o Proposition 9. The space W\{G) is a real Hilbert space. Proof. Note that C™^) is a linear subspace of the Hilbert space W2{G). Now use Proposition 10 from Section 2.1. □ In the special case where N = 1 and G = ]a, b[, let us briefly write W%(a,b) and Wl2{a,b) o instead of W2{G) and W2(G), respectively. The following example shows o that the functions u €W2(a, b) possess a simple structure. o Example 10. Let —oo < a < b < oo. If u eW2(a,b), then there exists a unique continuous function v: [a, b] —> R such that u{x) = v(x) for almost

132 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle all x G ]a,b[ and v(a) = v(b) = 0. In addition, we have the estimate max \v(x)\ < (6-a)M / u'2dx J < (b - a)i||u||lj2. Recall that, by Section 2.5.3, (u\v)ii2= / (uv + uv')dx and IM|i,2 = I / (u2 + u,2)dx\ , o for all u, v GWj^a, &). Proof. Uniqueness of v. If two continuous functions v, w: [a, b] —> R differ at a single point, then they also differ on a small interval J with meas( J) > 0. Hence "v(x) = w(x) for almost all x G ]a, &[" implies t>(x) = w(x) on [a, &]. Existence of v. First let W G Co°(a, 6). Then w(x) = / w'dy for all x G [a, &]. By the Schwarz inequality (6), Kx)| < / 1 • I^ldy <([ dy\ If \w'\2dy) < (b — a)2 ||k;||i52 for all x G [a, b], o Now let it ew\{a-> &)• Then, there exists a sequence (vn) in Co°(a, &) such that ll^n — wlli,2 ""^0 as n —> oo. o Since (vn) is a Cauchy sequence in W\{.ai &)> it follows from max |vn(a:) - vm(a;)| < {b - a)*\\vn-vm\\i,2 a<x<b that (vn) is also a Cauchy sequence in the Banach space C[a, b]. Thus, there is a function v G C[a, b] such that max \vn(x) — v(x)\ —> 0 as n —> oo. a<x<b

2.5. The Functional Analytic Justification of the Dirichlet Principle 133 Since vn(a) = vn{b) = 0 for all n, this implies v(a) = v(b) = 0. Finally, it follows from pb pb / (v — u)2dx= lim / (vn — u)2dx < lim \\vn — u\\\ 2 = 0 J a n^°°Ja n^°° that v{x) = u{x) for almost all x G ]a, b[. D The following example will be used in Section 2.7.3 in order to prove the convergence of the important method of finite elements. Standard Example 11. Let —oo<a<&<oo, and let the function u: [a, 6] -* R be continuous and piecewise continuously differentiable. Denote by C the set of points x where the classic derivative exists. Define the real function \ arbitrary otherwise. More precisely, we assume the following: (a) The function u is continuous on [a, b). (b) There exists a finite number of points aj with a = ao < cl\ < - - - < an = b such that, for all j, u is continuously differentiable on the open subintervals ]aj,aj+i[ and the derivative v! can be extended continuously to the closed subinterval [aj,aJ+i] (cf. Figure 2.8). Then, the function u has the following properties: (i) The function w is the generalized derivative of u on ]a, b[, i.e., w = v! on ]a,6[. (ii) ueW£(a,b). (iii) u ew\(a, b) iff u{a) = u(b) = 0. Proof. Ad (i). Divide the interval [a, b] into the subintervals [aj, aj+i] and use integration by parts as in Example 5. Ad (ii). Since u is continuous and w = vl is piecewise continuous and bounded, we get pb pb / u2dx < oo and / uf2dx < oo. J a J a

134 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle a\ Q>2 (a) ueW}(a,b) (b) ueW\{a,b) (c) veCl[a,b] FIGURE 2.8. Hence u G Wj (a,6). o Ad (iii). If u eW\(a, b), then u{a) = u(b) = 0, by Example 10. Conversely, let u(a) = u(b) = 0. Choose a number 77 > 0. By smoothing the function u at the corners, we obtain a function v G C1 [a, b] such that v vanishes in a neighborhood of the two boundary points x — a and x = b along with <2= [ {(u-v)2 + (u'-v')2}dx<r,2. J a The idea of the construction of the function v related to u is pictured in Figures 2.8(b) and (c). Letting ve(x) := / 0e(. J a x - y)v(y)dy, it follows from the density proof in Problem 2.13a that v£ G C§°(a,b), for sufficiently small e > 0, and v£ —> v in L2(a, &) as e —> +0. Differentiation and integration by parts yield ve0*0 = / <^(X - y)v(y)dy = - / <M^ - yW(y)dy. J a J a Hence This implies —> i/ in L2(a, 6) as £ —> +0. ||ve — v||i,2 —> 0 as 5 —> +0.

2.5. The Functional Analytic Justification of the Dirichlet Principle 135 Summarizing, we get \\u- ve||i,2 < \\u- v||i,2 + \\v - vc||i,2 < V, where v£ G Co°(a,6), for sufficiently small £ > 0. Set un := vj,. Since 77 > 0 is arbitrary, n un -* u in Wj (a, &) as n —> 00, o where un G Co°(a, 6) for all n. Hence it GW2(a5 &)• ^ ^.5.5 Generalized Boundary Values Definition 12. Let G be a nonempty bounded open set in R^, N > 1. If o u G^fC), then we say that the functiou u satisfies the boundary condition u = 0 on dG, (34) in the generalized sense. Remark 13 (Motivation of (34)). A very formal motivation is based on the o fact that the set Cq°(G) is dense in W\{G) and the functions u G Cq°(G) vanish on a boundary strip of G, i.e., u satisfies condition (34) in the classic sense. A more convincing motivation is obtained as follows: (a) Let G C RN with N = 1 and G = ]a, b[. Then, Example 10 tells us that (34) holds true in the "classic sense." (b) Let G C R^ with N > 2, and suppose that the boundary dG of the nonempty bounded open set G is sufficiently regular. Then it can be proved that f u2dO < const / I u2 + Ylidjuf I dx for all u G W%(G). (B) o This implies the following: If u £Wl(G), then there exists a sequence (un) in Cq°(G) such that un —> u in Wl{G) as n —> 00. By (B), / (it — un)2dO < const||w — un\\\ 2 —> 0 as n —> 00. Since un = 0 on <9G, we get tt2dO = 0, JdG

136 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle and hence u(x) = 0 for almost all x G dG, (34*) in the sense of the surface measure on dG. The elementary proof of the boundary inequality (B) can be found in Zeidler (1986), Vol. 2A, p. 247. This proof is based on the Schwarz inequality. 2.5.6 The Poincare-Friedrichs Inequality We want to prove the following Poincare-Friedrichs inequality: r r N C / u2dx < / y2(dju)2dx for all u ew\(G). (35) Jg JGj^i Proposition 14. Let G be a nonempty bounded open set in RN, N = 1,2, Then there exists a constant C > 0 such that inequality (35) holds. Example 15. We consider first the special case where N = 1 and G = ]a,b[ with —oo < a < b < oo. Step 1: Let u G C^{a, b). Then rX u{x) = / uf(y)dy for all x G [a, b]. J a By the Schwarz inequality (6), u{x)2 < ( / 1' \u'\dy\ < f dy f u'2dy, and hence po no / u2dx <(b- a)2 / u/2dx. J a J a Step 2: Let u G^^ b). Then, there is a sequence (un) in Co°(a, b) such that \\u — iin|| 1,2 ->0asn->oo. Hence un -+ u in L2(a, &) and u'n —> it7 in L2(a, &) as n —> oo. By Step 1, rb pb I u2ndx <{b- a)2 I u,2dx for all n. J a J a Letting n —> oo, this implies the special Poincare-Friedrichs inequality,

2.5. The Functional Analytic Justification of the Dirichlet Principle 137 V d FIGURE 2.9. rb P f o / u2dx <{b- a)2 I u,2dx for all u ew\{a, b). (35*) J a J a □ Proof of Proposition 14. Let N = 2. The general case proceeds analogously. Step 1: Let u G Cq°(G). As in Figure 2.9 consider a rectangle ft := [a, b] x [c, d] with G C int TZ. Note that it vanishes outside G. Then u(£ V)= u^fa y)dy for all fa rj) G ft. The Schwarz inequality yields l'Urjfay)dy) < dy uvfay)2dy fd <{d-c) uvfa y)2dy for all fa rj) G ft. Integrating this over ft, we get / u2dx <{d — c)2 I u2dx. Jn Jn This is (35). o Step 2: Let u eWli^). Then there is a sequence (un) in Cq°(G) such that \\v>n — ^||i,2 --* 0 as n —> oo. Hence un-*u in L,2(G) and djUn-+djU in L,2{G) as n —> oo, for all j. By Step 1, C / u2ndx < I T,j(djUn)2dx for all n. Jg Jg

138 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle Letting n —> oo, we get the desired inequality (35). □ In 1890, inequalities of type (35) were considered by Poincare in a famous paper on eigenvalue problems for the Laplace equation. In 1934, Priedrichs recognized that such inequalities represent the key to the functional analytic existence theory for linear elliptic differential equations. 2.5.7 The Existence Theorem for the Dirichlet Problem That's what it was really all about. Faust Parallel to Section 2.5.1, let us consider the generalized Dirichlet problem r N r 2_1 / Yl^u^dx - fudx = mml, u-g ew\{G), (36) along with the generalized boundary-value problem r N r / S^djudjvdx= \ fvdx for all v ew\{G), u-gew\{G). (37) Jg j==1 Jg o As defined in (34), the condition "u — g EWliG)" corresponds to the boundary condition u — g = 0 on <9G, in the generalized sense. Theorem 2.B (The Dirichlet principle). Let G be a nonempty bounded open set in RN, N = 1,2, We are given f e L2(G) and g e W$(G). Then, the following hold true: (i) The generalized Dirichlet problem (36) has a unique solution u G (ii) This is also the unique solution u G W\ (G) of the generalized boundary-value problem (37). Proof. We set X :=W\{G) and N lG a(u,v) := / 2_)djudjvdx, b\{v) := / fvdx, Jgjt[ Jg

2.5. The Functional Analytic Justification of the Dirichlet Principle 139 for all u, v G Wl{G). Introducing w := u — g, the original problem (36) can be written in the following form: 2~1a(w + g,u) + g) — b\{w + g) = min !, w G X. If we use a(w + g,w + g) = a(w, w) + 2a(w, g) + a(g, g) and &i (w + g) = bi(w) + &i(#), then this minimum problem is equivalent to 2~1a(w, w) - b(w) = min !, w G X, (36*) where we set b(w) := bi(w) — a(w,g) for all w G X. Furthermore, the generalized boundary-value problem (37) is equivalent to a(w, v) = b(v) for fixed w G X and all v £ X. (37*) We want to apply Theorem 2.A to problems (36*) and (37*). Step 1: Properties oi a:X x X -*R. Set H|2:=(/g, % 2 2. \Jg and recall that N 2 iMii,2= M"2+x>^) „ ,2 j <& 3=1 represents the norm on X. By the Schwarz inequality (14), for all v, w € X, we get /• N \a(v,w)\ <2_1 / V] |0jt/dju;|<* < 2"1 J2 ll^lbll^Hk < 2-1Ar||^||lj2||^||l52, (38) 3=1 i.e., a(-,-) is bounded. Obviously, a(-,-) is bilinear, and a(t>, k;) = a(w, t>) for all v, w G X, i.e., a(-,-) is symmetric. Finally, the Poincare-Friedrichs inequality (35) tells us that C f I v2 + ^(d^)2 J dx < (1 + C) / ^{djvfdx for all uGl

140 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle Hence C(l + C)"1 |M|?|2 < a(v, v) for all v G X, i.e., a(-, •) is strongly positive. Step 2: The functional &: X —> R. By the Schwarz inequality, \bi(v)\< [ |/^<||/||2H|2 Jg < \\fh\\v\\i,2 for all VGA". By (38), \a(v,g)\ < 2-1Ar||^||l52||^||l52 for all v G X. Hence \b(v)\ < const||v||i>2 for all v G X. (39) Obviously, &(•) is linear. By (39), b: X —> R is a linear continuous functional. Thus, the assumptions of Theorem 2. A are satisfied. Consequently, using Theorem 2.A, we obtain that problem (36*) has a unique solution w G X, and w is also the unique solution of (37*). Finally, set u = g + w. Then u G W^G), and u is the unique solution of both (36) and (37). □ 2.6 The Convergence of the Ritz Method for Quadratic Variational Problems Let us again consider the variational problem F[u) := 2~la{u, u) - b{u) = min !, u G X, (40) along with the corresponding variational equation a{u, v) = b(v) for fixed u G X and all v G X. (40*) In order to construct the fundamental Ritz method for solving approximately the problems (40) and (40*), let us consider the Ritz problem 2~1a(un, un) - b{un) = min !, un G Xn (41) and the Ritz equation a{un, vn) = b(vn) for fixed un G Xn and all vn G Xn, (41*) where Xn is a finite-dimensional subspace of the real Hilbert space X,

2.6. The Convergence of the Ritz Method 141 i.e., 0 < dim X < oo. Comparing (40) with (41), it turns out that the space X of the original problem is replaced with Xn. Thus, the Ritz method is based on a quite natural idea. Let {ein,..., e^n} be a basis of Xn. Then, the elements un and vn of Xn allow the following simple representation: N N Un ~ / J QinCin and Vn = / j Pin^iw z=l z=l Choosing vn := ejn, we obtain that the Ritz equation (41*) is equivalent to the following system of linear equations: N Y^ a™a{ein, ejn) = b(ejn), j = 1,..., N, (41**) for the N unknown real numbers ain,..., olnw We assume the following: (HI) X is a real Hilbert space. There exists a sequence (Xn) of finite- dimensional linear subspaces Xn of X such that (J Xn = X, n=l i.e., the union |J^=i Xn is dense in X. (H2) The bilinear form a:X xX -+ Ris bounded, symmetric, and strongly positive, i.e., there exist positive constants c and d such that |a(w,v)| < d\\u\\ \\v\\ and c\\u\\2 < a(u,u) for all u,v G X. (H3) The functional b: X —> R is linear and continuous. Theorem 2.C (The Ritz method). Assume (HI) through (H3). T/ien, t/ie following hold true: (i) Existence and uniqueness. T/ie original variational problem (40) /ms a unique solution u. This is also the unique solution of the variational equation (40*). We have the a priori estimate NI<c_1IH|. (42) (ii) The Ritz equation. For all n = 1,2,..., the Ritz problem (41) has a unique solution un. This is also the unique solution of the Ritz equation (41*).

142 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle (iii) Convergence. The Ritz method converges, i.e., un -> u in X as n —> oo. (iv) Rate of convergence. For all n = 1,2,..., \W~ un\\ < d~1c- distx(u,Xn). (43) (v) Error estimates. Suppose that we know a lower bound /3 for the minimal value of the original problem (40), i.e., F(u) > (3. Then, 2~1c\\u - un\\2 < F{un) - /? for all n = 1,2,... . Remark 1 (Discussion of Theorem 2.C). (a) Typical applications. For example, in elasticity we encounter the following situation: u = displacement of the elastic body (e.g., a beam or a plate); 2-1a(i£, u) = elastic potential energy of the body; b(u) = work of the outer forces. Furthermore: minimum problem (40) = principle of minimal potential energy, variational equation (40*) = principle of virtual power. A simple example will be considered in Section 2.7 (the deformation of a string). Applications of functional analysis to elasticity are studied in detail in Zeidler (1986), Vol. 4. In 1909, Ritz introduced his "Ritz method" in order to compute approximately the deformation of plates under the action of forces. (b) Lower bounds and error estimates via duality. In part (v) of Theorem 2.C we need a lower bound f3 for the minimal value F{u) in order to get error estimates. Such lower bounds can be obtained by using duality theory. Here, the basic idea is the following. Along with the original minimum problem F{u) = min!, ueX, (V) we also study a dual maximum problem G(v) = max!, veY, (V*) which has the crucial property that minimal value of (V) = maximal value of (V*).

2.6. The Convergence of the Ritz Method 143 Therefore, if u is a solution of (V), then, for each v eY and each w e X, we get G(v) < F(u) < F(w), i.e., G(v) is a lower bound and F(w) is an upper bound for the minimal value of (V). Such dual problems will be considered in Section 2.12. (c) The golden rule for the rate of convergence. Using information from approximation theory, it is possible to obtain estimates for the distance distx(u,Xn), which depend on the smoothness of the solution u in the case of boundary- value problems for elliptic differential equations (e.g., the Poisson equation or problems in elastostatics). Then, relation (43) yields information on the rate of convergence of the Ritz method. This will be explained with a simple example in the next section. Generally speaking, one has the following golden rule of numerical analysis: The smoother the solution u of the original problem and the smoother the functions in Xn, the faster is the convergence of the Ritz method. Tka. 4wn$~4ftrX^-thc faster io the convergence of the Ritbt-'methodT* A detailed discussion of this golden rule can be found in Zeidler (1986), Vol. 2A. (d) Finite elements. In engineering, the spaces Xn consist frequently of so-called finite elements, which are piecewise smooth functions. A simple example will be studied in the next section. More information can be found in Zeidler (1986), Vol. 2. The method of finite elements is one of the greatest achievements of modern numerical analysis. Proof of Theorem 2.C. Ad (i). This follows from Theorem 2.A. In addition, if u is a solution of the variational equation (40*), then a(u,u) ~ b(u). Hence c\\u\\2 < a(u,u) = b(u) < \\b\\ \\u\\. This implies c\\u\\ < \\b\\. Ad (ii). This follows from Theorem 2.A applied to the Hilbert space Xn. Note that each finite-dimensional linear subspace of a Hilbert space is again a Hilbert space. —? ^ hc\tui Ad (iii), (iv). The key to the proof is relation (44). Subtracting the Ritz equation (41*) from the variational equation (40*), we obtain the so-called orthogonality relation a(u — un, v) = 0 for all vGl,

144 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle Letting v := un, we get a(u — un, un) = 0, and hence a(u — un, u — un) = a(u — un, u — v) for all v £ Xn. (44) This yields c\\u — un\\2 < a(u — un,u — un) = a(u — un,u — v) < d\\u — un\\ \\u — v\\ for all v £ Xn. Hence d_1c||tt — un\\ < inf \\u — v\\ = distx(^,Xn). v£Xn By (HI), distx(^, Xn) —> 0 as n —> oo. Hence un —> u as n —> oo. Ad (v). For all v£X, F(u + v) = 2~la{u -f v, u + v) — b(u + v) = 2~1a(v, v) + (a(u, v) - b{v)) + 2~1a(u, u) - b{u). By the variational equation (40*), a(u, v) — b(v) = 0. This implies F(u + v) = 2~la{v, v) + F(u), and hence F{u + v) - F(u) > 2"1c||^||2 for all v £ X. If we set w := u + v and if F(u) > /?, then F(w)-0> 2-1c||it-^||2. D Definition 2. We set (u | v)e '•= o>(u, v) for all u, v £ X and call (• | -)e the energetic inner product to a(-, •). The energetic norm is defined through i \\u\\e •— (w | tt)|; for all u £ X. The linear space X equipped with (• | -)e is called the energetic space Xe to a(-, •). A physical motivation for this terminology will be given in the next section. Corollary 3. Let X be a real Hilbert space, and assume that the bilinear form a:X x X —> R satisfies assumption (H2) of Theorem 2.C. Then, the following are met:

2.7. Applications to Boundary-Value Problems 145 (i) The energetic space Xe is a real Hilbert space. (ii) The original norm || • || on X is equivalent to the energetic norm \\'\\e, i.e., C\W\\2 < IMll < d\W\\2 for all ueX. (45) (iii) A set is dense in the energetic space Xe iff it is dense in the original Hilbert space X. Proof. The inequality (45) follows from (H2). Obviously, (• | •)# is an inner product on X, since a(-, •) is bilinear and \\u\\e = 0 implies \\u\\ = 0, i.e., u = 0, by (45). Let (un) be a Cauchy sequence in Xe- By (45), this is also a Cauchy sequence in X, and hence it is convergent in X. Furthermore, (45) tells us that (un) is also convergent in Xe- Thus, Xe is a Hilbert space. Assertion (iii) follows immediately from (45). □ Corollary 4. Assume (HI) through (H3) of Theorem 2.C. Then, we get the following crucial error relations for the Ritz method: \\v> — UtiWe = distxE(u,Xn) for all n = 1,2,... . Proof. Apply Theorem 2.C to the energetic space Xe- This situation corresponds to c = d = 1. By Theorem 2.C(iv), ||w-Wn||js <distxE(u,Xn). But since un G Xn, we can replace "<" with "=" by the definition of the distance dist(-). □ 2.7 Applications to Boundary-Value Problems, the Method of Finite Elements, and Elasticity We want to solve the following boundary-value problem: -u"{x) = f(x), a < x < (3, (46) u(a) = u(0) = 0, ueC2[a,p], where —oo<a</3<oo. According to Section 2.5.1, the corresponding variational problem reads as follows: F(u) := / (2"V2 - uf)dx = min !, (47) Ja u(a) =«(/?) = 0, u€C2[a,(3}.

146 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle f(x0) u = u(x) a. xq j3 x FIGURE 2.10. This problem allows the following physical interpretation5 (cf. Figure 2.10): u(x) = deflection of a string at the point x under the vertical outer force density f (x) = f(x)e; F(u) = total potential energy of the string; (3 2 1uf2dx = elastic energy of the string; a (3 uf dx — work of the outer force density f with respect to the l I vertical displacement u(x) — u(x)e of the string = —(potential energy stored by the force density f). Let us also consider the following generalized variational problem: F{ u) := / (2~1u'2 - uf)dx = min !, u €W\{a, (3), (47*) J a where we replace the classical boundary condition "u(a) = u((3) = 0 and o u e C2[a,/?]" in (47) with the more general condition "u GW^(a,/?)." Observe that the solutions u of (46) through (47**) are assumed to have different smoothness properties. From the physical point of view, problem (47*) is quite natural, namely: o The functions u £W\{a^) have a finite elastic energy, i.e., I JOL uf2dx < oo, and u(a) — u{(3) = 0, in the sense of Example 10 in Section 2.5. 5That is, the force (J f(x)dx)e acts on the part of the string over the interval [7,6].

2.7. Applications to Boundary-Value Problems 147 Furthermore, we set r13 i (u I v)e '.= / u'v'dx and \\u\\e •= (u \ u)^, Ja where (• | •)# and || • \\e are called the energetic inner product and the energetic norm, respectively. Obviously, 2~x{u | u)e = 2_1||w|||; = elastic potential energy with respect to the displacement u. Finally, the variational equation reads as follows: dF{u + tv) I dt t=o I (u'v'-fv)dx = 0 for all v ew\(<x,0). (47**) Ja The variational equation (47**) represents the principle of virtual power. To explain this, consider the motion t \-* u + tv of the string, where t denotes time. Then, F(u + tv) is the total potential energy of the string at time t, and relation (47**) tells us that the time derivative of the energy at time t = 0 equals zero. In physics, the time derivative of energy is called power.® We will also need the integral formula u(x)= / g(x,y)f(y)dy, (48) Ja G(x,y):=l 0- where (P-y)(x-a) [{ a<x<y<f3 <*-g£-«> Ha<y<x</3 is called the Green function to the original boundary-value problem (46). Functions of this type were introduced by Green in 1828. The physical meaning of the Green function will be discussed in Section 2.8. 2.7.1 Existence and Uniqueness We set w||i,2 := / [vr +uu)dx and ||u||2 := 6For historical reasons, the principle of virtual power is frequently called the principle of virtual work.

148 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle Recall that / f(3 y Proposition 1. Let the force density f G L2(a,/3) be given. Then, the following hold true: o (i) The variational problem (47*) has a unique solution u eW^i0-^)- This is also the unique solution of the variational equation (47**). We obtain Nli,2 < const||/||2 for all f G L2(a,/3). (49) (ii) If f G C[a,(3], then the original boundary-value problem (46) has a unique solution u G C2[a, /?], which is given by the integral formula (48). This function u is identical to the unique solution of the two variational problems (47) and (47*). o Corollary 2. Set X :=W2(aiP)- Let Xe denote the linear space X equipped with the energetic inner product (• | •)#. Then, Xe is a real Hilbert space. By our physical motivation, it is quite natural to call Xe the energetic space of the string. The integral formula (48) explicitly relates the force density / to the displacement u. By Proposition 1 (ii), the integral operator from (48) is the inverse operator to the differential operator A from (46) given by A: D(A) C C2[a, /?] -* C[a, 0] along with An := -u" and D(A) := {u G C2[a,p]:u(a) = u(p) = 0}. Roughly speaking: Integral operators are frequently inverse operators to differential operators. In Section 4.5, we will use the integral formula (48) in order to reduce boundary-eigenvalue problems to integral equations and to solve them. In this connection, it is important that the Green function is symmetric, i.e., Q(x, y) = Q(y, x) for all x, y G [a, /?].

2.7. Applications to Boundary-Value Problems 149 It was one of the main goals of functional analysis in the twentieth century to generalize the results above to more general elliptic partial differential equations. The Ritz method for solving numerically the original problem (46) will be studied ahead. o Proof of Proposition 1. We set X :=w\{ai P) along with a(u,v) := / u'v'dx and b(v) := / fvdx. J a J a The norm on X is given by || • ||ij2- The variational problem (47*) is identical to 2~1a(it, u) - b(u) = min !, ueX, (V) and the variational equation (47**) is identical to a(u, v) — b(v) = 0 for fixed u G X and all v e X. (E) Step 1: Uniqueness of the solution u of the original boundary-value problem (46). Let ui and u^ be solutions of (46). Set w := u\ — U2- Then w" = 0, a < x < /?, w(a) = w(0) = O, weC2[a,l3}. Integration by parts yields 0 = / wnwdx = ~ w'2dx, J a J a and hence w; = 0on [a, /?]. Step 2: Existence for (46). Let / G C[a, /?]. We want to show that the function u from (48) represents a solution of (46). In fact, for all x G [a, /?], (/? - a)u(x) = (/?-<*)/ Q(x, y)l(y)dy = ((3-x) f (y-a)f(y)dy + (x-a) f (0-y)f{y)dy, J a J x and hence (/? - a)u'(x) = - [ (y- a)f(y)dy + (/? - x)(x - a)f(x) Ja + / ((3-y)f(y)dy-(0-x)(x-a)f(x), Jx

150 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle which yields (/? - a)u"(x) = -(x - a) f{x) - (/? - x)f[x) = -(/? - a)f(x). In addition, u(a) = u{f3) = 0. Step 3: Existence and uniqueness for (V) and (E). By the Schwarz inequality, \b(v)\<U f2dx) Uv2dx) < H/yMli.2 for all v € X Hence b: X —> R is a linear continuous functional with 11*11 < 11/112- Again by the Schwarz inequality, \a(u,v)\< I f u,2dx\ I f v,2dx\ < IMIi,2lMli,2 f°r aU u,v e x. Furthermore, it follows from the Poincare-Friedrichs inequality (35*) that / u2dx <(0- a)2 / uf2dx for all ueX, (50) J a J a and hence / (u2 + u'2)dy < (09 - af + 1) / u'2dy, i.e., there is a constant c > 0 such that cIMIi,2 ^ a(u,u) for all u G X. (51) By Theorem 2.C(i) in Section 2.6, problem (V) has a unique solution u, which is also the unique solution of (E). Moreover, IMIi,2 < const||6|| < const||/||2. Step 4'> Regularity of the solution of the variational equation (E). For the moment, let w denote the unique solution of the original boundary-value problem (46). By Standard Example 11 in Section 2.5, wex. Integration by parts yields fP a(w, vn) - b(vn) = / {w'v'n - fvn)da Ja

L 2.7. Applications to Boundary-Value Problems 151 {w" + f)vndx = 0 for all vn G C0°°(a, 0). (52) For each given v G X, there is a sequence (vn) in Cq°((*,0) such that vn —> v in X as n -^ oo. Letting n —> oo in (52) we obtain a(w, v) — 6(v) = 0 for all v € X. Since the solution of (E) is unique, we get w = u. By Step 3, w is also the unique solution of the variational problem (V), which corresponds to (47*). Finally, since w satisfies the side conditions of the variational problem (47) and w is a solution of the variational problem (47*), the function w is also a solution of (47). □ Corollary 2 follows from Corollary 3 in Section 2.6. 2.7.2 Finite Elements and the Ritz Method In order to solve approximately the given boundary-value problem -u"(x) = /(#), a < x < /?, (53) u(a) = u(0) = 0, ueC2[oL,0], by means of the so-called method of finite elements, let us divide the interval [a,0] into n + 1 equal subintervals, i.e., a0 = a < ai < • • • < an < an+i = /?, (54) where olj := ^+1 • ^ definition, a finite element ein:[a,0] -* R, i = l,...,n, is a piecewise linear function with e»n(o») = 1 and Zin{a>j) = 0 for all j ^ i (cf. Figure 2.11(a)). We also set Xn := span{ein,..., enn}. Then, itn G Xn iff n 1in = ^ Oiin^in With Qin, . . . , Qnn G R. 2=1 The point is that

152 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle 62 a a\ a,2 j3 a a i a,2 (3 (a) (b) FIGURE 2.11. Each basic function ein satisfies the boundary condition ein{a) ^in(P) = 0 of the original problem (53). Hence un{a) = un(P) = 0 for all un G Xn. (55) The function un G Xn is piecewise linear and un{ai) = azn for all i = l,...,n. Thus, the space Xn consists precisely of all piecewise linear functions, with respect to the nod points a, ai,..., an, /?, which satisfy the boundary condition (55) (Figure 2.11(b)). Along with the original boundary-value problem we consider the variational problem F(u):= (2"V2 ~uf)dx = mini, uGW^(a,/?). (56) J a This induces the Ritz problem F(un) = min !, un G Xn, (57) which represents a minimum problem with respect to the real variables &im • • • 5 OLnn. If un is a solution of (57), then d da F{un) = 0, j = l,...,n. jn This yields the Ritz equation, rP rP / u'ne'jndx= / ejnfdx, Ja J a un G Xn, j = l,...,n. (58) Explicitly, we get the linear system '\ pp pp X,a™ / e'ine'jndx = / ejnf dx, j = 1 ,...,n,

2.7. Applications to Boundary-Value Problems 153 for the unknown real coefficients ain,..., ann of un. Proposition 3 (The Ritz method via finite elements). Let f G C[a,/?]. Then, the Ritz method (58) converges to the unique solution u of the original o boundary-value problem (53), in the sense of the Sobolev space Wl(aiP)- That is, for each n = 1,2,..., the Ritz equation (58) has the unique solution un and \\v>n — wlli,2 --* 0 as n —> oo. More precisely, for all n = 1,2,..., we get the following error estimates: (/3-a)~2 max \u(x) - un(x)\ < \\u - un\\E < hn\\f\\2, (59) a<x</3 If \u(x)~un(x)\2dx\ =||u-un||2<^||/||2, (60) where hn denotes the mesh size hn := fef • The proof will be based on the following result. Lemma 4. (i) Xn C Xe for all n. (ii) For all u G C2[a, (3] with u(a) = u((3) — 0 and all n = 1,2,..., distxE(u,Xn) < ftnll^lb- (iii) U^=i Xn is dense in Xe- Proof. Ad (i). This follows from Standard Example 11 in Section 2.5. Ad (ii). Set n ^n(#) ."= yj^(aj)ejn(^) &nd v(x) := u(x) — vn(x). i=i Then the function vn is piecewise linear and v(aj) = 0 for all j = 0,..., n + 1. Consider a fixed subinterval [aj,aj+i], j = 0, ...,n. Since v(aj) = v(aj+i) = 0, it follows from the mean value theorem of calculus that v'(£) = 0 for some £ G [aj,aj+i]. Hence we get the key formula v'{x) = I v"{y)dy for all x G [aj,aj+i].

154 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle This implies rdj + i \v'(%)\ < / i • W'{y)\dy f°r aiix £ [aj>aj+i]- By the Schwarz inequality, raj+i raj+i \v'(x)\2 < dy \v"\2dy J a,j J a j = hn / \v"\2dy. Ja3 Hence raj+i raj+i raj+i / \v'\2dx<h2n \v"\2dy = hl \u"\2dy, Ja j Ja j J a j since v!^ = 0 on [a^aj+i], by the linearity of vn on [aj,aj+i]. Summing over j, we get / |</|2<te < /l2 / |</fdx. J a J a Recall that v — u — vn. Since vn G Xn, this implies ft fP [distxE(u,Xn)]2 < \\u - vn\\2E = / \v'\2dx <h2n I \u"\2dx. J a J a Ad (iii). The set Cg°(a,/3) is dense in X =w\{a,(3). By Corollary 3 in Section 2.6, C§°(a,(3) is also dense in Xe- Thus, assertion (ii) implies (iii), since hn —> 0 as n —> oo. D o Proof of Proposition 3. We set X :=W^2(aJ^) an(^ 6(w) := / fudx. Ja It is convenient to use the energetic space Xe, which is equal to the linear space X equipped with the energetic inner product •P (u | v)e '•— / u'v* dx. Ja Recall that \\u\\e = {u \ u)^. Along with the variational problem 2-1(u\u)E-b(u) = mml, u e XE, (V) and the variational equation (u | v)e - b(v) = 0 for fixed u e Xe and all v £ Xe, (E)

2.7. Applications to Boundary-Value Problems 155 we consider the Ritz problem 2~1(un | un)E - b(un) = min !, un G Xn, (Vn) and the corresponding Ritz equation {un I vn)E - b(vn) = 0 for fixed un G Xn and all vn G Xn. (En) Problems (V), (Vn), and (En) correspond to (56), (57), and (58), respectively. By (50), Nh <(P~ <x)\\u\\e for all u G XE. According to the Schwarz inequality along with (50), \b(u)\ = I Ja fudx < H/II2NI2 <(P- <*)\\f\\2\\u\\E for all u G XE. Kence \\b\\E < (p - a)\\f\\2. Ad (59). Applying Theorem 2.C to (V) through (En), we obtain the convergence of the Ritz method along with the following error estimates: ||w-'Un||js =distxE(u,Xn) < hn\\u"\\2, n = 1,2,... , by Lemma 4. It follows from Proposition 1 that the solution u of (V) is also a solution of the original boundary-value problem (53), i.e., — u" = f on [a,/?]. Hence \\u-un\\E<hn\\f\\2. (61) By Example 10 in Section 2.5, max \u{x) -un{x)\ < (/? - a)^!^7 -i4||2 = (P ~ <x)*\\u -un\\E. a<x<(3 Ad (60). Let us consider the following two functions: w = solution of the original variational problem (V) with f := u — un\ wn = solution of the Ritz equation (En) with f := u — un. Replacing u, un, f with w, wn, u — un in (59), respectively, we get the key estimate ||W ~ ^n||jS < hn\\u - Wn||2. (62) Choosing v := u — un in the variational equation (E), we obtain 11^ "~ ^n||i — / (u ~ un)2dx — (w I u — un)E. J a

156 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle From (E) and (En) with v := wn and vn := wn we get (u | wn)E - b(wn) = 0 and (un \ wn)E - b(wn) = 0, respectively. This yields the decisive orthogonality relation (un-u\ wn)E = 0. Hence \\U ~ Un\\l = {U-Un\w- Wn)E. By the Schwarz inequality along with (61) and (62), \\u - un\\l < \\u - un\\E\\w - wn\\E < hl\\fh\\u - un\\2. This implies (60). □ 2.8 Generalized Functions and Linear Functionals We want to discuss the physical meaning of the Green function Q from Section 2.7 and its relation to the theory of generalized functions. 2.8.1 Special Force Densities Let us consider the basic boundary-value problem (46) in the special case where the force density fy^£(x)e is located around the given point ye]a,0[. That is, let us study the following boundary-value problem: -u"{x) = fy,e(x), a<x<p, u(a)=u(p) = 0, ueCPfap], where /y?e: [a,/?] —> R is continuous with f /^_/^° ifxe[y-e,y + e] for sufficiently small e > 0, along with (63) I (3 fVi£(x)dx = 1 (cf. Figure 2.12). By (48), the solution u = uy,£ of (63) is given through hAx)= G(x,z)fyi£(z)dz. (64)

2.8 Generalized Functions and Linear Punctionals 157 CxL I I' 1 *+ a y-s y y + s (3 FIGURE 2.12. Jy,> G(.,y) a y-e y y + e j3 (a) (b) FIGURE 2.13. In terms of physics, uy,e (x) represents the deflection of a string at the point x under the vertical outer force density *y,e\x) = Jy,e\x)®'> with the unit vector e, i.e., the force = Ij fyAx)dxJ (65) acts onto each subinterval [7, #] of [a,/3] (cf. Figure 2.13(a)). Let us now consider any fixed sequence {fy,e)e>o, &nd let us study the limiting process e —> +0 (cf. Figure 2.14). Since the Green function Q: [a,/?] x [a,/?] —> R is continuous, the mean value theorem along with (64) tells us that uy,e(x) = g{x, y), where y-e<y<y + e. This yields the classical key relation G(x, y) = lim Uyi6(x) for all x, y G ]a, /?[. (66) -z—1—v 1—- -<—e. a y 6y FIGURE 2.14.

158 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle Along with (65) we get the following physical interpretation of the Green function Q\ For fixed y G ]a, /3[, the function #H-» Q(x,y) describes the deflection of a string caused by the vertical unit force F = e, (67) which is concentrated at the point y (cf. Figure 2.13(b)). 2.8.2 Formal Approach of Physicists via the Dirac 6-Function In order to describe formally the point force F from (67) by a "force density 6y" physicists introduce the Dirac 6y-function defined by along with jfW"{j Jj^ («,) and they formally write fy,e(%) -* 6y(x) as £ —► +0 for all x e [a,/?] (68c) (cf. Figure 2.14). By (68b), /6 p6 fVi£(x)dx= / Sv(x)dx, for all subintervals [7,6] of [a,/?]. Finally, with e —> +0 it follows formally from (63) and (66) that -^|J^=W, «<*</?, (69) for each y £ ]a,/?[. Obviously, there is g(a,y)=g(/3,y), no classic function 6y that satisfies relations (68a) and (68b). In the following we want to show that 6y can be regarded as a generalized function. The theory of generalized functions allows us to justify rigorously equation (69).

2.8 Generalized Functions and Linear Functional 159 Let </>:[a,/?] —> R be continuous. Formally, we have "(cj)6y)(x) = (j)(y)6y(x)" since "8y(x) = 0 for x =^ y." Hence /•£ /•£ / <t>{x)6y{x)dx = (j>(y) / 6y(x)dx = 0(2/). (70) The rigorous definition of 6y will be based on this formal relation. 2.8.3 Rigorous Approach via Generalized Functions In this section, G denotes a nonempty open set in R^, N > 1. Definition 1. Let x = (£i,..., £;v) and cfy := d/d£j. By a multiindex a = (ai,..., ajv)? we understand a tuple of nonnegative integers ai,..., a;v. We set |a| := ai H + a^ and dau:=d^'^d%Nu, i.e., a u ■ For a = (0,..., 0), we set dau := it. Proposition 2 (Integration by parts). For all u, v G Cq°(G) and all mul- tiindices a, / udavdx = (-l)|a| / (fl^vdx. (71) Jg Jg Proof. Use repeatedly the classic formula / udjvdx = - / (dju)vdx for all u, v G Cq°(G). Jg Jg For example, if <9ai£ = didjU, then / (didju)vdx = — / djudivdx = / u(djdiv)dx, Jg Jg Jg for all «,«€ Cg°(G). This is (71). □ We set V{G) := C0°°(G). Let </>n, 0 G V{G) for all n. We write <j)n -» 0 in ©(G) as n -► oo (72) iff

160 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle (i) there exists a compact subset K of the open set G such that 0n(x) = 0 for all x G G — K and all n, and (ii) we have the uniform convergence da(j)n z3 da(j) on K for all multiindices a, i.e., max \da(j)n(x) — da(j)(x)\ —> 0 as n —> oo for all multiindices a. x6KT If G = ]a, /?[, then we set £>(a, /?) := V(G). Fundamental Definition 3. By a generalized function U G V'(G), we understand a linear, sequentially continuous functional U:V(G)-*R, with respect to the convergence (72), i.e., U{a4 + hj)) = aU((/)) + bUty) for all 0, i\) G ©(G), a, & G R, and, as n —> oo, 0n -^ 0 in P(G) implies U((j)n) -> U(</>). The theory of generalized functions (also called distributions) was created by Laurent Schwartz around 1950. We want to show first that a broad class of functions can be identified with generalized functions. More precisely, we want to show that L2(G) C V{G), (73) i.e., the Hilbert space L,2(G) can be identified with a linear subspace of the linear space W{G). Standard Example 4. For u G L2(G), we define U((j)) := f u(x)cj)(x)dx for all 0 G V{G). (74) Jg Then (i) U is a generalized function, i.e., U G V'{G). (ii) If u = v in the Hilbert space L2(G), then U = V in P7(G).

2.8 Generalized Functions and Linear Functional 161 (iii) The map u\-^U from L2{G) into W(G) is injective. Proof. Ad (i). Obviously, the functional U:V(G) —> R is linear, and (j)n —> <j> in V(G) as n -^ oo implies (j)n —> 0 in L2(G), and hence U(4>n) —> ?/(</>) as n —> oo. Ad (ii). It follows from tt(x) = ^(x) for almost all x G G that / u(x)(j)(x)dx = / ^(x)0(x)dx for all 0 G D(G). Jg Jg Ad (iii). Let it, v G L2(G), and suppose that [/(</>) = / tt(x)0(x)dx = / v(x)(j)(x)dx for all <j> G £>(G). This implies / (u(x) - v(x))(j)(x)dx = 0 for all <j> G £>(G). Jg By the variational lemma from Section 2.2.4, we get u = v in L2(G). □ Standard Example 5. Let y G G. We set «y(0) := 0(2/) ^ all 0 G V(G). (75) Obviously, this is a generalized function, which we call the Dirac 6y-distribution. In the special case where G = ]a,/3[, definition (75) is motivated by the formal relation (70) of physicists and by (74). General Strategy 6. The basic definitions in the theory of generalized functions are chosen in such a way that they are generalizations of the corresponding definitions for functions via relation (74). As a typical example, let us consider the derivative of a generalized function. Definition 7. For a generalized function U G W(G), the derivative daU is defined through (daU)((j>) := (~l)WU(da(j)) for all (j> G V(G), (76) and for all multiindices a.

162 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle In order to motivate this formula, assume that U corresponds to the function u G V{G) and V corresponds to dau, in the sense of Standard Example 4, i.e., for all </> G V(G), U{(j)) = [ ucj)dx and V(</>) = [ (dau)<j)dx. Jg Jg Integration by parts yields / (dau)(j)dx = (-l)W / uda(j)dx. Jg Jg Hence V = dau. Proposition S.IfUe V{G), then daU G V'{G) for all multiindices a. Proof. The functional daU: V{G) -* R is linear. In fact, for all </>, i/j G V{G) and all a, b G R, (daU)(a$ + fa/0 = {~l)MU{ada(j) + bda^) = (-l)H (aU(da$) + bU(da^)) = a(daU)(<j>) + b(daU)(*/;). Furthermore, as n —> oo, it follows from K - 0 in ©(G) that <9a0n -* <9a</> in D(G), and hence f/(cP</>n) - u(da<i>), i.e., (d"U){<l>n) - (^f/)(0). □ In contrast to classic functions, generalized functions possess derivatives of arbitrary order. Standard Example 9. For fixed y G R, define the function tt:R —> R through Then £/'=«„ in£>'(R), where U denotes that generalized function that corresponds to u. Proof. For all </> G D(R), /OO tt(x)0(x)c?x. -CXD

2.8 Generalized Functions and Linear Functionals 163 Hence /•oo U'{4>) = -U{4>') = - / 4>'{x)dx = 4>(V) = Sy(<f>), since the function <j> vanishes outside some compact interval. □ Definition 10. Let Un, U G V'{G) for all n. We write Un-+U as n -* oo in V'{G) (77) iff f/n(0) -> U((j)) as n -> oo for all 0 G £>(G). Proposition 11. If un, u G L2 (G) and all n and un —> u in L,2(G) as n —> 00, £/ien relation (77) fto/ds £n/e. Proof. Observe that £>(G) C L2(G). Hence, for all 0 G £>(G), / un(j)dx —> / ./G ^G tt(/)dx as n —> 00. □ G Standard Example 12. Let —00 < a < (3 < 00 and 2/ G ]a, /3[. Suppose that we are given a family (/y,e) of continuous functions /y?e: ]a, /?[—> R as considered in Section 2.8.1. Then, Fyi6 -* 8y as e —> +0 in D(a, /?), where Fy?e denotes the generalized function that corresponds to /y?e. This is a rigorous formulation of (68c). Proof. Let <j> G T>(G). By the mean value theorem, there is a y G [2/—£, 2/+e] such that ^/,e(</>) = / fyAx)^ix)dx = ( / /y,e(«)^J 0(2/) = 0(2/)- This yields hmoFv,£(cl>) = cj>(y). D Applications to electrostatics can be found in Problem 2.9.

164 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle 2.8.4 Applications to the Green Function Proposition 13. Let Q = Q(x,y) denote the Green function as defined in (48). Then, for each y G ]a,/?[, -U" =6y <mV(a,0), where U denotes the generalized function that corresponds to the classic function u(x) := G(x,y) for all x G [a, (3]. This result rigorously justifies equation (69). Proof. For simplifying computations, we set a — 0 and (3 = 1. The general case proceeds analogously. Let <f> G D(0,1). Then Jo Q(x,y)(j)(x)dx. Jo By (48), U"(</>)= [ g(x,y)ci>"{x)dx Jo = [ (l-y)x(l)"(x)dx+ [ {l-x)y<l)"{x)dx. JO Jy With 0(0) = <j)'(0) = (f)(1) = <j)'(l) = 0, integration by parts yields t/"W = (1 - v)v4>'(v) - /V - v)4>'{x)dx Jo - (1 - y)y<t>'{y) + / y<j)f(x)dx Jy = -(1 - y)(S>{y) - y(S>{y) = -<j>{y) = -«y(0). D 2.8.5 Generalized Derivatives Recall that G denotes a nonempty open set in R^, N > 1. Definition 14. Let u,w G L2(G), and let a denote any multiindex. We write w = dau (78) iff this relation holds for the corresponding generalized functions, i.e., W = DaU. Explicitly, / wcj)dx = (-I)'"' f uDacj)dx » for all </> G V(G). (79) Jg Jg

2.9 Orthogonal Projection 165 If (78) holds true, then we call the function w a generalized derivative of the function u of type da. This generalizes Definition 3 in Section 2.5.2. Proposition 15. The generalized derivative w from (78) is uniquely determined up to the values of w on a set of N-dimensional measure zero. Moreover, w is uniquely determined as an element of the Hilbert space L2{G). Proof. Let u,v,w G L2(G), and let w = dau as well as v = dau. By (79), / (v - w)</>dx = 0 for all 4> G £>(G), Jg and the variational lemma from Section 2.2.3 implies w(x) = v(x) for almost all x G G. Hence w = v in L2(G). D 2.9 Orthogonal Projection We consider the minimum problem ||ifc-v|| =min!, v G M, (80) and make the following assumptions: (H) Let M be a closed linear subspace of the real or complex Hilbert space X, and let u G X be given. Figure 2.15 shows the geometrical meaning of (80), i.e., we seek the foot v of a perpendicular from the point u to the plane M. Let M1- denote the orthogonal complement to M, that is, by definition, ML := {weX:(w\v) = 0 for all v G M}. Theorem 2.D (The perpendicular principle). Assume (H). Then, the minimum problem (80) has a unique solution v, and u — v G M1-. Corollary 1 (Orthogonal decomposition). // (H) holds, then there exists a unique decomposition of u of the form u = v + V v G M, we ML.

166 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle FIGURE 2.15. Proof of Theorem 2.D. Since \\u — v\\2 — (u — v I u — v) = (u I u) — (u I v) — (v I u) + (v I v), problem (80) is equivalent to 2~1a(v, v) - b(v) = min !, v G M, (80*) where a(v,w) := Re(v \ w), b(v) := 2_1[(^ | v) + (v \ u)] — Re(u \ v). Note that a{v,v) = (v \ v) for all v G X. By the Schwarz inequality, |a(v,w)| < \\v\\ \\w\\, a(v,v) > \\v\\2, \HV)\ ^ ll^ll II^IIj for all v,w G X. First let X be a real Hilbert space; then M is also a real Hilbert space. It follows from Theorem 2.A that problem (80*) has a unique solution v. Hence the original problem (80) has also a unique solution v. Now let X be a complex Hilbert space. Then, X becomes a real Hilbert space with respect to the new inner product (v | w)* := Re(v \ w) for all v, w G X. Again by Theorem 2.A, problem (80*) has a unique solution v, and hence (80) has a unique solution v. Finally, let X be a Hilbert space over K. We want to show that u — ve M1-. Since v is a solution of (80), we get \\u - v\\2 < \\u - (v + Aw) ||2 for all w G M, AG K. Hence (u — v\u — v)<(u — v\u — v) — X(u — v\w) — X(w \u — v) + \X\2(w | w).

2.10 Linear Functional and the Riesz Theorem 167 Suppose that u — v ^ 0 and w ^ 0. Letting A := [M2 > we Set 0 < — |(ia — v | tf)|2, and hence (u — v \ w) — 0 for all w E M. This remains true if ix — v = 0. □ Proof of Corollary 1. The existence of such a decomposition follows from Theorem 2.D. To prove the uniqueness of the decomposition, let u = v\ + wi, v\ G M-1, w\ G M be a second decomposition of u. Then 0 = (v — vi) + (w — wi), v — v\ e M, w — w\ e M1-. Hence 0 = (v — v\ \w — w\) = —(v — v\ \v — vi), i.e., v = v\ and w = w\. D Corollary 2 (The Pythagorean theorem). If u is orthogonal to v, i.e., (u | v) = 0, then \\u + v\\2 = \\u\\2 H- \\v\\2 (cf. Figure 2.6(b)). Proof. \\u + v\\2 — (u + v | u + v) = (u \ u) + (u \ v) + (v \ u) + (v \ v) = (u | u) -f (v | v). D 2.10 Linear Functional^ and the Riesz Theorem Theorem 2.E (The Riesz theorem). Let X be a Hilbert space over K, and let X* denote the dual space of X. Then, f G X* iff there is a v G X such that f(u) = (v\u) for all ueX. (81) Here, the element v of X is uniquely determined by f. In addition, 11/11 = HI- (82) Proof. Step 1: Uniqueness of v. It follows from (v | u) = (vi | u) for all u G X that (v — v\ | u) = 0 for u = v — v\, and hence v = V\.

168 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle Step 2: Existence of v. Let / G X* with / ^ 0. The null space N(f):={ueX:f(u)=0} is a closed linear subspace of X. In fact if f(un) = 0 for all n and un —> u as n —» oo, then /(ix) = 0, by the continuity of /. According to the orthogonal decomposition theorem (Corollary 1 in Section 2.9), there exists an element uQ G N(f)-1 with uQ ^ 0. Otherwise we would have N(f)± = {0}, and hence N(f) = X. But this is impossible because of / ^ 0. Since u0 0 N(f), we get f(uo) ^ 0. Without any loss of generality we may assume that f(uo) = 1. This implies f(u — f(u)uo) = 0 for all u £ X, i.e., u — f(u)uo G N(f). Hence we obtain the orthogonal decomposition u = w + f(u)uo, w G N(f), u0 G Ntf^. (83) Inner multiplication by Uq yields (uq I u) = f(u)(uo I Uq) for all u £ X. This implies (81) with v := j£fa. If / = 0, then (81) holds with v = 0. Step #: Conversely, if / is given through (81), then / G X*. In fact, / is linear because f(au + /?w) = (v \au + j3w) = a(v \ u) + j3{v \ w) = af(u) + 0f(w) for all v,w e X, a, (3 G K. Furthermore, the continuity of / follows from \f(u)\ = \(v | u)\ < \\v\\ \\u\\ for all ueX. (84) Step 4- By (84), ||/|| < \\v\\. Furthermore, f(v) = \\v\\ \\v\\. Hence ||/||= sup |/(u)| = |M|. □ lltill<l Equation (83) tells us the following fundamental geometrical fact: If f is a nonzero linear continuous functional on a Hilbert space, then the null space N(f) of f is a closed plane and its orthogonal complement N(f)± has dimension one, i.e.. dim N(f)-1 = 1.

2.12 Duality for Quadratic Variational Problems 169 2.11 The Duality Map Definition 1. Let X be a Hilbert space over K. We define the duality map J:X ->X* of X through J(v) := /, where / is given by (81). Using the notation (f,u) = f(u) for / G X* and u G X, this means (J(v),u) := (v | u) for all u, v G X. Proposition 2. The duality map J is bijective, continuous, and norm preserving, i.e., \\J(u)\\ = \\u\\ for all u G X. If X is a real Hilbert space, then J is linear. If X is a complex Hilbert space, then J is antilinear, i.e., J(av + j3w) = aJu + J3 Jw for all a,/3eC, u,w G X. Proof. This follows from Theorem 2.E. □ The duality map will be used critically in Section 5.4 in connection with the energetic extension of symmetric operators. This allows important applications to mathematical physics. 2.12 Duality for Quadratic Variational Problems Along with the original minimum problem F(u) := a(u, u) - 2b(u) = min !, ueX, (85) let us consider the dual maximum problem F*(v):=-a(v,v) = maxl, vev0 + Y, (85*) where vq G Z is a fixed solution of the following equation: a(vo, w) = b(w) for all w G X. We make the following assumptions: (HI) Let X be a linear closed subspace of the real Hilbert space Z. (H2) Let a: Z x Z —» R be a bounded, symmetric, positive bilinear form.

170 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle (H3) Let a:X x X —>Rbe strongly positive, i.e., cIMIz — a(^'u) ^or a^ u ^ ^ an<^ fixed c > 0. (H4) The functional b: X —» R is linear and continuous. Finally, we set Y := {u G Z: a(u, w) = 0 for all w G X}. Theorem 2.F (Duality). The original minimum problem (85) has a unique solution Uq. The element Uq is also the unique solution of the dual maximum problem (85*), and the extremal values of (85) and (85*) are the same, i.e., F(0 = F*«). Moreover, we have a(u -v,u-v) = F(u) - F*(v) for allueX, vev0 + Y. (86) Corollary 1 (Error estimates). For all u G X and v G vo + Y, we get F*(v)<F(u0)<F(u) (87) and c\\uq - u\\2z < a(u0 - u,uq - u) < F(u) - F*(v). (88) In numerical analysis, one computes u and v in Corollary 1 as solutions of the Ritz method for (85) and (85*), respectively. The Ritz method for the dual problem (85*) is also called the Trefftz method. Proof. By Theorem 2.A in Section 2.4, problem (85) has a unique solution uq that satisfies the variational equation a(u, Uo) = b(u) for all u € X. By construction of Y, a(u, v — Vo) = 0 for all u G X, v Gvo + Y, and the choice of ^o yields the key relation a(u, v) = a(u, vq) = b(u) for all u G X, v Evq + Y.

2.12 Duality for Quadratic Variational Problems 171 Hence 0 < a(u — v1u — v) = a(u, u) — 2a(u, v) + a(v, v) = a(u, u) — 2b(u) + a(v, v) = F(u) - F*(v) for all u G X, v G v0 + Y. (89) This implies F*(v) < F{u0) for all vev0 + Y. Furthermore, we shall show that uq G vq + Y and F*(uo) = F(uo). (90) Thus, Uq is a solution of the dual problem (85*). To prove (90) observe that a(uo,w) = b(w) and a(vo,w) = b(w) for all w G X. Hence a(uo — Vq^w) = 0 for all w G X, i.e., uq — Vq G Y. Furthermore, letting u = v = Uq in (89), we get (90). Corollary 1 is an immediate consequence of Theorem 2.F. □ Example 2. Let X be a linear closed subspace of the real Hilbert space Z. For given vq G Z, we consider the minimum problem F(u) := ||v0 - ^ll2 ~ INH2 = min !, u e X, (91) together with the dual maximum problem F*(w) := -||v0 - ^||2 = max !, we X1-, (91*) where || • || denotes the norm on Z. Then, problem (91) has a unique solution uq, and vq — Uq is the unique solution of (91*). Moreover, F(uo) = F*(v0-uo). (92) The geometrical meaning of the result is pictured in Figure 2.16. Relation (92) is identical to the Pythagorean theorem \\vo\\2 = \\Uq\\2 + ||v0 - ^o||2, ^0 G X, Vq - Uq G X1. Proof. Use Theorem 2.F with a(u,v) := (u | v)z and b(u) := (vq \ u)z> Here, Y = X±. D

172 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle ^X FIGURE 2.16. 2.13 The Linear Orthogonality Principle The following three existence principles are mutually equivalent: (i) the existence principle for quadratic minimum problems (Theorem 2.A); (ii) the perpendicular principle (Theorem 2.D); (iii) the Riesz theorem (Theorem 2.E). These three principles represent variants of the linear orthogonality principle in Hilbert spaces. In the preceding sections we have already proved that (i) =* (ii) =* (in). It remains to show that (iii) => (i). To this end, we consider the minimum problem 2~1a(u, u) - b(u) = min !, ueX, (93) as in Section 2.4. Let us introduce the energetic inner product on X through (u | v)e '= a(u, v) for all u, v G X. By Definition 2 in Section 2.6, the energetic space Xe consists of the set X equipped with (• | •)#, and Corollary 3 in Section 2.6 tells us that Xe is a Hilbert space. Moreover, there are positive constants a and j3 such that cHMU < \\u\\x < /3|MU for all u e X. By assumption, the linear functional b: X —» R is continuous. Hence |fc(t*)| < ll&ll Nix < /?||fc|| Nb for all ueX. That is, b(-) also represents a linear continuous functional on Xe- By the Riesz theorem, there is a v G Xe such that b(u) = (v | u)e for all u G XE.

2.14 Nonlinear Monotone Operators 173 Consequently, problem (93) can be written in the following form: 2~1(u | u)e — (v | u)e = nrin !, u G Xe- This is equivalent to the problem (u — v\u — v)e = (u I u)e — 2{u | v)e + (v | v)e = niin !, u G XE, which has the unique solution u = v. 2.14 Nonlinear Monotone Operators We want to solve the nonlinear operator equation Au = z, ueX. (94) We make the following assumptions: (HI) The operator A:X —» X is strongly monotone on the real Hilbert space, i.e., by definition, there is a constant c > 0 such that (Au — Av | u — v) > c\\u — v\\2 for all u, v G X. (H2) The operator A is Lipschitz continuous, i.e., there is a constant L > 0 such that \\Au - Av\\ < L\\u - v\\ for all u, v G X. Theorem 2.G. For each given z G X, problem (94) has a unique solution u. This theorem was proved by Zarantonello in 1960. It marks the beginning of the modern theory of monotone operators that allows many applications to nonlinear mathematical physics. This can be found in Zeidler (1986), Vols. 2, 4, and 5. Proof. We will use the Banach fixed-point theorem. The idea of our proof is to replace the original equation (94) by the equivalent fixed-point problem u = Bu, ueX, (95) where Bu := u — t(Au — z) for fixed real t > 0. If X = {0}, then the statement is trivial. Let X ^ {0}. For all u, v G X, \\Bu-Bv\\2 = \\u-v\\2-2t(Au-Av \ u-v) + t2\\Au-Av\\2 < m||^-i;||2, (96)

174 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle where ra=: l-2tc + t2L2. By (96), m > 0. If t = 0 or t = |§, then m = 1. This implies /c := v^- < 1 f°r all * £ Therefore, \\Bu — Bv\\ < k\\u — v\\ for all u, v G X, i.e., the operator 5 is ^-contractive for each t G ]0, ■£§[. By the Banach fixed-point theorem in Section 1.6, problem (95) has a unique solution u. □ In addition, it follows from the Banach fixed-point theorem that, for each given uq G X and each fixed t G ]0, jrt[, the iteration method un+i =un- t(Aun - z), n = 0,1,... converges to the unique solution u of the original problem (94). Moreover, we have the error estimates ||^ — un\\ < kn(l — fc)-1||^i — Uo\\ for n = 1,2,... . 2.15 Applications to the Nonlinear Lax-Milgram Theorem and the Nonlinear Orthogonality Principle We want to solve the equation a{u, v) = b(v) for fixed u G X and all v G X. (97) We make the following assumptions: (HI) Let b: X —» R be a linear continuous functional on the real Hilbert space X. (H2) Let a:IxI-4Rbea function such that, for each w G X, v k+ a(w,v) represents a linear continuous functional on X. (H3) There are positive constants L and c such that, for all u, v, w G X, c||ix — v||2 < a(ix, ia — v) — a(v, ix — v) and |a(w, w) — a(v, w)\ < L\\u — v\\ \\w\\. „ 2c

Problems 175 Theorem 2.H (The nonlinear Lax-Milgram theorem). Problem (97) has a unique solution. Proof. By (H2) and the Riesz theorem in Section 2.10, for each w G X, there is an element called Aw such that a(w, u) = (Aw | u) for all u G X. This way we get an operator A: X —» X. It follows from (H3) that c\\u — v\\2 < (Au — Av | u — v) for all u,v E X, i.e., A is strongly monotone. Furthermore, \(Au — Av | w)\ < L\\u — v\\ \\w\\ for all u,v,w G X. Hence ||Aia —Av|| = sup \(Au — Av | w)\ < L\\u — v\\ for all ix,v e X. \\w\\<l Again by the Riesz theorem, there is a z G X such that b(u) = (z | u) for all uGX Consequently, the original problem (97) is equivalent to the operator equation Au = z, ueX. (98) It follows now from Theorem 2.G in the preceding section that equation (98) has a unique solution u. □ In the special case where a: X xX —► R is bilinear, bounded, and strongly positive, i.e., a(w,w) > c||w||2 for all w G X and fixed c > 0, assumptions (H2) and (H3) are satisfied. Then, Theorem 2.H is called the linear Lax-Milgram theorem. If, in addition, a(-, •) is symmetric, then problem (97) is identical to the variational equation from Theorem 2.A. By Section 2.13, it is motivated that Theorem 2.H can be regarded as a nonlinear orthogonality principle on Hilbert spaces.

176 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle Problems In Problems 2.9ff the importance of generalized functions for mathematical physics will be studied on an elementary level. Moreover, in Problems 2.12ff we will study in detail a smoothing technique that plays a fundamental role in modern analysis (Friedrichs' mollification). 2.1. Weierstrass' classical counterexample from 1870. Consider the minimum problem F(u) := f {xu'{x)fdx = min !, u G C^-l, 1], u(-l) = 0, u(l) = 1. (P) Use the sequence , x 11 arctan nx ^ n un(x) := o + o 1 ' n = 1,2, ..., 2 2 arctan n in order to show that this variational problem has no solution. Recall that C1[—1,1] denotes the space of continuously differentiable functions u:[-l,l] ->R. Solution: Set M := {u G C^-l, l]:u(-l) = 0 and u(l) = 1}. Then, problem (P) can be written in the following form: F(u) = min !, ue M. Since un(—1) = 0 and un(l) = 1, we get un G M for all n. Explicitly, w< a - 1 f1 n2x* *[Un) ~ 2n2 • arctan n J_x (1 + (nz)2)2 = * r y2 d < i r y2 d 2n2 - arctan n J_n (1 + y2)2 ~ 2n2 • arctan n J_00 (1 + y2)2 Hence F(un) —» 0 as n —» oo. Since F(ix) > 0 for all ix G M, this implies inf F(u) = 0, i.e., (ixn) is a minimal sequence for (P). Suppose now that u is a solution of (P). Then, F(u) = 0, weM, and hence xu'(x) = 0 for all EG [-1,1]. This implies v!(x) = 0 on [—1,1], i.e., ia(sc) = const. But, this contradicts the side condition u(—l) = 0 and u(l) = 1.

Problems 177 This example was given by Weierstrass to show that a minimum problem in the calculus of variations need not always have a solution, namely, The infimum of the functional F on the set M is not attained at some point u of M. We shall discover in Chapter 2 of AMS Vol. 109 that the reason for this bad structure of (P) is related to the fact that the Banach space C1[—1,1] is^ not reflexive. A detailed historical discussion can be found in Zeidler (1986), Vol. 2A, Sections 18.7 through 18.9. 2.2. The classical Hilbert space if. By definition, the space if consists of all the sequences (un)n>i with un G K for all n G N and oo Y^\Un\2 < OO. n=l The linear operations are defined as in Problem 1.5 for K°°. Show that if is an infinite-dimensional Hilbert space over K equipped with the inner product oo (u | v) := ^2,unvn, n=l where u := (un) and v := (vn). As usual, the bar denotes the conjugate complex number. Hint: Apply the limiting process N —» oo to the classical Schwarz inequality IN I / N \5 /JV \i \^unvn\<l^2\un\2\ f^KI2] , |n=l I \n=l / \n=l / which corresponds to the Hilbert space KN (cf. Standard Example 1 in Section 2.2). Hilbert introduced the space I2 in 1906. He used this space in order to establish his general theory of integral equations (cf. Hilbert (1912)). The notion of an abstract Hilbert space was introduced by von Neumann in 1929. 2.3. Simple identities. Let X be a pre-Hilbert space over K with the inner product (• I •). Show that the following hold true: (i) If K = R, then 4(u I v) = \\u + v\\2 - \\u - v\\2 for all u,v e X. (99a)

178 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle (ii) If K = C, then 4(u | v) = \\u+v\\2-\\u-v\\2-i\\u+iv\\2+i\\u-iv\\2 for all u,v G X. (99b) (iii) Appolonius' identity. If K = R, C, then \\w - u\\2 + \\w - v\\2 = 2~1\\u - v\\2 + 2\\w - 2~1(u + v)\\2 (99c) for all u,v,w e X. 2.4. The Banach space C[a,b], Let —oo < a < b < oo. Show that the Banach space C[a, b] equipped with the usual maximum norm \\u\\ = maxa<x<6 \u(x)\ is not a Hilbert space. Hint: Prove that the parallelogram identity is violated. Use the same method in order to show that the Banach space C of complex numbers equipped with the norm (|x + iy\[= \x\ + \y\ for all x + iy e C is not a Hilbert space. 2.5.* The role of the parallelogram identity. Let X be a normed space over K. Show that X is a pre-Hilbert space iff the parallelogram identity holds, i.e., 2||ifc||2 + 2||v||2 = ||u-v||2 + ||^ + ^||2 for all %v G X. Hint: Use (99a) and (99b). Cf. Jordan and v. Neumann, On inner products in linear metric spaces, Annals of Math. (1935), 719-723. 2.6. Complexification of real Hilbert spaces. Let X be a real pre-Hilbert. As in Problem 1.9, consider the complexification Xc of the space X, where Xc consists of all the elements u + iv with u, v G X. Show that (i) Xc becomes a complex pre-Hilbert space with the inner product 4- (u + iv \w + iz) := (u\ w) — i(v \ w) + i(u \ z)(v \ z) for all u + iv, w + iz G Xc. (ii) If X is a real Hilbert space, then Xc is a complex Hilbert space. 2.7. Orthogonal complements. Let L be a linear subspace of the Hilbert space X over K. Set LLL := (L1)1. Show that

Problems 179 (a) L = LLL. (b) Lis closed iff L = LLL. 2.8. The Ritz method. By Section 2.7.1, the variational problem (2"V2 - u cos x)dx = mini, ugC2[0,tt], u(0) = u(tt) = 0 (V) is equivalent to the boundary-value problem u"(x) + cosx = 0 on [0, tt], u(0) = u(tt) = 0, (B) which has a unique solution u. Explicitly, u(x) = cosx + 27r_1x — 1. Use the Ritz method in order to compute an approximate solution U2n of (V), by making the ansatz 2n U2n{x) = ^2, Ck Sin kx' fc=l Determine the coefficients ci,..., c2n. Show that (^2n) converges uniformly on [0,7r] to the solution u of (V). Hint: Cf. Zeidler (1986), Vol. 2A, p. 94. 2.9. Applications of generalized functions to mathematical physics. In the following, we set 6 := <S0, i.e., 6(</>) = <p{0) for all 0 G C^(RN). We want to use very elementary arguments in order to explain the importance of generalized functions for mathematical physics. In fact, the theory of generalized functions allows us to justify many classical heuristic arguments of physicists (for example, see Problems 2.9b and 2.9f on applications to electrostatics). 2.9a. A special fundamental solution. Show that the equation U{n) =« onl, n = l,2,... , has a solution U G £>'(R) that corresponds to the function u{x) = [&y. ifa^°> 10 if x < 0. Hint: Use a similar argument as in Standard Example 9 from Section 2.8.

180 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle 2.9b. The electric field of a charged point. Let us first consider the classic basic equation of electrostatics in a vacuum: -€QAu = p on R3. (100) Here, p denotes the charge density and £o denotes the so-called dielectricity constant in a vacuum. The function u is called the electrostatic potential caused by the charge density p. If there is a particle P at the point x of charge Q, then the force K acts on P, where K = QE(x) with E(x) = -grad u(x). Here, the vector E(x) is called the electric field at the point x. Suppose now that p corresponds to a point of charge q at the origin x = 0. Then, formally we get P{X) too iix = 0 and JR3 p(x)dx = q. Thus, p corresponds to the Dirac distribution q6. Consequently, let us replace the original equation (100) with the equation -e0AU = q6 on R3, (101) in the sense of generalized functions. Show that equation (101) has a solution U G £>'(R3) that corresponds to the function «<*> = i^R • <102> Moreover, show that —grad U corresponds to the classical vector field This is the classic electric field caused by a charged point at the origin with charge q. The corresponding force K = QE(x) is the Coulomb force. Solution: Ad (101). To simplify notation, set q = 47T and £o = 1- Let Ufa) = f &?. for all 4> e C0°°(R3). ./k3 \x\ By (101), we have to show that -(AC/)(0) = 4tt<S(0) for all <p G C£°(R3). Since A = d\ + <9f + $35 the definition of derivatives of generalized functions yields -t/(A0) = 4tt<S(0) for all 0 G C^(R3). Explicitly, this means that ^P^-dx = 4^(0) for all <f> € C%°(R3). (104)

Problems 181 To prove this, set G := {x G M3:0 < r < \x\ < R}. Choose R so large 2 ' that <j>(x) = 0 if \x\ > §. Using spherical coordinates, we get dx = r2 sin d dr dd d<fi. Hence f &pz)_dx= f f f ^rsmddrdddcj). Jr3 \x\ Jo Jo Jr=o Thus, JR3 \X\ r-+OjG \x\ Integration by parts yields Jg r Jg \mJ JdG \\x\ dn 9n \\X\J J where ^ denotes the derivative in the direction of the outer unit normal vector, i.e., ^ = —^ for \x\ = r. Since Af-rM =0onG and 4> vanishes in a neighborhood of the set {x: \x\ — R}, we get By the mean value theorem, there are points y and z with \y\ = \z\ = r such that J = 4ir(-r</>r(y) - (f)(z)) —► -47T0(O) as r —► 0. This is (104). Ad (103). By the definition of the derivative of generalized functions, we have to show that / E(x)(j)(x)dx = -(grad U)(4>) = C/(grad <j)) J®* = / ?^(b forall0GCo-(R3), JR3 Air€0\x\ where E(x) = ^^ui- However, this follows as above by using integration by parts. 2.9c. Special fundamental solutions. Justify Table 2.1.

182 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle Table 2.1 Differential equation Fundamental solution (a; G I3) -AC/ = 6 on R3 u(x) = -Vr 47r|x| (potential equation) (potential) {—|x|^/4t 8(7Tt)3/2 if * > 0 0 if t < 0 (heat equation) (temperature) Utt ~ AC/ = S on R4 U(<f>) = j- [ t'1 I [ <f>(x, t)dOx | dt. 47r Jo \J\x\=t J (wave equation) 2.9d. Convolution of generalized functions. Let Li)o(Riv) denote the set of all measurable (e.g., continuous) functions f:RN —> 1^ that vanish outside some compact set and that are integrable, i.e., jRN \f(x)\dx < oo. Let F denote the generalized function corresponding to /, i.e., F(<f>) = f f(x)(f)(x)dx for all <j) G C% Jr" oo/irpiV )• For all generalized functions Ug V'(Rn) and all / G Lit0(RN), we define the convolution U * F through (/ (U*F)((f)) = U( / f(x)(f)(x + y)dx) forall0GCooo(RiV). Show that (i) U*FeW( (ii) 6*F = F. (iii) Da{U*F) = (DaU)*F for all derivatives Da. (iv) (/3iC/i + (32U2)*F = PxiUx+F) + P2(U2^F) for all /?i,/?2 G R. Convince yourself that this follows simply from the corresponding definitions.

Problems 183 2.9e. The importance of fundamental solutions. Let L denote any linear differential operator of order m = 1,2,... with real coefficients, i.e., L:= J2 a<*D(*' (105) \&\<m where aa G R for all a. Suppose that U G V'(RN) is a fundamental solution of L, i.e., LU = S on R*. Let / G Li^R^)- Show that the convolution V = U * F is a solution of the nonhomogeneous equation £7 = F on R*. Solution: By Problem 2.9d, L(U*F) = (LU)*F = 6*F = F. 2.9f. Applications to electrostatics. We are given the charge density p G Li)0(R3) H L2(R3) (e.g., p is continuous and vanishes outside some large ball B). Let p G £>'(R3) denote the generalized function corresponding to p. Show that the generalized function V G £>'(R3) corresponding to the classic function /* nlni\rlii x G R3, (106) v(x) = I /R3 47re0|a-2/r is a solution of the fundamental equation of electrostatics: -e0AV -p on R3. (107) The function v from (105) is called the classic volume potential. Recall from classic analysis that v is a classic solution of (107) provided p is sufficiently smooth (e.g., p G C^R3)). Solution: Using spherical coordinates, it follows as in Problem 2.9b that •It dy IggJb \x~y\2 SUp / -j rx < 00, where B and G denote arbitrary balls in R3. By the Schwarz inequality, |/f^|2</p(y)2%/r^<oo) Ur3 \x-y\\ JB Jb F-2/12 provided p vanishes outside the ball B. Thus, the function v is well defined on R3 and bounded on each ball. Let / := /R3. For all (j) G Cq°(R3), set V{4>) := / v{x)4>{x)dx and p(</>) := / p(x)<p(x)dx.

184 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle By Problem 2.9b, U is a fundamental solution of (106), i.e., — EqMJ = 8, and Problem 2.9e tells us that the convolution W := U * p is a solution of (106). Explicitly, for all <f> e C^°(R3), W{<t>) = U (jp{x)<j>{x + y)dx\ = J ^-j-r (jp(x)<P(x + y)dx\dy = l^\{lp{Z-V)<t>{z)dZ)dV- By the Tonelli theorem (cf. the appendix) and by the substitution z := x—y, we get W((j)) = f (/^4 "ff^) <KZ)*Z = [ v(z)(/)(z)dz for all <j> G C0°°(R3). Thus, W = V. 2.9g. Generalized plane waves. Let x := (^1,^2^3) £ R3, t G R, and consider the wave operator 0:-££-<*-<*-* Let n := (ni,712,^13), where n2 = n2 + n\ + n§ = 1. Let the function ip:R —» R be given such that ij) G Lijioc(R) (e.g., ij) is continuous). Then, the function u(x, t) := i>{nx - ct), x G R3, t G R, (108) is called a p/ane wawe, where nx := ni£i + 712^2 + ^3^3- Observe that the function u is constant for nx — ct = const. That is, ij) is constant on planes that move with the velocity c in the direction of n. Show that the generalized function U corresponding to u is a solution of the wave equation DC/ = 0 on R4. (109) Observe that u is a classic solution of (109) if i\) is sufficiently smooth (e.g., 1> G C2(R)). Hint: Approximate i\) by polynomials ij)n. Then, D^n = 0 in the classical sense. Cf. Zeidler (1986), Vol. 2, p. 1050. 2.9h. A special tensor product. In classic analysis, the tensor product <fi <8) ip of the two functions <fi = <fi{%) and ij) = ijj(y) is defined to be the function (<l>®il>)(x,y) :=(f)(x)^(y). Let U G T>'(RN) and 6 G £>'(R). We define the tensor product U <g> «(n> through (CW(n))(x) = E/(*(n)(x)), n = 0,l,..., (110)

Problems 185 for all functions x = x(xi 0 with X £ Co°(Riv"fl). Here, 6 acts on the time variable t. To motivate this definition, let us formally consider the product (u®8)(x,t) = u(x)6(t). If U denotes the generalized function to u, then formally we get (U®8)(x)= u(x)6(t)x(x,t)dtdx = f u(x)x(x,0)dx. This is (110) for n = 0. Show that the tensor product (110) yields a generalized function, i.e., U®6^ eVf(RN+1). 2.9i. The generalized initial-value problem for the wave equation in R3. Using the notation from Problem 2.9g, the classic initial-value problem for the wave equation reads as follows: Du = f on R4, (111) u(x,0) = ^o(^) and ut{x,Qi) = u\{x) on R3. For given functions uq (initial state) and U\ (initial velocity), we are looking for a function u = u(x, t) such that (111) is satisfied. Set R\ := {(x, t) eR4:t> 0}. We are given feC(R\), uoeC^R3), Ul e C(R3). Suppose that the function u G C2(int R%) n CX{R%) is a solution of (111). Set u(x,t) = 0 and f{x,t) = 0 outside R\. Let U, C/o, C/i, F denote the generalized functions corresponding tou,uo,ui,f, respectively. Show that U is a solution to the following equation: nU = F + U0®6' + U1®6 on R4, where supp U C R4.. (Ill*) Here, supp U C R4. means that U(<f>) = 0 for all functions 4> G C£°(R4) which vanish on an open neighborhood of R\. Problem (111*) is called the generalized problem to (111). In physics, the right-hand side / corresponds to some outer force. In problem (111*), the initial conditions from (111) are replaced with additional outer forces C/o (8) S' and U\ (8) <$, where the appearance of the ^-distribution is responsible for the fact that these additional forces only act at the initial time t = 0. Hint: Use integration by parts. Cf. Zeidler (1986), Vol. 2, p. 1054. 2.10.* The general tensor product Let U G T>'(RN) arid V G V'(RM). Then, there exists exactly one generalized function W G T)'(RN+M) such that W(<f>®t/>) = U(<l>)V(tl>) for all 0 G CHR^) and ^C0°°(RM),

186 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle We call W := U (8) V the tensor product of U and V. Explicitly, (U ® V)(X) = U(V(X(x, •)) for all X € C^°(RN+M), where V acts on the function y h-> x(x, y). Study the proof in Hormander (1983), Vol. 1, Section 5.1. 2.11.* The existence theorem for fundamental solutions. To each (nonzero) linear differential operator L with constant coefficients (cf. (105)), there exists a fundamental solution [/, i.e., the equation LU = 6 on RN has a solution U G V'(RN). The proof of this famous Malgrange-Ehrenpreis theorem from 1955 can be found in Yosida (1980), Chapter 6. This proof relies on properties of the Fourier transform (the Pay ley-Wiener theorem). 2.12. Smoothing of functions by using mean values (Friedrichs' mollification). The point of departure is the integral u£(x):= 4>£(x-y)u(y)dy, (112) where <j>e(x) := e~N4>(e~1x) along with M \ - /ce-t1"!*!2)"1 if x G RN and \x\ < 1 0[X)' \0 ifxGR"and|x|>l. Then (i) j*eCf(RN). (ii) 4> > 0 on R*. (iii) JRN (j)(x)dx = 1 for a suitable choice of the constant c > 0. Hence: (i*) 4>E e C$°{RN) and <f>e(x) = 0 if \x\ > e for all s > 0. (ii*) 4>£ > 0 on RN for all e > 0. (iii*) jRN (f)£(x)dx = 1 (see Figure 2.17 for N = 1). Let u G L2(G), where G is a nonempty open set in RN, N > 1. We set w(rc) = 0 outside G. Show that (a) u£ eC°°(RN) foralU>0.

Problems 187 FIGURE 2.17. (/?) u£ G L2(G) for all s > 0. (7) u£ —» u in L2{G) as £ —* +0. Solution: Ad (a). Consider the ball 5:= {x GR^.'lx-xol < 1} around the given point #o, and consider the set B£:={yeG:dist(B,y)<e}. Since 0e(x — y) = 0 for all points x,y G M^ with |x — y\ > e, from (112) we get the key formula u£{x) = I <fi£(x — y)u{y)dy for all x G B. (H3) Jb£ By the Schwarz inequality (16), we obtain \u(y)\dy= / 1- |w(2/)|d2/ < (J dyY (J \u(y)\2dyY < 00, since JB dy = meas(i?e) < 00 and u G L2(Miv) implies ix G L2{B£). Thus, the function y »-> |rx(2/)| is integrable over f?e. First let iV = 1. For all x e B, y e B£, and fc = 0,1,2, ...,£> 0, we obtain |^*)(x-y)ti(y)| <const(fc,e)|ti(y)|, (114) where <fi£ ' denotes the kth derivative. In this connection, note that the function <jye ' is continuous on R, and hence it is bounded on compact sets by the Weierstrass theorem (Proposition 8 in Section 1.11). In particular, <jy£ ' is bounded on each ball. Applying standard theorems on parameter integrals (see the appendix) to (113), the majorant condition (114) tell us that the continuous derivative u£ ' exists on 5, where u£k\x) = f 4k)(x ~ V)<y)dy for all x G B, k = 0,1,... . Jb£ Jbp

188 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle Since the center xq of the ball B is arbitrary, this implies (a). In the case where N = 2,3,..., we use the same argument with respect to partial derivatives. Ad (/?). Set / := JRN and / / := JrnxRn- Using the substitution z := e~1(x — y), it follows from (112) that ue(x) = / <p{z)u{x — ez)dz. (115) Observe now the identity <pu = <$*{<$* u). Thus, by the Schwarz inequality (16), we get \u£(x)\2 < J(f)(z)\u(x-ez)\2dz, noting that / <p{z)dz = 1. Substituting y := x — ez, we obtain J (J<f>(z)\u(x - ez)\2dx) dz = J(t>(z) (j \u(y)\2dy) dz = / \u{y)\2dy < oo. Therefore, it follows from the Fubini-Tonelli theorem (see the appendix) that / \u£(x)\2dx < / ( / <P(z)\u(x - ez)\2dz j dx = I [ I <fi(z)\u(x — ez)\2dx J dz < oo, and hence u£ G L2(Miv). Ad (7). Let B := {z e RN: \z\ < 1}. Recall that 4> = 0 outside and fB <f>(z)dz = l. By (115), u£{x) = / u(x — ez)(p(z)dz, Jb and hence ue(x) — u(x) = / (u(x — ez) — u(x))<p(z)dz. Jb The Schwarz inequality (16) yields \ue{x) — u(x)\2 < C / \u(x — ez) — u(x)\2dz, Jb where C is a positive constant. By the p-mean continuity of the Lebesgue integral with p = 2 (see the appendix), for each rj there is an £q > 0 such that / \u(x — ez) — u(x)\2dx < 77, Jg

Problems 189 for all z G B and all s: 0 < e < Sq. Thus, it follows from the Fubini-Tonelli theorem (see the appendix) that / \u£(x) — u(x)\2dx <C I I I \u(x — ez) — u(x)\2dz j dx = C / ( / \u(x — ez) — u(x)\2dx 1 dz < Cmea,s(B) • 77, for all e: 0 < e < £o- Hence / \u£(x) — u(x)\2dx —* 0 as e —» +0. This is (7). 2.13. Density (Proof of Proposition 7 in Section 2.2). Let G be a nonempty open set in R^, N > 1. 2.13a. Show that the set C°°(G) is dense in L2(G). Solution: This follows immediately from Problem 2.12 (a)-(j). 2.13b. Show that C$°(G) is dense in L2(G). Solution: Case A: The nonempty open set G is bounded. Let C be a compact set7 with G C G, and let ix G L2{G). We set / n r u(x ) onG on G - G. Then / |w — i;|2dx = / |ix|2dx. Jg Jg-c By the absolute continuity of the integral (see the appendix), the right- hand integral is arbitrarily small provided the measure of the set G — G is sufficiently small. Thus, for each given 77, we can choose the set G in such a way that \\u — v\\=[j \u — v\2dx j < 77. By Problem 2.12, there is a function ve G C°°(RN) such that \\v — v£\\ <V for all e: 0 < e < £o- Next let us show that ve G Gq°(G) for sufficiently small s. In fact, since v = 0 outside G — G, it follows from (112) that v£(x)= / <p£(x-y)v(y)dy. Jc 7C C G means that C is a subset of G, and C C G means that C is a proper subset of G, i.e., G C G and C ^ G.

190 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle FIGURE 2.18. Hence v£{x) = 0 for all x G G with dist(x, C) > e because of <j>e(x — y) = 0 for |x — y\ > e. Since C is a compact subset of the open set G, there is an open set H such that C CHCH CG (see Figure 2.18).8 Consequently, if we choose the number e sufficiently small, then dist(x, C) > e for all x G G — H, and hence v£(x) =0 for all x G G - H, i.e., v£ G Cq°(G). Summarizing, \\U - Ve|| < \\U - V\\ + \\V - Ve\\ < 277, i.e., Cg°(G) is dense in L2(G). Case B: The open set G is unbounded. Then, for each r\ > 0, there is an open ball B such that / |ix|2dx < t?2, where H :—,GC\B and if ^ 0, by a well-known property of the Lebesgue integral. Applying Case A to the nonempty bounded open set H, there is a function v£ G C$?(H), and hence v£ G C^°(G), such that /■ u — ve\ dx < r\ . Since ve = 0 on G — H, we get ||^-<ue||2= / \u\2dx+ / \u-v£\2dx < ry2, ./G-i/ ^if 8In fact, for each point x, there exists an open ball B around x such that B C B C G. Since C is compact, a finite set of such balls already covers C. Call the union of these balls H.

Problems 191 i.e., C%°(G) is dense in L2(G). 2.13c. Show that C(G) is dense in_L2(G). Solution: Note that G£°(G) C C[G) and use Problem 2.13b. 2.14. Show that both C^°(G)C and C(G)C are dense in L2(G)C. Hint: Use the same arguments as above. 2.15. Separability (Proof of Corollary 8 in Section 2.2). 2.15a. Let G = ]a, b[ be a bounded open interval in R. Show that L2(G) is separable. Solution: Let u G L2(G) and e > 0 be given. By Problem 2.13c, the set C[a, b] is dense in L2(G), i.e., there is a function v G C[a, 6] such that . i 2 — v\\ = I \u — v\2dx J < e. By the Wezerstfrass approximation theorem (Proposition 2 in Section 1.25), the set of polynomials with real coefficients is dense in the Banach space C[a, 6], i.e., there is a real polynomial p such that \\v - p\\* := max \v(x) - p(x)\ < e. a<x<b Let us introduce M := set of all polynomials with rational coefficients. By the proof of Corollary 3 in Section 1.26, for each polynomial p, there is a polynomial q G M such that Ib-tfll* <s Hence \\v - g||* < \\v - p||* + \\p - g||* < 2e. This implies \\v -q\\ = I / \v -q\2dx j < (b-a)i\\v - q\\* < (b - a)*2e. Summarizing, for each s > 0, there is a q G .M such that lb - ?ll < lb - v\\ + lb - <l\\ <£ + {b- a)*2et That is, the set M. is dense in L2(G). Since the set M is countable, the space L2(G) is separable. 2.15b. Let G be an unbounded open interval in R, e.g., G = R. Show that L2(G) is separable.

192 2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle Solution: There exists a sequence (Gn) of bounded open intervals G such that Gi C G2 C • • • C G and oo G = (J Gn. n=l Define <»(*):= {j;[ R — Gn, and A^oo := {XnT- q e M and n = 1,2,...}. Let ix G L,2(G) and £ > 0 be given. There exists a bounded interval J with J C.G and / |?x|2Gte < £2, by a well-known property of the Lebesgue integral. Choose some interval Gn such that J C Gn C G. Then /c |^|2dx < £2. G—Gn By Problem 2.15a, there is a polynomial q E M. such that |ix- g|2dx < e2. JG, Hence ll^-Xng||2= / \u\2dx+ / |^-g|2dx < 2e2. JG-Gn JGn Consequently, the countable set Moo is dense in L2(G), i.e., L2(G) is sep- araWe. 2.16. Let G be an open interval in R. Show that L%(G) is separable. Use an analogous argument as in Problem 2.15, based on polynomials with complex coefficients. 2.17. Let G be a nonempty open set in R^, N >1. Show that L2(G) and L^{G) are separable. Hint: Use the same arguments as above. Replace the one-dimensional Weierstrass approximation theorem with the iV-dimensional Weierstrass approximation theorem (cf. Problem 6.19b). 2.18.* The Sobolev embedding theorems. An elementary approach to the Sobolev embedding theorems can be found in Zeidler (1986), Vol. 2, Section 21.3. Study these proofs. The appendix to Zeidler (1986), Vol. 2, contains a

Problems 193 summary of important material related to the Sobolev embedding theorems (relation to the theory of generalized functions, interpolation theory, etc.). We also recommend Gilbarg and Trudinger (1983). 2.19. A formal relation for the 6-function. Let /:l->lbea (71-function that has precisely the zeros x\,..., xn. In addition, assume that f'{xj) ^ 0 for all j. Use the formal definition of the <$-function to show that N 6(f(x)) = J2S(X - xj)\f(xj)\~1 for a11 x £ R- i=i This relation is frequently used by physicists (cf. Standard Example 18 in Section 5.24). For example, if a ^ 0, then c/ 2 2\ $(x — o) + Six + a) 6(x2 - a2) = -± T-r-r-^ for all x G R. 2|o| Solution: Let (j> G Co°_(R), and let gj be the inverse function to / in a sufficiently small neighborhood of Xj such that f(gj(y)) = y> Then, for sufficiently small e > 0, = £/ %)^(5i(y))l5-(y)My =E^(o))i^(o)i = E^)i//(^)r1-

Hilbert Spaces and Generalized Fourier Series The interplay between generality and individuality, deduction and construction, logic and imagination—this is the profound essence of live mathematics. Any one or another of the aspects can be at the center of a given achievement. In a far-reaching development all of them will be involved. Generally speaking, such a development will start from the "concrete ground," then discard ballast by abstraction and rise to the lofty layers of thin air where navigations and observations are easy; after this flight comes the crucial test of landing and reaching specific goals in the newly surveyed low plains of individual "reality." In brief, the flight into abstract generality must start from and return to the concrete and the specific. Richard Courant (1888-1972) Let /:l->Rbea function of period 27T. Then, the corresponding classical Fourier series reads as follows: oc f(x) = 2^"1ao + 2_2 ak cos ^x "*~ ^k sin kx, (1) with the so-called Fourier coefficients ak := 7T-1 / /(x) cos kxdx, k = 0,1, 2,... , J — 7T bk:=7r'~1 / f(x)sinkxdx. J — IT

196 3. Hilbert Spaces and Generalized Fourier Series From the physical point of view, relation (1) tells us that the 27T-periodic "oscillation" / can be represented as a superposition of simple "harmonic oscillations" h{x) := coskx, sinfcx, (2) of period ^, where k = 1,2, The Fourier coefficients a& and bk correspond to the amplitudes of the harmonic oscillations (2). For example, if \dk\ is large for some fc, then the harmonic oscillation x h-> cosfcx contributes strongly to the oscillation /. Therefore, physicists are interested in such periods ^l for which |a&| or \bk\ are large with respect to the other Fourier coefficients. For example if /: R —» R has the period 2tt with f(x) := \ (| - \x\) for - *" < x < n, then n/ x cos3x cos5x ^ f(x) = cos x H -^ \ T2 1 for all x G R. Figure 3.1 represents schematically the corresponding superposition. In the nineteenth century, many mathematicians studied in detail the convergence of Fourier series. In 1876, du Bois-Reymond obtained the surprising result that there are continuous functions / for which the Fourier series / does not converge at each point x G R. This counterexample shows that the classical convergence of infinite series is not the right concept for solving the fundamental convergence problem for Fourier series. In 1907, Fischer and Riesz proved independently that A natural answer to the convergence problem for (1) can be given in terms of the Hilbert space L2(—7T, 7r). In this chapter we will show that this is a special case of an abstract result on complete orthonormal systems in Hilbert spaces. In particular, it turns out that, for each function / G L2(-7T,7r), the Fourier series (1) converges in the Hilbert space L2(—7r,7r), i.e., lim ||/-sn|| =0, n—>oc where n sn(x) := 2_1ao + V^afcCOste + bksinkx, k=l and || • || denotes the norm on L2(—7r,7r). Explicitly, lim / {f{x)-sn{x))2dx = 0.

3. Hilbert Spaces and Generalized Fourier Series 197 n/Ka, , — 7T 7T (a) cos 3 a: AAAAA (b) (c) FIGURE 3.1. We shall also show that this corresponds to the convergence of the Gauss method of least squares. In the nineteenth century, mathematicians and physicists also used more general series expansions than (1) of the following form: f(x) = YlCkfk^'> x gG' (3) k=0 where G is a nonempty open set in R^, and the functions /o, /i,... satisfy the so-called orthogonality relation (// 'fe I fm) = / fk{x)fm{x)dx = Skm, k, m = 0,1,... Jg (4) Here, 6 km =0iik ^ m and 6 km = 1 if fc = ra. Observe that (• | •) represents the inner product in the Hilbert space L2(G). Using (4), it is easy to compute the unknown coefficients c& formally. Namely, multiplying (3) with /m(x) and integrating over G, it follows formally from (4) that Cm = (/ | fm) = / f(x)fm(x)d3L Jg m = 0,1,... . Our abstract results will show that the infinite series (3) converges in the Hilbert space L2(G) provided the set of all the real linear combinations ^2a^fki oto,...,aneR, n = 0,l,... k=0

198 3. Hilbert Spaces and Generalized Fourier Series is dense in L,2(G). This is a quite natural result. In Chapters 4 and 5 we shall show that series expansions of the form (3) are closely related to eigenvalue problems for integral equations and differential equations where /o, /i, • • • are the corresponding eigenfunctions. In terms of physics, for example, the infinite series (3) represents the oscillations of an elastic body as a superposition of "eigenoscillations." Moreover, in quantum mechanics the eigenfunctions fk from (3) correspond to bounded states of atoms and molecules with a well-defined energy, whereas (3) describes the superposition of a state / by means of the energy states fk- The continuous version of the Fourier series (1) is given through the Fourier integral /oo a(k)eikxdk, -oo < x < oo, (5) -OC with /OO f(x)e-ikxdx, -oo<k<oo. (5*) -OC The function a(-) is called the Fourier transform of /(•). From the physical point of view, formula (5) describes the superposition of an arbitary, not necessarily periodic "oscillation" / by means of the simple "harmonic oscillations" e-ikx _ ^cog fcx _ i gin faj ^ of period ££ with real k, and {2n)~^a{k) is the amplitude of the harmonic oscillation (6). Differentiating formally equation (5), we get df(x) /OO ika(k)eikxdx. -OO dx Thus, the Fourier transform of the differential operator is given through the multiplication operator a(k) h-> ika(k). This is the most important property of the Fourier transformation. The Fourier transformation allows the reduction of differential equations to purely algebraic problems. For example, suppose we want to solve the differential equation /'(*) = AC*), (7)

3.1 Orthonormal Series 199 where f\ is given. Applying formally the Fourier transformation to (7), we get ika(k) = ai(k), where a and a\ denote the Fourier transform of / and /i, respectively. Hence and the solution / of (7) is obtained by the Fourier transiormation (5). In classical mathematics, the crucial difficulty of this method was caused by the lack of convergence for the Fourier integrals (5) and (5*). For example, if f(x) = 1, then /oo e~ikxdx. -OO But this integral does not exist. In Section 3.7 we shall show that The extension of the Fourier transformation to generalized functions of the class S'{RN) overcomes the classical difficulties. In order to describe this formally, choose f(x) :=6y(x), where 6y denotes the "Dirac function" from Section 2.8.2. By (5*), /oo Sy(x)e-ikxdx = (2ir)-ie-iky. -OO Thus, in the formal language of physicists, the "Dirac function" 6y has the Fourier transform k h-> (2tt)~ * e~~lky. In particular, the "Dirac function" 60 has the constant Fourier transform a(fc) = (27r)-i. A rigorous approach will be considered in Section 3.7. 3.1 Orthonormal Series In this section, we make the following assumption: (H) Let X be a Hilbert space over K = R, C, and let {uq^ui, ...} be a finite or countable orthonormal system in X, i.e., by definition, (uk I um) = 8km for all k, m. (8)

200 3. Hilbert Spaces and Generalized Fourier Series Our goal is to study the convergence of the so-called abstract Fourier series oc U = ^T(un\ U)un. (9) n=0 We also set 771 n=0 The numbers (un \ u) are called the Fourier coefficients of u. Definition 1. Assume (H). The finite orthonormal system \uq, ..., u^} is called complete in X iff N u = y^(^n I u)un for all u e X. (10) 71=0 The countable orthonormal system {uo,ui,...} is called complete in X iff the infinite series (9) converges for all u e X, i.e., u = lim sm for all u E X. m—>oo Proposition 2. The finite orthonormal system {uq, ... ,un} is complete in the Hilbert space X over K iff it is a basis of X. Proof. Let {un} be a basis in X. Then, N u = \] cnun for all u e X, (10*) n=0 where the coefficients Co,..., cm GK depend on u. Using (8), we get N (Uk | U) - Y Cn{Uk \Un)=Ck, fe = 0, 1, . . . , N. n=0 This implies (10), i.e., {un} is complete. Conversely, let {un} be a complete orthonormal system; then {un} is a basis of X, by (10). In this connection, note that {uq, ... ,?xjv} is linearly independent, since (10*) with u — 0 implies that c& = 0 for all fc. □ Corollary 3. Let {un} be a countable orthonormal system in the Hilbert space over K. Assume that the infinite series oc u = 2^ cnun with cn e K for all n n=0

3.1 Orthonormal Series 201 is convergent for some fixed u G X. Then, cn — (un \ u) for all n. Proof. Using (8) we get / 771 \ 771 (uk \u)= lim I uk | \l cnun = lim V" cn(uk \ un) = ck. 771—>00 V *—-^ / 771—>OQ *—-^ D 71=0 n=0 This motivates the ansatz (9). Let us give a second motivation for (9). In order to get a good approximation of u by a linear combination cqUq + hcmixm, we set /(co,---,cm) := / v CnUn n=0 and we consider the following minimum problem: /(co, • • •, Cm) = min !, c0,..., cm E : according to the least-squares method of Gauss. (ii) Proposition 4. ylsswfte (H). Tften, £fte unique solution of (11) is given through the Fourier coefficients cn = (un \ u), n = 0,..., m. Proof. By (8), (771 771 \ u ~ mCnUn \u~^2ckUk) n=0 k=0 J 771 771 771 = {u\u)- Y^cn{un I w) - Y^ck{u I ixjfe) 4- Y^cncn. Hence 71=0 fc=0 n=0 f(c) = \\uf - ]T \(un I u)|2 + ^ \(un | «) - cn|2. (12) n=0 n=0 The smallest value of / is attained for cn = (un \ u), n = 0,..., m. D In particular, it follows from (9*) that \\u - sm\\2 < f(c) for all c G Km and all ra. (13) By (12), i 11 o ii n9 - Y2 \(Un i ^)i2 for a11 u e x and a11 m' (14) n=0

202 3. Hilbert Spaces and Generalized Fourier Series Hence we obtain the following Bessel inequality: m 2_] \{un | u)\2 < \\u\\2 for all u G X and all ra. (15) n=0 Proposition 5 (Convergence criterion). Let {un} be a countable orthonormal system in the Hilbert space X over K. Then, the series oo , 22 cn^n-> cn eK for all n, n=0 is convergent iff the series Yl™=o \cn\2 ^ convergent. Proof. Let Sm := ^=0cnun. By (8), H^ra+fc — ^n m+k / j cn^"n n=ra+l m+k n=ra+l (16) for all ra, k = 1, 2, If ]T)n |cn|2 is convergent, then (5m) is a Cauchy sequence. Hence (Sm) is convergent, i.e., ]Cnf^fLjs convergent. Conversely, if ]T)n cnixn is convergent, then (5m) is a Cauchy sequence, and hence ]T)n |cn|2 is convergent, by (16). D It follows from Proposition 5 and the Bessel inequality (15) that for each u e X the Fourier series is convergent, i.e., there is some v G X such that oo V = Yl(Un I U)Un' 71=0 However, it is possible that v ^ u. But if the orthonormal system {un} is complete, then v = u for all u G X. Theorem 3.A. Le£ {^n} &e a countable orthonormal system in the Hilbert space X over K. Then, the following two conditions are equivalent: (i) The system {un} is complete in X. (ii) The linear hull of {un} is dense in X. Proof, (i) => (ii). This is obvious, by Definition 1. (ii) => (i). For given e > 0, there exist coefficients Co,... ,cm G K such that /(c) / j cn^"n <€. 71=0

3.2 Applications to Classic Fourier Series 203 We also may assume that m is sufficiently large, by letting cn = 0 for large n. Choosing s = £, r = 1,2,..., it follows from (13) that there exists a subsequence (smr) such that ||t*-*mr|| < -, r = l,2,... . (17) r Since a Fourier series is always convergent, the sequence {sm} is convergent, i.e., Sm —» v as m —» oo. Letting r —» oo, it follows from (17) that v = u. □ Corollary 6. Let {un} be a countable complete orthonormal system in the Hilbert space X over K. Then, the following hold true: (i) For all u, v e X, oc (u | v) = Y^ cn(u) cn(v) (the Parseval equation), (18) 71=0 where cn(w) := (un \ w). (ii) For all u G X, the Bessel inequality is replaced with the so-called special Parseval equation oo 112 = £|(«nl«)l2. (is*) n=0 (iii) // (un | u) = 0 /or a// n and /wred uGX? then u = 0. Proof. Ad (i). By (8) and (9), (m m \ m Y^cn(u)un | Y]ck(v)un I = lim Y]cn(^)cn(i;). n=0 fc=0 / n=0 Ad (ii). This is a special case of (i). Ad (iii). This follows from (9). □ 3.2 Applications to Classic Fourier Series Recall that the inner product in the Hilbert space L2(—7r,7r) is given through (u | v) = / u(x)v(x)dx. J —TV

204 3. Hilbert Spaces and Generalized Fourier Series For all x G [—7r,7r], we set uq{x) := (2tt)~^ and ^2m-i(#) •= ^~^ cosrax, U2m{%) •= ^~^ sin rax, m = 1,2,... . Proposition 1. The set {uq, u\, ...} forms a complete orthonormal system in the Hilbert space L2{—7T, 7r). This proposition tells us that for each u G L2(—7r,7r) the Fourier series oo u=^T cnun, cn := (un I w), (19) n=0 converges in L/2(—7T, 7r). This is identical to the classic Fourier series oc u(x) — 2_1ao + V^afcCoste + ^smfcx, (19*) fc=i where bk := n * / ia(sc) cos fcx dx, J— IT :=7T~1 / ia(sc) sin kxdx, k = 0,1, 2,... . In fact, (^2m I u)u2m{x) = 7T 1 [Jl^ u(x) sinmxdx J sin rax, m = 1,2,..., and so on. Consequently, Proposition 1 implies the following corollary. Corollary 2. For each u G L2(—tt, 7r), £/ie classic Fourier series converges in L2(—7T, 7r), i.e., /.*■ / ™ \2 lim / j ti(x) — 2_1ao — ^Jafccoste + fefesin^x J dx = 0. The proof of Proposition 1 will be based on the following classic approximation theorem due to Weierstrass (see Lemma 3). Let T denote the set of all trigonometric polynomials, i.e., p G T iff 771 p(x) := 22 an cos nx + f3n sin nx, 71=0 where m = 0,1,..., and all the coefficients an, j3n are real numbers. It follows from the classical addition theorems for sin(-) and cos(-) that p,q eT implies pq G T.

3.2 Applications to Classic Fourier Series 205 We also set ll/llc[a,6] := max |/(x) a<x<b Lemma 3. For each function f G C[—7T, tt] with /(—tt) = f{it) and each e > 0, there exists a function p &T such that ll/-p||c[-7r,7r] <e. Proof. Step 1: Let / be even, i.e., f(—x) = f(x) for all x G [—tt,tt]. The function <j>{x) := cosx is strictly decreasing on [0, tt}. Since y h-> f{<p~1{y)) is continuous on [—1,1], it follows from the Weierstrass approximation theorem (Proposition 2 in Section 1.26) that there exists a polynomial p(y) = cq + c\y + • • • + cnyn such that max |/(^_1(y))-p(y)l <£ -1<2/<1 Letting y = cosx, this implies max \f(x)-q(x)\ < e, 0<X<TT where q(x) := p(cosx), and hence q G T. Since / and q are even, we also get max \f(x)-q(x)\ < e. — TT<X<TT Step 2: Let / be odd, i.e., f{—x) = —f(x) for all x G [—7r,7r], and let /(0) = /(tt) = 0. Set [0 if0<x<8 or tt - — <S < £ < 7T. Finally, let #(x) := —#(—£) if — tt < x < 0. Since / is uniformly continuous on [—7r, 7r], we get max \f(x)-g(x)\ < - —7r<:r<7r Z for sufficiently small <S > 0. Applying Step 1 to the even continuous function x i-> Q^l on f—7T, 7rl, it follows that there exists a a G T such that sin a; L 7 ■" * max -tt<x<tt 9(x) / \ V ' - q{x) smx <2- Setting r(a;) := ?(a;) sin a;, we obtain that r £T and max |0(a;)-r(aO| < t- —7r<a;<7r Z

206 3. Hilbert Spaces and Generalized Fourier Series By the triangle inequality, max \f(x) — r(x)\ < e. — TT<X<TT Step 3: In the general case, we use the decomposition /(*) = 2~\f{x) + f{-x)) + 2-\f{x) - f(-x)), and we apply Steps 1 and 2 to the even part and odd part, respectively. Observe that f(x) — f{—x) = 0 for x = 0, tt. D Corollary 4. The set T of trigonometric polynomials is dense in L2(-7r,7r). Proof. Let u G L2(—7T, 7r), and let e > 0 be given. By Proposition 7 in Section 2.2, the set C[—7r,7r] is dense in L2(—7T, tt). Thus, there exists a continuous function /: [—7T, tt] —> R such that \\u-f\\=y\u(x)-f(x))2dx^ 2 < €. Changing continuously the function / near the point x = 7T, we may assume that /(—7r) = /(7r). By Lemma 3, there exists a function q E T such that ||/-g||<(27r)i max \f(x) - q(x)\ < e. ■~TT<X<7T By the triangle inequality, \\u — q\\ < 2e. D Proof of Proposition 1. We first show that {un} forms an orthonormal system, i.e., (un | Uk) = / un(x)uk(x)dx = 8nk for n, k = 0,1,... . J —TT (20) Using e±mx _ ^cog nxJLj/ sm na.^ relation (20) follows from „inx _i_ p—inx n = l,2,..., pina; „—inx COo (Mb — T i Mil / t»«A/ — —— —• 2 ' 2% and / Jrnxj _ ^ C 1 c CIX — ; , m = ±l,±2,... zm by a simple computation. By Corollary 4, the set T = span{?xo,^i,...} is dense in L,2(—tt, 7r). It follows from Theorem 3.A that the orthonormal system {un} is complete in L2(—7T, 7r). □

3.3 The Schmidt Orthogonalization Method 207 3.3 The Schmidt Orthogonalization Method Proposition 1. In each separable Hilbert space X over K with X ^ {0}, there exists a complete orthonormal system. Proof. By assumption, there exists an at most countable set {vq,V\, ...} that is dense in X. We may assume that ^T^O. Set _ ^o Uo - W Suppose that we have already constructed uo,...,un such that {uq, ..., un} forms an orthonormal system. Then, let n wn . _ k=0 If wn+i 7^ 0, then we set Un+l := Vn+1 - Y^iUk I Vn+l)Uk' (21) lFn+l|| Thus, (um I ifcn+i) = 0 for m = 0,..., n, and (ifcn+i I ^n+i) = 1- If wn+i = 0, then we use ^n+2, and so forth. This way we obtain an orthonormal system {um}. By induction, it follows that all the vm are finite linear combinations of the un. Hence the linear hull of {um} is dense in X. If {um} is countable, then Theorem 3.A tells us that {um} is complete. If {um} is finite, then span {um} = X, since each finite-dimensional linear subspace of a Hilbert space is closed, by Corollary 7 in Section 1.12. Thus, {un} is again complete. □ Method (21) for constructing the orthonormal system {un} is called the Schmidt orthogonalization method. The following two results will be used critically in the next section. Proposition 2. We assume the following: (i) Let {vo, vi,...} be a sequence in the Hilbert space X over K such that ^o> • • • 5 ^m are linearly independent for each m = 0,1, (ii) Let the span {vo, ^i,...} be dense in X. (iii) Let {uq,u\, ...} be a countable orthonormal system in X such that u0 = a0v0, aQ > 0, (22) and n Un+1 = Oin+l^n+l + ^ OLkVk, Q^n+l > °> (23) k=0 for all n = 0,1,... and appropriate coefficients ^GK,fc = 0,...,n.

208 3. Hilbert Spaces and Generalized Fourier Series Then, the following hold true: (a) The system {un} is obtained from {vn} by means of the Schmidt orthogonalization method. (b) The system {un} is complete in X. Proof. Ad (a). It follows from (22) that «o = Tnjnr, and hence Foil Let n > 1. By (23), vk G span{^0,..., un} for k = 0,..., n. (23*) Hence n m=0 where f3o,...,/3n G K are appropriate coefficients. Using (uk | ^n+i) = 0 for k = 0,..., n, we get 0k = -an+i(wjb I vn+i). By (21), un+i = Tj n and wn+i ^ 0. lFn+l|| Thus, according to (21*), ^n+i corresponds to the Schmidt orthogonalization. Ad (b). Since span {vn} is dense in X, it follows from (23*) that the set span {un} is also dense in X. By Theorem 3.A, {un} is complete. □ Corollary 3. Let {vo, v\,...} be a sequence in the Hilbert space X over K. Suppose that (vn | u) = 0 for all n and fixed u G X implies u = 0. (24) Then, the set span {vo, ^i, • • •} is dense in X. Proof. Let S := span{v0, vi,...}. By (24), (S)-1 = {0}. Thus, Corollary 1 from Section 2.9 tells us that X = S. □ 3.4 Applications to Polynomials Standard Example 1 (Legendre polynomials). For n = 0,1,..., let vn(x) := xn, -1 < x < 1.

3.4 Applications to Polynomials 209 Applying the Schmidt orthogonalization method to {vn}, we get the complete orthonormal system {un} in the Hilbert space L2(—1,1). Explicitly, -M-*^^-1'*- — ^ (25) The polynomials 1 dn , 2 are called the Legendre polynomials. Proof. Step 1: We show that the system {un} defined by (25) represents an orthonormal system in L2(—1,1), i.e., (un | Um) = / un(x)um(x)dx = 6nm for n, m = 0,1,... . To this end, we set wn(x):=£^(x2-l)n, n = 0,l,.... Let n > m > 0. Then, -^(x2-l)n = 0 forx = ±l and r = 0,1,... ,n - 1. dxrK J ' ' ' Thus, integration by parts yields ^ wn(x)wm(x)dx = - jf_i ^^(x2 - i)n^r(*2 - i)mdx = ... /I rfm+n (x2 - l)n~—r~(x2 - l)mdx = 0, _i ; dxm+nK J since m + n > 2m. Similarly, t«„(a;)t«m(i)<fc = (-1)™ / (i2 - 1)™^j(i2 " l)™<fc Step #: The set span {i;n} is dense in L,2(—1,1). This follows from the density of the set C[— 1,1] in L2(—1,1) and from the fact that span {vn} is dense in the Banach space C[—1,1] by the Weierstrass approximation theorem (cf. the proof of Corollary 4 in Section 3.2).

210 3. Hilbert Spaces and Generalized Fourier Series Step 3: The assertion follows now from Proposition 2 in Section 3.3. In fact, condition (23) is satisfied, since un is a polynomial of nth degree with a positive coefficient at xn. □ Standard Example 2 (Hermitean functions). For n = 0,1,..., let vn(x) := xne 2 5 —oo < x < oo. Applying the Schmidt orthogonalization method to {^n}, we get the complete orthonormal system {un} in the Hilbert spaces L2(—oo, oo) and L§ (—00,00). Explicitly, un(x) = ane 2 Hn(x), n = 0,1,... , (26) with the Hermitean polynomials Hn(x):=(-l)ne*2?^-, n = 0,l,..., and an := 2~t (n!)~ 2 n~*. The functions ixn are called Hermitean functions. These functions play an important role in quantum mechanics (cf. Section 5.14.3). Proof. Step 1: We show that the system {un} denned by (26) forms an orthonormal system in Lf (—00, 00) with K = R or K = C, i.e., /oc un(x)um(x)dx = 6nm for n, m = 0,1,... . (27) -00 A simple computation shows that u'n(x) + (2n+ 1 -x2)un(x) =0, -00 < x < 00, n = 0,1,... , (28) um(x) + (2m + 1 - x2)um(x) = 0, -00 < x < 00, m = 0,1,... . (29) Observe that, for all a > 0 and k = 0,1,..., lim e~a*V = 0. (30) £—»±OC Consequently, integration by parts yields /oo pN u'n(x)urn{x)dx = lim / u'n{x)urn{x)dx -00 N-*ooJ_N \N fN = lim ^n(x)^m(x) — lim / un(x)u'(x)dx N-+oq \-N N-*ooJ_N /oo un(x)u'm(x)dx, n, m = 0,1,... . ■00

3.4 Applications to Polynomials 211 Similarly, /oo /»oo Un(x)um(x)dx = / un(x)u!^(x)dx, n, m = 0,1,... . -oo J—oo Thus, multiplying (28) and (29) by um and un, respectively, we get /oo un(x)um(x)dx = 0, n, m = 0,1,... . -oo This implies (27) for n^m. Furthermore, if we use (30), then integration by parts yields /. oo oo /oo jnp-x~ poo Hn~l p~x {-l)2nH^\x)e-x2dx ) 2nn\e-x2dx = 2nn\(7r)i. This yields (27) for n = m. Step 2: A classic lemma. Let the function /:R —» C be integrable, i.e., X^o \f{x)\dx <oo. Suppose that J — c f(x)e~lkxdx = 0 for all k G R. Then, f(x) = 0 for almost all x e R. The proof of this well-known uniqueness theorem for the Fourier transformation can be found in Rudin (1966), p. 200. Step 3: We want to show that span {vn} is dense in Lf (—oo, oo). According to Corollary 3 in Section 3.3, we have to show that if u G Lf (—oo, oo) and /°° x2 xne~22~u(x)dx = 0 for all n = 0,1,... , (31) -oo then u = 0. To this end, let M := {k G C: |Im fe| < 1} and set /OO 2 e~^~u(x)e~ikxdx for all k e M. -oo Formally, /OO 2 e-^tx(x)Hx)ne-tocte for all k G M, n = 0,1, 2,... . -oo (32)

212 3. Hilbert Spaces and Generalized Fourier Series For all x G R and k G M, we get \e~^~u{x){^-ix)ne~ikx\ < e"^"|x|ne,:E,|^(x)| I x2 I < const(n) e~~^(x) , n = 0,1,... . (33) 2 Since xne'T and u are elements of Lf (—oo, oo), J— c e *4 ia(#) dx < oo. Thus, the majorant condition (33) justifies formula (32) (cf. Parameter Integrals in the appendix). Consequently, the function g is analytic on the strip M. By (31) and (32), ff(»)(0) = 0 for all n = 0,1,... . Hence g(k) = 0 for all k G M. By Step 2, u(x) = 0 for almost all x G R. Step ^: The assertion follows now from Proposition 2 in Section 3.3. □ 3.5 Unitary Operators Definition 1. Let X and Y be Hilbert spaces over K. The operator U: X —» Y is called unitary iff £/ is linear, surjective, and (Uv | CM = (v I w) for all v, w G X (34) Proposition 2. // £/ie operator U:X —» V as unitary, then U is bijective and continuous, and 11^11 = 11^11 for all v eX. (35) Moreover, there exists the inverse operator U^.Y —» X, which is also unitary. Proof. Equation (35) follows from (34) with v = w. By (35), C/ is continuous. If Uz = Uw, then U(z — w) = 0, and hence z — if — 0, by (35). That is, U is bijective. Finally, equation (34) implies (a \ b) = (U~1a | C/_16) for all a, 6 G Y, i.e., C/_1 is unitary. □

3.6 The Extension Principle 213 3.6 The Extension Principle Proposition 1. Suppose that (a) X and Y are Banach spaces over K. The linear operator A: D C X —» Y satisfies \\Au\\ < C\\u\\ for all u G D, (36) where C > 0 is a constant (b) The set D is a linear dense subset of X. Then, the following hold true: (i) The operator A can be uniquely extended to a linear continuous operator A: X —» Y such that (36) holds for all u G X. (ii) //, in addition, A is compact on D, then so is the extended operator A:X ->Y. Proof. Ad (i). Step 1: Existence. Let u G X — D. Since D is dense in X, there exists a sequence (un) in D such that un —» u as n —» oo. In particular, (un) is a Cauchy sequence. By (36), \\Aun- Aum\\ <C\\u Hence (Awn) is also a Cauchy sequence, i.e., (Aun) converges. We define An := lim Aun. (37) n—use We have to show that this definition is independent of the choice of (un). To this end, let (vn) be another sequence in D such that vn —» u as n —* oo. Then ||Aifcn - Avn|| < C\\un - vn\\ -> 0 as n -> oo. Hence Aun —* yhx as n —» oo. A passage to the limit shows that (36) holds for all u G X and the operator A is linear on X. Step 2: Uniqueness. Each linear continuous extension A: X —» Y of the operator A: D —* y satisfies (37). Hence the extension is unique. Ad (ii). Let (un) be a bounded sequence in X. Since D is dense in X, there exists a bounded sequence (vn) such that un — vn —»Oasn—»oo. By assumption, the operator A: D —» y is compact. Thus, there exists a convergent subsequence (Avn'). Since A^n/ = A(un' - vn>) + Avn/,

214 3. Hilbert Spaces and Generalized Fourier Series the sequence (Aun') is also convergent. □ Standard Example 2 (Unitary operators). Let A:D C X —» D be a linear surjective operator such that (Av | Aw) = (v\w) for all v,w e D, (38) where D is a linear dense subspace of the Hilbert space X over K. Then, this operator can be uniquely extended to a unitary operator A.X-+X. Proof. It follows as in the proof of Proposition 2 in Section 3.5 that A: D —» D is bijective and \\Au\\ = HA-^H = ||u|| for all u € D. According to Proposition 1, the operators A:D —> D and A~X\D —» D can be uniquely extended to linear continuous operators A:X —* X and B:X —* X, respectively. Using (37), a passage to the limit shows that the two relations (38) and ABv = v for all v G D remain true for all v,w £ X. Hence AB(X) = X, i.e., A is surjective. □ 3.7 Applications to the Fourier Transformation Definition 1. The space S consists precisely of all the C°°-functions u: R -» C with \\u\\Piq < oo for all p, # = 0,1,... , (39) where |HU:=sup(l + |xnV|^)(x)|. (40) xeu n=Q Let un, u e S for all n. We introduce the convergence s un —► u as n —» oo by means of ll^n — u\\p,q --* 0 as n —* oo for all p, # = 0,1,2,... . The operator A: S --> 5 is called sequentially continuous iff, as n —* oo, un —► ix implies Aun —> Au.

3.7 Applications to the Fourier Transformation 215 Obviously, u G S implies \uW(x)\ < constjn^) on R for all n,p = 0,1,... . (41) 1 + \x\p Therefore, the functions u from the linear space S are called rapidly decreasing at infinity. In particular, if u,v G <S, then / u^n\x)dx\ < oo for all ra = 0,1,... , \J — oo I and integration by parts yields /oo .N pN u'{x)v{x)dx = lim u(x)v(x)\ — / ^(^^(rcjcte -oo iV-+±oc \-N J_N u{x)v'{x)dx. (42) Obviously, formula (42) remains true if u G S and v G C1(R) along with \v(x)\ + |v;(aO| < const on R. Each || • \\Ptq represents a norm on S. In contrast to a linear normed space, S is equipped with a countable set of norms. Example 2. Set u{x) := e *2 . Obviously, u G 5. Moreover, e 2 /ex -c e-%-e-ifcxda, for all A; €R. (43) In terms of the Fourier transformation introduced ahead, relation (43) tells us that u(k) = (Fu)(k) for all k G R, i.e., u(-) is a fixed point of the Fourier transformation F:S -+ S. Proof. Set Formally, J — C e-^-e-ikxdx for all A; el. (44) /OC -c e-~e-lkx(~-ix)ndx for all k G R, n = 0,1,... This can be justified rigorously because of the major ant condition j — ( e-*re-lkx(ix)n dx < e 2 b|nefa; < oo

216 3. Hilbert Spaces and Generalized Fourier Series for all k G R, n = 0,1,..., (cf. Parameter Integrals in the appendix). Integration by parts yields /»CXD J — CXD /»CXD =_ye «/ —CXD Hence By(44),/(0) = (27r)-4. Let i r g 2,"':r x2 2 fce fc2 = e 2 {—ix)dx -ikxdx = 7(0) -fc/(fc) for all k G for all keR. lis now study the so-called Fourier transformation a(k) = (2tt)~ /»CXD u(a;)e-ifexda; for all k G R, D (45) along with the inverse Fourier transformation /CXD a(k)eikxdk for all ieR. (46) -CXD We set Fu := a. Proposition 3. T/ie following hold true: (i) Tfte Fourier transformation F:S->S is linear, bijective, and sequentially continuous. (ii) The inverse transformation F'^.S-^S is also sequentially continuous. Explicitly, F~x is given through (46). (iii) For all u,v e S, /CXD /»CXD u(x) v(x) dx = (Fu)(k) (Fv)(k) dk. (47) -CXD J — CXD (iv) For all u G S and all keR, we have kpa(k) = (-i)p(Fu(p))(k) (48a) and aM(k) = (-i)qF[xqu(x)](k), (48b) where p,q = 0,1,

3.7 Applications to the Fourier Transformation 217 Corollary 4. Let u G S. Then Fu = 0 implies u—0. Proof. Step 1: We prove (48). Let u G S. Formal differentiation yields /oc u(x)(-ix)qe-ikxdx, -OC for all k G R, q = 0,1, This can be justified rigorously by using the majorant condition /oc poo \u(x)(-ix)qe-ikx\dx < / \u(x)\\x\qdx < oo for all k € E -oc J — oo (cf. Parameter Integrals in the appendix). Moreover, integration by parts yields kpa{q\k) = (2tt)— / u(x)(-ix)q(-i)-p -- dx J—oo dx* = (27r)-*(-iy|~ { £!*(*)(-<*)*} {-i)-ve~ik*dx. (48*) This implies (48). Step 2: We want to show that F: S —> S is sequentially continuous. Let 9 < m. By (48*), /oo P (l + |xP)^|?/r)(x)|cte -°° r=0 -oc 1 + N2 ^' ^ ^ f°° dx < const / ||ifc||m+2>P i , I |2 ^ const||^||m+2,p. J—oo *■ ' Pi This implies IMIp.g < const(p,g)||ifc||9+2>p. Recall that a = Fu. Since F: S —* 5 is linear, we get 11Fix - Fun\\p,q < const(p, q)\\u - un\\q+2,P for all p, q = 0,1,... , and all u, un G 5. Consequently, as n —* oo, ixn —► u implies Fun —> Fu. Step 3: Let us prove the following key formula: /oo poo a(k)v(k)eikydk= / &(*)«(« +j/)dz, (49) -oo J — oo

218 3. Hilbert Spaces and Generalized Fourier Series provided a = Fu and b = Fv with u,v G S. In fact, by the Fubini theorem, /oo /»oo / /»oo \ a{k)v{k)eikydk = (2ir)~ */ v{k)eiky I / e-ife:Eu(a:)<fc Idk -oo J — oo \J — oo I* / /oo / /»oo w u(x)l / e-^^-^v^jda? ^oo x ^oo = / u(x)b(x — y)df = / b(z)u(z + y)dz J— oo J—oo (cf. Iterated Integration in the appendix). 2 2 Step ^: Let £ > 0 and choose v(x) = e~§~2~~. Using the substitution - fc2 z = ex, it follows from Example 2 and b = Fv that b(k) = £-1e 2^. By (49), Z400 e2fc2 r00 z2 / a(fe)e ~^~el*ydk = e L / e *e*u(z + y)dz J—00 «/ — 00 /OO 2 e-^u(et-\-y)dt. (50) -00 -00 Observe that roc ^ ^o< / e 2 |^(^ + 2/)|eft < const / J— 00 J — c e 2 d£ < 00, for all y G R and £ > 0, since u G <S. Thus, letting £ —* +0 in (50), it follows from the Lebesgue dominated convergence theorem (cf. the appendix) that /OO /»00 2 a(k)eikydk = (2-K)~1u(y) / e-V<ft = u(p) for all y € R. -OO J —OO (51) Instead of (51), let us write Ga := u. This way we obtain the linear sequentially continuous operator G:S —> S. Observe that G is obtained from F by replacing e~lkx with ezfc:E, and use the same argument as in Step 2. Step 5: We want to prove that the linear operator F: S —» S is bijective. By (51) with a = Fu, we get GFu = u for all u G 5. (52) Replacing e~2fc:E with elkx, the argument from Step 4 yields FGu = u for all u e 5. (53) By Proposition 6 in Section 1.20, F:«S —» 5 is bijective and F-1 = G.

3.8 The Fourier Transform of Tempered Generalized Functions 219 Similarly as in Step 2 it follows that F~X:S —» S is sequentially continuous. Step 6: Let us prove (47). By (49) with y = 0, /OC />OC (Fu^k^F^bX^dk = / u(z)b(z)dz for all u,beS. -oc «/ — oc Set 6 = it). Then /oc t»(*)e*teda; = (Fw)(k) for all A; € R. -OC Hence /oo /»oo {Fu){k){Fw){k) dk= u{z)w{z) dz for all u, w G 5. -oc «/—oc This is (47). □ Proposition 5. The Fourier transformation F:S —> S can be uniquely extended to a unitary operator F:L%(R)-*L%(R). Proof. By the definition of <S, CS°(R)cc$cl£(R). Since the set C^°(R)C is dense in l£(R), so is the set S. By (47), (F^ | Fv) = (u\v) for all u,v E S, where (• | •) denotes the inner product in L^R). The assertion follows now from Standard Example 2 in Section 3.6. 0 / □ 3.8 The Fourier Transform of Tempered Generalized Functions Definition 1. The set Sf consists exactly of all the linear, sequentially continuous mappings T:S-+C, i.e., we have T G Sf iff T(au + /3v) = aTu + /3Ti; for all a, /? G C, u, v G 5,

220 3. Hilbert Spaces and Generalized Fourier Series and, as n —* oo, un —> u implies Tun —» Tu. The elements T of S' are called tempered generalized functions (or tempered distributions). Definition 2* Let T G S'. The Fourier transform FT of T is denned by (FT)(u) := T{Fu) for all u G S. Proposition 3. The operator F:S' —> S' is linear and bijective. s s Proof. Let T G S'. If un —> u, then Fun —> Fu, as n —► oo. Hence (FT)(un) = r(Fixn) -> T(Fix) = (FT)(u) as n -> oo. Consequently, FT eS'. Let S G 5'. Define Tu = SiF^u) for all u G 5. Then, (<FT)(t4) = SiF^Fu) = S{u) for all u G 5. Hence FT = 5, i.e., F: 5' —» 5' is surjective. Moreover, let FT = 0. Then T(Fix) = 0 for all u G 5, and hence Tv = 0 for all v G 5, i.e., T = 0. Thus, F:«S' -+ 5' is bijective. Let a, /? G C and T, 5 G 5'. Then F(aT + /?S)(u) = (aT + 0S){Fu) = aT(Fu) + /?S(Fu) - a(FT)(ti) + 0{FS){u) for all u G S. Hence F(aT + /3S) = aFT + /3F5, i.e., F: S' -> 5' is linear. D Standard Example 4. Let v: R —> C be a measurable bounded function (e.g., v is continuous and bounded, i.e., \v(x)\ < const for all x G M). Define /oo v(rc)w(a;)da; for all u G 5. -oo Then, T € <S'. Proof. Let un —> u as n —* oo. Then An := sup(l + x2)\un(x) — u(x)\ —» 0 as n —* oo. xeR

3.8 The Fourier Transform of Tempered Generalized Functions 221 Hence \T{un - u)\ < f°° J^i (1 + x2)\un{x) - u{x)\dx J — OO "*■ ' X < const An —» 0 as n —» oo, i.e., Tun —» Tu as n —» oo. D Standard Example 5. Let ?/ G R. Define the tempered delta distribution 6y through 6y(u) :— u(y) for all u G S. Then (i)*y€S'. (ii) F^ = (2tt)-4 "e-***." (iii) «y = (27r)-*F-1 ("e"^"). Here, the tempered generalized function "e~lky" corresponds to the classic function a(k) = e~lky for all k G R and fixed 7/ G R, in the sense of Standard Example 4. That is, /oo e~ikvu{k)dk for all ue5. -OO o Proof. Ad (i). Let un —► ix as n —* oo. Then sup \un{x) — u{x)\ —» 0 as n —* oo. :r£R Hence 6y(un) —» <$2/(t0 as n —* oo. Ad (ii). For all u G 5, y'/i v~/>00 (F«„)(u) = «y(F«) = (Fu)(|/) = / u{k)e~ivkdk. Ad (iii). Observe that F\Sf —» 5' is bijective. □ Remark 6 (The language of physicists). Instead of (ii) and (iii) from Standard Example 5, physicists formally write /oo e~ikx6(x - y)dx for all k, y e R, (54) -OO

222 3. Hilbert Spaces and Generalized Fourier Series and /oo eik(x-y)dk for all Xj y € R> (55) -OO Formally, (54) follows from the "naive interpretation" J_ f(x)6(x — y)dx = /(y) of the Dirac delta function 6. Moreover, (55) follows from (54) by means of the inverse Fourier transformation if we regard 6 as a "classical function." Formulas (54) and (55) are frequently used in quantum physics. Applying the Dirac calculus from Section 5.21, physicists "elegantly obtain" (55) in the following way: /°° eik(*-y)dk = ^(x | fc)(fc | y) = (x | y) = 6(x - y). -oo i_ Problems 3.1. Density. Let M be a subset of a Hilbert space X over K. Show that the set span M is dense in X iff (u | v) = 0 for all v € M implies ix = 0. 3.2. The Parseval equation. Let (un)n>i be an orthonormal system in the separable Hilbert space X over K. Show that (un) is complete iff ]P \(un | u)|2 = |H|2 for all ue X. n>l 3.3. A fundamental completeness theorem. Let —oo < a < b < oo. We are given a measurable function /: ]a, b[—* K (e.g., / is continuous) such that |/(a;)| < Ce-^ for all x G M and fixed a > 0 and C > 0. Show that the linear hull of the system {#n/(x)}n=o,i,... is dense in the Hilbert space Lf(a,b). Hint: Use a similar argument as in the proof of Standard Example 2 in Section 3.4. Cf. Kolmogorov and Fomin (1975), Section 8.4.3. 3.4. The completeness of the system of the Laguerre functions. Starting from the system xne"^, n = 0,l,... , £<EM, the Schmidt orthogonalization method yields a system of functions Ln(x)e~^, n = 0,l,..., x€R. (56) Show that the following are true:

Problems 223 (i) System (54) forms a complete orthonormal system in Lf (0, oo). (ii) Explicitly, Ln(x) := ^—^- e-x — (e-xxn), n = 0,1,... , x G R. Hint: Use a similar argument as in the proof of Standard Example 2 in Section 3.4 and use Problem 3.3. 3.5.* Properties of the Fourier transform. Let /: R —» R be a measurable function (e.g., / is continuous) such that JR \f(x)\dx < oo. Assume that / f(x)e~ixtdx = 0 for all t G M. Then, f(x) = 0 for almost all x G R. Study the proof in Rudin (1966), p. 200. 3.6. Applications to density. Problem 3.5 can be used in order to prove the density of certain sets in X := L%(R) via the Fourier transformation F: X —► X. Let D denote the set of all the Gaussian functions uaAx) := e'^x~a)2 for all x G R, where a G R and /? > 0. Show that D is dense in X. Solution: Since F: X —» X is a unitary operator, it is sufficient to show that F(D) is dense in X. Observe that, for all fcGl, /oo -OO - (2b)^e~iakw(k), where w(fc) := e w . To prove that span F(D) is dense in X, by Problem 3.1, we have to show that /oo (Fua^){k)v(k)dk = 0 for all a G R, 0 > 0, (57) -oo implies v(k) = 0 for almost all k G R. In fact, from (57) we get /oo eiakw(k)v(k)dk = 0 for all a G R. -OO Since v, w G I^Wj we Set J^ |^(&)w(fc)M& < oo. Thus, Problem 3.5 implies that v(k)w(k) = 0 for almost all fcGl, and hence v(k) = 0 for almost all fcGM. 3.7.* The fundamental Payley-Wiener theorem. Let us consider the Fourier transform /oo -oo

224 3. Hilbert Spaces and Generalized Fourier Series Then, for each fixed R > 0, the following two statements are equivalent: (i) The function F: C —> C is holomorphic and, for each N = 1,2,..., there is a constant Cn > 0 such that \F(z)\ < CN(1 + \z\)-NeR\lm *l for all zeC. (ii) The function / belongs to Cq°(R)c and vanishes outside the interval Study the proof in Yosida (1980), Chapter 6. 3.8. The tensor product X ® Y. Let X and Y be linear spaces over K, and let X* denote the space of all linear functional u*: X —» K. Define (u®v)(u*,v*) :=u*(u)v*{v), (58) for all u G X, v G Y, u* G X*, v* eY*. Obviously, u®v is a bilinear form on X* x y*. Furthermore, let X (8) y denote the set of all possible finite linear combinations ^2uj®vk, (59) where Uj e X, Vk € Y. Thus, each element of X (g) Y is a bilinear form on X* x y* given by (59), i.e., Y^uj ®vk> (u*,v*) = Y^(uj ®vk)(u*,v*), j,k J j,k for all u* e X*, v* e Y*. Naturally enough, if a, b G X (g) y, then we say that a is identical to b iff the corresponding bilinear forms are identical, i.e., a = b iff a(u*,v*) = b(u*,v*) for all u* G X*, v* G y*. (60) Observe that different expressions (59) may correspond to identical bilinear forms, i.e., the representation (59) is not unique for the elements of X®Y. Show that (i) X (g) y is a linear space, by means of the natural linear operations for bilinear forms. More precisely, if a, b G X (8) Y and a, /? € K, then aa + /36 is given by (aa + 0b)(u*,v*) := aa(^*,t;*) + /36(^*,t;*) for all u* G X>* G y*.

Problems 225 (ii) The symbol "(8)" behaves like a product, i.e., for all u, v G X,w,z G Y, and a, /? G K, we get the following distributive laws: (ptu + /?v) (8) w — a(w ® w) + j3{v (8) if) and u (8) (aw + /?z) = a(u <g> w) + /?(m (8) z). (iii) Let {i^i,..., u^} and {^i,..., vm} be a basis of the finite-dimensional linear spaces X and Y. Then, {(uj®vk):j = l,...,N, fc = l,...,M} forms a basis of the tensor product X (8) Y. (iv) Let X and F be Hilbert spaces. Set (u®v \w®z) := (u\ w)(v | 2;), and generalize this definition to linear combinations in a natural way by letting ]P Uj (8) Vk \^Wr<g)Zs J := ]P Prs{Uj ®Vk\Wr® Zs)- 3>>k ri8 I 3,k,r,s Then, we get an inner product on X (8) Y. (v) Consider situation (iii). If X and Y are finite-dimensional Hilbert spaces and {uj} and {vk} is an orthonormal basis of X and Y, respectively, then {(uj (8) vjt)} is an orthonormal basis of the Hilbert space X (8) Y\ Solution: Ad (i), (ii). Use elementary computations. Ad (iii). Define Ur ^2ajuj I := ar and V« ( Yl@kVk ) ''= @s' By (ii), each a € X <g>Y can be represented as a linear combination of the elements u3- ®Vfc. Moreover, it follows from ^ay*.^ ®vfc) = 0 3,k that 0 = ]T]&jk(u<j (8) Vfc)(w*, v*) = ars for all r, s.

226 3. Hilbert Spaces and Generalized Fourier Series Ad (iv). We first want to show that the definition of (• | •) is independent of the choice of the representatives. For given w E X and z E Y, define w* G X* and z* G Y* by w*(u) := (w | u) and z*(v) :— (z \ v), for all u G X and uGF, respectively. According to (58), (u®v)(w*,z*) = w*(u)z*(v) = (w | ifc)(z \v) = (w ®z \u®v). (61) Thus, ix (8) v = 0 implies (iu ® 2 | ^ (8) v) = 0 for all w, z G X. Now let a,b,cE X ®Y. Suppose that a = b. Then a — b = 0. Using linear combinations, it follows from (61) that (c\a-b) = 0. Thus, a = b implies (c\ a) = (c\b). Furthermore, one checks easily that (aa + 0b\c) = a(a \ c) + 0(b | c), (a\b) = (b | a), for all a, 6, c G X (g) y and a, /? E K. Finally, we have to show that (a | a) > 0 and (a | a) = 0 implies a = 0. (62) To this end, let a = ]T] ajk{uj®Vk)> Set L := span{iXj} and M := span^}. Choose an orthonormal basis {e^} and {/&} of L and M, respectively. By (ii), Since (ej <g> A | er <g> /s) = (e,- | er)(A | /s) = £j A*> (a\a) = Y^PjkPjk. By (iii), a = 0 iff /3ifc = 0 for all j, k. This yields (62). 3.9. The tensor product X ® Y ® Z. Let X, Y, and Z be linear spaces over K. Similarly to Problem 3.8, we set (ia® v ®w)(u*,v*,w*) := u*(u)v*(v)w*(w) for aHueX,v£Y,w€Z,u* eX*,v* e Y, w* E Z*. Thus, u <g> v<g> is a trilinear form on X*xY*x Z*. By definition, the tensor product X®Y®Z consists of all possible finite linear combinations j,k,m where Uj E X, Vk E y, wm E Z. Show that

Problems 227 (i) X (g) Y (g) Z is a linear space. (ii) The symbol "(g)" behaves like a product. That is, for all u,v E X, y G y, z e Z, and a, /? € K, we get (aw H- /?v) (8) y ® z = a(w (8) y <8) z) + /?(v ®y®z), and so on. (iii) If {uj}, {vk}, {vim} form a basis of the finite-dimensional linear spaces X, y, Z, respectively, then {uj <g>Vk<8>wrn} forms a basis of the tensor product X (8) y (8) Z. (iv) Let X, y, Z be Hilbert spaces. Define (u (8) v (8) w | a? <8) y <8> 2r) := (w | #)(v | y)(w \ z) and extend this definition to linear combinations in a natural way. Then, we get an inner product on X (8) Y (8) Z. (v) If {v>j}, {vk}, {wm} form orthonormal bases of the finite-dimensional Hilbert spaces X, Y, Z, respectively, then {uj (8) Vk (8) wm} forms an orthonormal basis of the Hilbert space X ®Y (8) Z. It will be shown in Section 2.22 of AMS Vol. 109 that Tensor products describe composite states of elementary particles.

4 Eigenvalue Problems for Linear Compact Symmetric Operators The validity of theorems on eigenfunctions can be made plausible by the following observation made by Daniel Bernoulli (1700-1782). A mechanical system of n degrees of freedom possesses exactly n eigensolutions. A membrane is, however, a system with an infinite number of degrees of freedom. This system will, therefore, have an infinite number of eigenoscillations. Arnold Sommerfeld, 1900 In 1900 Fredholm had proved the existence of solutions for linear integral equations of the second kind. His result was sufficient to solve the boundary-value problems of potential theory. But Fred- holm's theory did not include the eigenoscillations and the expansion of arbitrary functions with respect to eigenfunctions. Only Hilbert solved this problem by using finite-dimensional approximations and a passage to the limit. In this way he obtained a generalization of the classical principal-axis transformation for symmetric matrices to infinite-dimensional matrices. The symmetry of the matrices corresponds to the symmetry of the kernels of integral equations, and it turns out that the kernels appearing in oscillation problems are indeed symmetrical. Otto Blumenthal, 1932 A great master of mathematics passed away when Hilbert died in Gottingen on February 14, 1943, at the age of eighty-one. In retrospect, it seems that the era of mathematics upon which he impressed

230 4. Eigenvalue Problems for Linear Compact Symmetric Operators the seal of his spirit, and which is now sinking below the horizon, achieved a more perfect balance than has prevailed before or since, between the mastering of single concrete problems and the formation of general abstract concepts. Hermann Weyl, 1944 In this chapter we want to study the following eigenvalue problem: An = \u, u G X, AeK, u ^ 0, (1) on the Hilbert space X'over K, along with applications to integral equations and boundary-value problems. Each solution (u, A) of (1) is called an eigensolution of A, where u is called an eigenvector and A is called an eigenvalue of A, respectively. Recall that K = R or K = C. The set of all the eigenvectors u that correspond to a fixed eigenvalue A is called the eigenspace to u. By definition, the eigenvalue A has finite multiplicity iff the corresponding eigenspace has finite dimension. In this chapter we will assume that A: X —> X is a linear compact symmetric operator. We want to show that such operators possess a complete orthonormal system of eigenvectors. In the next chapter we will study problem (1) for more general symmetric operators A: D(A) C X —> X, along with applications to the Laplace and Poisson equations, the heat equation, the wave equation, and the Schrodinger equation in quantum mechanics. 4.1 Symmetric Operators Definition 1. The linear operator A: D(A) C X —> X on the Hilbert space X over K is called symmetric iff the domain of definition D(A) is dense in X and (Au | v) = (u | Av) for all u, v G D(A). Proposition 2. Let A: D(A) C X —> X be a linear symmetric operator on the Hilbert space X over K. Then (i) (Au | u) is real for all u G D(A). (ii) All the eigenvalues of A are real (iii) Two eigenvectors of A with different eigenvalues are orthogonal. (iv) Let {ui,U2, • • •} be a finite or countable complete orthonormal system of eigenvectors of A. Then the corresponding system {Ai, A2,...} of eigenvalues contains all the eigenvalues of A.

4.1 Symmetric Operators 231 Proof. Ad (i). Let u G D(A). Then, (Au | u) = (u | Au) = (Au | u). Ad (ii). It follows from (1) that \(u | u) = (u | Ait) = (Ait | u) = X(u | it) with (u\u) =fi 0. Hence A = A. Ad (iii). Prom Au = Xu and Av = \iv along with A, \i G R a id A ^ /jl it follows that (A - /x)(w | v) = (Au \v) - (u\ Av) = 0, and hence (u \ v) = 0. Ad (iv). Since {un} is complete, we have N u = y^(^n | u)un for all it G X, (2) n=l where iV is a natural number or "N — oo." Let Au = Xu with u =^ 0 and A ^ An for all n. By (iii), (wn | u) = 0 for all n. Hence u = 0, by (2). This contradicts w ^ 0. □ Proposition 3. Let A: X —> X be a linear continuous symmetric operator on the Hilbert space X over K with X ^ {0}. Then sup \{Au | «)| = \\A\\. ||u||=i Proof. Set ol :— su.p|jlfc||==|_ \(Au \u)\. Since A is linear, \(Av \v)\ < a\\v\\2 for all v G X. By the Schwarz inequality, \(Au | w)| < \\Au\\ \\u\\ < \\A\\ \\u\\2 for all ueX. Hence a < \\A\\. To prove that ||A|| < a, we set v± := Xu ± A_1Aw, A > 0. It follows from (A2w | u) = (Aw | Au) and ||Aw||2 = (Aw | Au) that ||Am||2 = ^[(Av^ | v+) - (A<;_ | v-)} ^^VKf + llv-H2) = 2-1^(A2||w||2 + A-2||Aw||2). Assume first that Aw ^ 0. Letting A2 = \\Au\\ and \\u\\ = 1, we find that ||Aw||2 < a||Aw|| if Au ^ 0. Hence \\Au\\ < a for all u £ X with ||w|| = 1. This implies ||A|| < a. □

232 4. Eigenvalue Problems for Linear Compact Symmetric Operators 4.2 The Hilbert-Schmidt Theory Theorem 4. A. Let A: X —> X be a linear compact symmetric operator on the separable Hilbert space X over K with X ^ {0}. Then, the following hold true: (i) The operator A has a complete orthonormal system of eigenvectors. (ii) All the eigenvalues of A are real, and each eigenvalue A ^ 0 of A has finite multiplicity. (iii) Two eigenvectors of A that correspond to different eigenvalues are orthogonal. , />- / (iv) If the operator A has a countable set of eigenvalues (e.g., X = 0 is not an eigenvalue of A and dim X = oo), then the eigenvalues of A form a sequence (An) such that Xn —> 0 as n —> oo. Proof of Theorem 4.A under the additional assumption (A). Suppose that (A) An = 0 implies u = 0 and let dim X = oo. Since X ^ {0}, this implies A ^ 0 and hence ||A|| ^ 0. Step 1 is decisive. Step 1: Variational problem for constructing an eigensolution. We consider the maximum problem \(Au | u)\ = max !, ||u|| = 1. (3) By Proposition 3 in Section 4.1, the maximal value is equal to ||A||. Thus, there exists a sequence (vn) with ||i;n|| = 1 for all n such that \(Avn | vn)\ -► ||A|| as n -► oo. Set an := (Avn \ vn). Since the real sequence (an) is bounded, there exists a convergent subsequence of (an). Consequently, there exist a subsequence, again denoted by (vn), and a real number Ai such that (Avn | vn) —> Ai as n —> oo. Therefore, |Ai| = ||A||>0. This implies ||-A^n|| < \\A\\ \\vn\\ = |Ai|, and hence 0 < \\Avn - Ai?;n||2 = ||Avn||2 - 2X1(Avn | vn) + X\ ^ 0 as n ^ oo.

4.2 The Hilbert-Schmidt Theory 233 The operator A is compact. Thus, there exists a subsequence, again denoted by (vn), such that (Avn) converges. Since Avn — \\vn —> 0 as n —> oo, and Ai ^ 0, the sequence (vn) also converges to a certain element m, i.e., Vn —> u\ asn-> oo. This implies Au\ — \\U\ =0, iti G X, ||wi || = 1, i.e., (wi,Ai) is an eigensolution of A. Step 2: Induction. Let Y :={ueX:(u\u1) = 0}. Then, Y is a closed linear subspace of X. The fcey to our induction argument is the relation A(Y) C F, (4) i.e., the Hilbert space Y is invariant with respect to the operator A. In fact, let u G Y. Then (Au | u\) = (u | Aui) = Xi(u | ui) = 0, and hence Au G Y\ Since dim X = oo, y ^ {0}. Furthermore, A ^ 0 on 7. Otherwise there would exist an element u G y with u y^ 0 and Ait = 0, which contradicts assumption (A). Therefore, we may apply Step 1 to the restricted operator A:Y —> Y. This way we obtain the eigensolution (u2, A2), i.e., Au2 = \2U2, w2e7, ||^2|| = 1. According to Step 1, |A2| is equal to the norm of A: Y —> y. Hence |A2| = sup ||Av||. llv||=i,vGy By Step 1, |Ai| = ||i4||= sup \\Av\l \\v\\=i,vex and hence |Ai| > |A2| >0. We now set Z := {u G y: (it | it2) = 0} and continue this procedure. This way we obtain a countable system {un, Xn} of eigensolutions, i.e., Aun = \nun, n = 1,2,... , |Ai| > |A2| > • • • > 0,

234 4. Eigenvalue Problems for Linear Compact Symmetric Operators where {un} is an orthonormal system. Step 3: We show that An —> 0 as n —> oo. Otherwise, |An| > const > 0 for all n, and hence the sequence (\~lun) is bounded. The operator A is compact Since A{\~1un) = un, n = 1,2,... , the sequence (un) contains a convergent subsequence. But this is impossible, since (un \ Um) = 0 for n ^ m, and hence \\un — ^m||2 = ||^n||2 + ||^m||2 = 2 for all n, m with n/m, i.e., no subsequence of (un) is Cauchy. Step 4: We show that oo An = yZ ^n{un | u)un for all it G X (5) n=l To this end, let ra Wm:=u- ^2(un I ^)wn> ra = 1,2,... . n=l Set V := {ue X: (u \ Uj) = 0, j = 1,..., m}. By Step 2, A(F) C V, and |Am+i| equals the norm of A: V —> V, i.e., Il-A^ll < |Am+i| ||v|| for all v G V. Obviously, Wm G V, and hence P^m|| < |Am+i| ||wm||. Since ra ||wm||2 = IM|2 — /J |(^n | u)\2 < \\u\\2 for all m, n=l we get ||-A^m|| < |Am+i| H^ll —> 0 as m —> oo. Hence ra -A^m = -A^ — Z^(Un I ^'Vi^n —► 0 as TTl —> OO. n=l This is (5).

4.2 The Hilbert-Schmidt Theory 235 Step 5: We show that oo U = Y^(Un I u)un- (6) n=l By Proposition 5 in Section 3.1, each Fourier series is convergent, i.e., there is some v G X such that v = Y2(Un I u>>Un' n=l Hence Av = ^2(un | it)A^n = ^2(un | u)\nun = Au. n=l n=l By assumption (A), it follows from A(v — u) = 0 that v — u = 0. Step 6: We show that each eigenvalue A ^ 0 of A has finite multiplicity. Since {itn} forms a complete orthonormal system in X, A is identical to some Am, by Proposition 2 in Section 4.1. For simplifying notation, assume first that A = Ai. Since An ^ 0 and An —> 0 as n —> oo, there exists a number N such that Ai = • • • = Ajv and Xj ^ Ai for all j > N. Let Au = Xu, u ^ 0. By (5), oo AT Xu = ^2 ^n{un | U)un = Ai ^(^n | ^)ltn, n=l n=l since two eigenvectors of A with different eigenvalues are orthogonal, i.e., (un | u) = 0 for all n > N. Hence {^i,...,^n} forms a basis of the eigenspace to A, i.e., A has the multiplicity N. The same argument applies to the general case where A = Am. The proof of Theorem 4.A is complete under the additional assumption (A). □ Proof of Theorem 4. A under the additional assumption (B). Suppose that (B) Au = 0 implies u = 0 and let dim X = N, where N = 1,2, In this case, it follows from Step 2 that there exist eigensolutions Aun = Xnun, n = l,...,M, where S := {u\,..., um} is an orthonormal system. Let M be the largest possible number. The construction from Step 2 shows that M = N. Hence the system S with M = N is complete.

236 4. Eigenvalue Problems for Linear Compact Symmetric Operators This finishes the proof of Theorem 4.A under the additional assumption (B). □ Proof of Theorem 4. A. Finally, let us consider the general case. We may assume that A = 0 is an eigenvalue of A. Otherwise, we meet assumption (A) or (B). Set N(A) := {ue X:Au = 0}. Then, N(A) ^ {0}. Since the operator A is continuous, N(A) is a closed linear subspace of X. In fact, if Aun = 0 for all n and un —> u as n —> oo, then An — 0. By Proposition 1 in Section 3.3, there exists a complete orthonormal system {wk} in N(A). Hence Awk = 0 for all k. Moreover, it follows from Corollary 1 in Section 2.9 that for each u G X there exists the unique decomposition u = w + z, weN(A), zeN(A)1-. (7) Recall that N(A)1- := {z € X: (z | w) = 0 for all w G N(A)}. Thus, N(A)1- is a closed linear subspace of X. We have these conditions: (a) The operator A maps N(A)1- into N(A)1-. (b) If Az = 0 with z G N(A)1-, then z = 0. Ad (a). Let z e N(A)1-. Then (Az \w) = (z\ Aw) = 0 for all w G N(A), and hence Az G N(A)1-. Ad (b). If Az = 0 with z G N(A)1-, then z G N(A) n iV^)-1. By the uniqueness of the decomposition (7), z = 0. We now apply Theorem 4.A with the additional assumption (A) to the restricted operator AiNiA)1- ^N(A)1-. This way we get a complete orthonormal system {un} of eigenvectors of A on N(A)1-. Recall that {wk} forms a complete orthonormal system in N(A). By (7), for each u G X, U = W + Z = Y2(Wk I W)Wk + Y2(Un I ^n' fe n

4.3 The Predholm Alternative 237 Since w G N(A) and z G N(A)1-, we get (wk \ z) = 0 for all k and (un | w) = 0 for all n. Hence u = Y^(wk I u)wfc + Yl(Un I ^n* fe n Consequently, {^1,^1,^2,^2, • • •} represents a complete orthonormal system of eigenvectors of A. The proof of Theorem 4. A is complete. □ 4.3 The Fredholm Alternative Let us consider the equation \u - An = 6, u G X, (8) along with the homogeneous problem f4v-> = 0, ve-Y. (8*) Let us make the following assumption: (H) The operator A:X —> X is linear, compact, and symmetric on the separable Hilbert space X over K with X ^ {0}. Theorem 4.B. Assume (H). We are given b G X and the number A G K iwtfi A ^ 0. Tften, tte original equation (8) ftas a solution iff (b\v) =0 for all solutions v of (8*). (9) Corollary 1. Assume (H). We are given b G X and A G K with A ^ 0. Suppose that equation (8) ftas a£ mos£ one solution. Then, the following hold true: (i) There exists the linear continuous operator (XI — A)~l:X —> X. (ii) Equation (8) ftas tte unique solution u = (XI — A)~1b. This corollary tells us that the following important principle holds true for the original problem (8): Uniqueness implies existence. Corollary 2. Assume (H) with K = C. Then, the following are met:

238 4. Eigenvalue Problems for Linear Compact Symmetric Operators (i) If dim X < oo, then the spectrum a(A) of the operator A consists precisely of all the eigenvalues of the operator A. (ii) If dim X = oo, then the spectrum cr(A) consists of all the eigenvalues of A together with the point A = 0. Proof of Theorem 4.B. Suppose first that the operator A has a countable system of nonzero eigenvalues. By (5), there exists an orthonormal system {un} in X such that oo Au-^2, An(^n | U)un, (10) n=l along with Aun = \nun for all n. The system {An} contains all the nonzero eigenvalues of A, and An —> 0 as n —* oo. (11) Furthermore, (un | Au) = (Aun | u) = Xn(un I u) for all u G X and all n. (12) Case 1: Suppose that A ^ An for all n, i.e., equation (8*) only has the trivial solution v = 0. Then, equation (8) has at most one solution. Let it be a solution of (8). Then, u = A _1 ^&+]PAnKi I u)un \ • By (8) and (12), (A — An)(^n I u) = (un I (XI — A)u) = (un | b) for all n. (13) Hence f 1 An it = A 1 < b + 5^ an(?/n I 6)itn f , where an := -—^—. I n=l J A"An Conversely, it follows from the Bessel inequality £|(Un|&)|2<||6||2 (14) along with An —> 0 as n —> ooHhat |an| < const for all n, and hence oo X]KKI&)l2^constll&ll2- n=l

4.3 The Predholm Alternative 239 By Proposition 5 in Section 3.1, the series (14) is convergent. It follows from (10) with u = b and (14) that {OO "j OO Ab + ^2 anXn(un | b)un > = ^ A_1An(l + an)(un \ b)un n=l J n=l and OO \u = b + ]P an(un | b)un. n=l Hence ~ Am-Am = 6, rtf.i u^ ^ \ki I i.e., u is a solution of (8). Since this solution is uniquef the inverse operator (XI - A)"1: X —> X exists, In addition, (14) tells us that OO INI2 = |A-2| ll&H2 + £2qb|(«b I 6)|2 + a2 |(W„ | 6)|2 n=l oo <|A-2| ||6||2 + ^conSt|(Un|6)|2 n=l < const ||6||2. This implies ||u|| < const H&H for all b e X. Hence the linear operator (XI — A)_1:X —> X is continuous. Case 2: Suppose A is an eigenvalue of A, i.e., A = Am for some m. To simplify notation, let us assume that A = Ai. Then Ai = A2 = • • • = A;v for some natural number N and An ^ Ai for all n > N. If u is a solution of (8), then it follows from (13) that (un I b) = 0 for all n = 1,..., N. (15) This is equivalent to condition (9). Hence u = A"1 < b + J2 an(un I b)un > , an = n . (16) I n=^+i J A"An Conversely, let condition (15) be fulfilled. As in Case 1, one checks easily that u from (16) satisfies Xu — An = 6, i.e., u is a solution of (8). Finally, observe that if the operator A has only a finite number of nonzero eigenvalues An, then the series from (14) and (16) reduce to finite sums En ... . □ Proof of Corollary 1. If equation (8) has at most one solution, then k -Xu — Au = Xv — Av implies u = v.

240 4. Eigenvalue Problems for Linear Compact Symmetric Operators By the linearity of A, this is equivalent to the fact that Xw—Aw = 0 implies w = 0. The assertion follows now from Case 1 in the preceding proof. D Proof of Corollary 2. Let A ^ An for all n and A ^ 0. By Corollary l(i), the point A G C belongs to the resolvent set of A. Ad (i). Statement (i) follows from Problem 1.4. Let us give a different proof. If A = 0 is an eigenvalue of A, then 0 G cr(A), by Section 1.25. If A = 0 is not an eigenvalue of A, then Av = 0 implies v = 0. By Theorem 4.A, the operator A has a complete orthonormal system {un} of eigenvectors. Set u-^Y^K1^ I b)un. Then, Au = b and \\u\\ < const ||6|p Thus, the operator A~lm.X —> X is continuous, i.e., A = 0 belongs to the resolvent set of A. Ad (ii). Suppose first that the operator A has only a finite number of nonzero eigenvalues. Since all these eigenvalues have finite multiplicity and dim X = oo, it follows from Theorem 4.A that there is some v ^ 0 for which Av = 0, i.e., A = 0 belongs to the spectrum &(A). Suppose now that the operator A has a countable set {An} of nonzero eigenvalues. Then, An —+ 0 as n —+ oo. Since the spectrum cr(A) is compact, the limit point A = 0 belongs to cr(A). D 4.4 Applications to Integral Equations We want to study the following integral equation: A(x, y)u{y)dy = Xu(x), a < x <b. (17) / J a Let —oo<a<6<oo. We are looking for eigensolutions X G R and u G 1/2 (a, b) with u ^ 0. To this end, we assume the following: *t Cxc (HI) The function A: [a, b] x [a, b] —> R is continuous. (H2) The function A is symmetric, i.e., A(x, y) = A(y, x) for all x, y G [a, b]. We set X := L2(a, b) along with the inner product fb (u | v) := / it(x)^(x)dx. We also define the integral operator (Au)(jc) := / A{x,y)u(y)dy. J a

4.4 Applications to Integral Equations 241 Then, the original equation (17) can be written in the following form: Au = Xu, AgR, ueX, u^O. (17*) Proposition 1 (Eigensolutions). Under assumptions (HI) and (H2), the following hold true: (i) The original integral equation (17) has a countable system of eigen- functions {ui,v,2,...}, which forms a complete orthonormal system in the Hilbert space L<i(a, b). (ii) Two eigenfunctions u and v of (17) which correspond to different eigenvalues are orthogonal in Z,2(a, 6), i.e., (u | v) = 0. (iii) Each nonzero eigenvalue of (17) has finite multiplicity. (iv) If the integral equation (17) has a countable number of eigenvalues (e.g., X = 0 is not an eigenvalue of (17)), then all the nonzero eigenvalues of (17) form a sequence (Xn) with Xn —> 0. ^ -» t^"**->t (v) For eac/i eigenvalue X ^ 0 of (17), the eigenf unctions u are continuous on [a, 6]. By (i), for each u G L<i(a, 6), the Fourier series oo ^ = J^(^n | u)un (18) n=l converges in L<i(a, 6), i.e., ,6 / m \ 2 lim / I u(x) — y^(iAn | u)un(x) I ate = 0. ra—»oo / \ ^—' / Ja \ n=l / Corollary 2 (Classical convergence). Suppose that the function u allows the following representation: u(x) = / A(x,y)v(y)dy for all x G [a, 6], ./a where v G L2(a,b). Tften, £/ie Fourier series oo ^(x) = X^n I ^M^) (18*) n=l converges absolutely and uniformly on the interval [a, 6].

242 4. Eigenvalue Problems for Linear Compact Symmetric Operators In addition, we have (un \ u) — 0 if the eigenvector un corresponds to the eigenvalue A = 0. Proposition 1 follows immediately from Theorem 4.A by using the following result. Lemma 3. Assume (HI) above and let X := Li{a, b). Then, the following hold true: (a) The operator A: X —+ X is linear and compact. (b) IfueX, then An e C[a, b}. (c) //, in addition, (H2) holds, then the operator A:X —> X is symmetric. Proof. Ad (a), (b). Let u G X, and let \\u\\ denote the norm of u in X. By the Schwarz inequality, 2 \\u\\. j\-\u{y)\dy<(j\y\ [j['l«(»)N = (*-<*)* Set v := An, i.e., v(x) = / A(x,y)u(y)dy for all x G [a, 6]. J a Since the set [a, b] x [a, 6] is compact, the function A is uniformly continuous on [a, b] x [a, 6]. Thus, for each e > 0, there is a (5 > 0 such that x, z G [a,6] and \x — z\ < 6 implies a := max \A(x,y) — A(z,y)\ <e. a<y<b Hence \v(x) - v(z)\ <a \u{y)\dy < e(b - a)* ||u||, (19) J a for all x, z G [a, 6] with |sc — z\ < 6. This proves the continuity of the function v on [a, 6]. Moreover, we also get fb max |i;(sc)| < max |*4(#,y)| / \u(y)\dy < const \\u\\. (20) This implies ||Aw|| = ( / |^(x)|2dxj < const |H|. (21)

4.4 Applications to Integral Equations 243 Obviously, the operator A:X —» X is linear. Consequently, relation (21) tells us that A: X —> X is continuous. Let M be a bounded set in X. Then, it follows from (19) and (20) along with the Arzeld-Ascoli theorem (Example 7 in Section 1.11) that the set A(M) is relatively compact in C[a,b]. Each relatively compact set in C[a, b] is also relatively compact in X = 1/2(a, b). In fact, if vn —> ^ in C[a, 6] as n —* oo, then \\vn ~ v\\ = I / (vn(ic) ~ v{x))2dx I < max \vn(x) — v(x)\(b — a)2 —> 0 as n —> 00. a<#<6 Thus, the set A(M) is relatively compact in X, and hence A: X —> X is compact Ad (c). It follows from the Tonelli theorem1 (cf. Iterated Integration in the appendix) that, for all u, v G X, (Au I v) — / I / A{x, y)u(y)dy J v(a:)da: = / ( / "4(^ y)v{x)dx J u(y)dy = (w | Av), since ^4(x, y) = A(y, x) for all #, y G [a, 6]. □ Proof of Corollary 2. We are given u G £2 (a, b). Let An denote the eigenvalue to the eigenvector un, i.e., Aun = \nun, Xn G R. Observe that An = 0 is possible for some indices n. By (18), 00 Au = ^(tin I Aw)i^n. (22) n=l This series converges in I/2(a, 6). Observe that (ttn I Alt) = (Attn I It) = Xn(un I W). 1 Observe that pb pb / WaJ,y)I \u(y)\dy < const / \u(y)\dy J a J a (/V 2„ < const I / \u(y)\ dy ) < 00.

244 4. Eigenvalue Problems for Linear Compact Symmetric Operators We now want to study the classical convergence of the series oo V\itn | Au)un(x) for all x G [a, 6]. n=l Let k,m > 1. By the classic Schwarz inequality (10) from Chapter 1, fc+ra k+m ]P |(wn | Aw)wn(a?)| = ]P \(Un I ^)An^n(^)| n=k n=k l l (k+m \ 2 /fe+m \ 2 £lKM|2) (ElA-u"(x)l2) for all xe [a, 6]. (23) n=fc / \n=k / By the Bessel inequality (15) from Chapter 3, it follows that for all x G [a, 6] and k,m > 1, fc+ra fc+ra A:=]T|AnUn(z)|2-£ n=fc n=fe fc+ra = Y,\(A(x,-)\un)\2<\\A(x, n=k / A(x,y)un(y)dy Ja = / |^4(x,y)dy|2dy < (b - a) I sup |^4(x,y)| J < const. Ja \a<x,y<b J Since the Fourier series J2n(un \ u)un converges, it follows from Proposition 5 in Section 3.1 that the series oo Eikni2 n=l is also convergent. Thus, for each e > 0, there exists a number no(e) such that k+m 2~J \(un | u)\2 < e for all k > no(e), m > 1. n=k By (23), k+m y^ |(^n I A^)^n(^)| < A • e < const • £, n=fe for all A; > no(e), m > 1, and x G [a, 6]. This proves the absolute and uniform convergence of the series ^^=1(^n I Au)un{x) on [a, 6]. □ We now study the following nonhomogeneous integral equation: rb / ^4(x,y)u{y)dy — \u{x) = h(x), a <x <b. Ja (24)

4.5 Applications to Boundary-Eigenvalue Problems 245 To this end, we need the corresponding homogeneous equation b A{x, y)v(y)dy - Xv(x) = 0, a < x < b. (24*) Proposition 4 (The Predholm alternative). Assume (HI) and (H2). Let the function h G L<i{a, b) and the real number X ^ 0 be given. Then, the following hold true: (i) If X is not an eigenvalue of the homogeneous integral equation (24*), then the original equation (24) has a unique solution u G L<i{a, b). (ii) If X is an eigenvalue of (24*), then (24) has a solution u G I/2(a, 6) h(x)v(x)dx = 0, for all the eigenfunctions v corresponding to X. (iii) Ifhe C[a, b], then each solution u of (24) is continuous on [a, 6]. This follows from Theorem 4.B along with Lemma 3. Observe that equation (24) can be written in the following form: An - Xu = 6, u G X, A G R, where X := L<i{a, b) and (h \ b) := Ja h(x)v(x)dx. 4.5 Applications to Boundary-Eigenvalue Problems Let us consider the following boundary-eigenvalue problem: —U"{x) = IJLU(X), 0 < X < 7T, u(0) = u(tt) = 0, u G C2[0,tt], fi G R. This problem can be written in the following form: Au = fiu, /i G R, ueD(A), where An := — u" and D(A) := {u G C2[0,tt]:u(0) = u(ir) = 0}. / / (25) (25*)

246 4. Eigenvalue Problems for Linear Compact Symmetric Operators Let us also study the following integral equation: u(x) = nf Q(x,y)u(y)dy, 0 < x < it, u G Z/2(0,7r), (26) Jo where _/i^£ if0<s<y<7r G(X>y)-\(!^)v ifo<y<x<n. Observe the following: The Green function Q is continuous and symmetric on [0, tt] x [0,7r]. We set X :=Z,2(0,7r) and (u | v) := / u(x)v(x)dx for all u, v G X. Jo Lemma 1. (i) The linear operator A: D(A) C X —> X is symmetric. (ii) Two eigenf unctions u and v of (25) which correspond to different eigenvalues are othogonal in X, i.e., (u \ v) = 0. (iii) Each eigenvalue \x of (25) is positive. (iv) The original boundary-eigenvalue problem (25) is equivalent to the integral equation (26). Proof. Ad (i). Integration by parts shows that, for all u, v 6 D(A), /»7T /»7T (Au \v) = I (~-u")vdx = — w'v|q + / ufvf dx /»7T /»7T = / wVda; = W|q — / uv" dx = — J^ uv"dx = (u | Av), (27) since u and ?; vanish at the boundary points x = 0 and # = tt. Ad (ii). This follows from Proposition 2 in Section 4.1. Ad (iii). Let Au = /m, where \i G R, u G £>(.A), and it ^ 0. By (27), /i(it | it) — (Au \u) = I u,2dx > 0. Jo Hence ji > 0.

4.5 Applications to Boundary-Eigenvalue Problems 247 Ad (iv). If it is a solution of (25), then u is also a solution of (26), by Proposition l(ii) in Section 2.7.1. In this connection, set / := \iu. Conversely, if it is a solution of (26), then it follows from Lemma 3(b) in Section 4.4 that u G C[0,7r], and again Proposition l(ii) in Section 2.7.1 tells us that it is a solution of (25). □ Recall that eigenvalues of multiplicity one are called simple. Proposition 2. (i) The original problem (25) has precisely the eigenvalues fin = n2, n= 1,2,... , which are simple. (ii) The normalized eigenfunction un to nn with (un \ un) = 1 is given by un{x) = 7r~2 sinnx, n = 1,2,... . (iii) For each function u G D(A), the Fourier series oo u{x) = Y2(un I u)un(x) (28) n=l converges absolutely and uniformly on the interval [0,7r]. The same is true for u\x) = ][>„ I u)u'n{x). (28*) n=l For each function u G D(A) with u" G D(A), the series oo u"{x) = £>n | «)<(*) (28**) n=l converges absolutely and uniformly on the interval [0,7r]. (iv) For each function u G X, the Fourier series (28) converges in X := £2(0,7r), i.e., {ui,v,2,...} forms a complete orthonormal system in X. Applications of these results to the vibrating string can be found in Section 5.12. Proof. Ad (i), (ii). By Lemma 1, each eigenvalue \i of (25) is positive. The general solution of the differential equation —u" = /j,u, /i > 0, is given through u(x) = Csin^x + jDcos/i2#,

248 4. Eigenvalue Problems for Linear Compact Symmetric Operators where C and D are real constants. Prom u(0) = 0 we get D = 0. Moreover, the second boundary condition u(n) = 0 along with u ^ 0 implies //2 = ft, ?2 = 1,2, ... . Ad (iii). Let u G jD(A). It follows from Proposition l(ii) in Section 2.7.1 with / := u" that u(x) = - G(x,y)u"(y)dy. Jo Thus, the assertion for (28) follows from Corollary 2 in Section 4.4. Let us prove (28*). Set (Bf)(x) := / G(x,y)f(y)dx for all x G [0,tt]. Jo It follows from (26) that u'n(x)=fjbn / Gx(x,y)un(y)dy Jo since Gx is piecewise continuous and bounded on [0,7r] x [0,7r]. Prom the symmetry condition G(x,y) = G(y,x) for all x,y G [0,7r], we get (?/n | It) = (Un | -Bit") = (BlAn I -u") = -^{Un | u"), because of un = —/j,nBun, by (26). Hence OO OO I pn ^T \{un | u)u'n{x)\ = ^T Uun | u") I Gx(x,y)u>n(y)dy n=l n=lI J° The uniform convergence of this series on [0,7r] follows as in the proof of (23), since Gx is bounded on [0,7r] x [0,7r]. This justifies the formal differentiation of (28) in order to get (28*). Finally, let us prove (28**). Observe that (Un | -u") = (Un | Au) = (Aun | U) = fin(Un | U). Since u" G D(A), relation (28) remains true if we replace u with u". Using un = -VnUn, we get OO OO U"(X) = Yl(Un I u")un{x) = Y2(Un I U)Un(x)' n=l n=l Ad (iv). We have to prove that span {^i,^2?---} is dense in £2(0,71"). Then, the assertion follows from Theorem 3.A in Section 3.1.

4.5 Applications to Boundary-Eigenvalue Problems 249 The set Co°(0,7r) is dense in 1/2(0,tt). Let v G 1/2(0,7r), and let e > 0 be given. Then, there is a function u G Co°(0, tt) such that ||i; — u\\ = I / (v(x) — u(x))2dx I < e. Let us now study the following nonhomogeneous boundary-value problem: —u"{x) = jiu{x) -h /(#), 0 < X < 7T, (29) 129 u(0) = u(tt) = 0, u<eC2[0,tt]. To this end, we need the corresponding homogeneous problem: —u"(x) = fiu(x), 0 < X < 7T, u(0) =u(tt) = 0, ugC2[0,tt]. Recall that (29*) has the following eigensolutions: Hn = n2, un{x) = 7r~2 sinnx, n=l,2,.... Proposition 3 (The Predholm alternative). We are given the function f G C[0,7r] and £fte rea/ number /j,. Then, the following hold true: (i) J//i is not an eigenvalue of (29*), tten tte original equation (29) ftas a unique solution u. (ii) J//i is an eigenvalue of (29*), i.e., /i = /j,n for some n = 1,2,..., tten equation (29) ftas a solution u iff the so-called resonance condition \ f(x)un(x)dx = 0 ./o is satisfied. Proof. It follows as in the proof of Lemma 1 that the boundary-value problem (29) is equivalent to the integral equation u(x)= [ G(x,y)(jjiu(y) + f(y))dy, 0 < x < tt, ugL2(0,tt). (30*) Jo This can be written as u(x) = /i / g(x, y)u(y)dy + b(x), 0 < x < tt, u G L2(0, tt), (30)

250 4. Eigenvalue Problems for Linear Compact Symmetric Operators where b(x) := Jq Q(x,y)f(y)dy. The corresponding homogeneous integral equation u(x) = n G(x, y)u(y)dy, 0 < x < it, Jo is equivalent to (29*). In particular, we get un(x) = fin / g(x, y)un(y)dy, 0 < x < tt, n = 1,2,... . (31) Jo Finally, recall that the inner product on 1/2(0, tt) is given by (/ I 9) := / f(x)g(x)dx. Jo Ad (i). This follows from Proposition 4(i) in Section 4.4. Ad (ii). It follows from Proposition 4(ii) in Section 4.4 that (30) has a solution iff (b | un) = 0. By (31), {b\un) = I G{x,y)f{y)dy) un(x)dx = / (/ G(x,y)un(x)dx) f{y)dy= fi-1un{y)f{y)dy = /^n1(/ I Un), since G(x,y) = G(y,x) for all #,y G [a, 6]. Therefore, (6 | itn) = 0 iff (fK) = 0. Observe that u G £>(.A). By (iii), there exists a function w £ span{iti, i*2, • • •} such that Hence A := max \u(x) — w(x)\ < e. \u — w|| = I / (^(#) — w(#))2Gfo j <7r2A<7r2£. This implies \\V — W\\ < \\v — u\\ + ||w — W\\ < S + 7T2£. Thus, the set span {iti, it2,. •.} is dense in 1/2(0,7r). D

Problems Problems 251 Let —oo < a < b < oo. 4.1. Integral equations with degenerate kernels. Consider the integral equation rb K(x,y)u(y)dy = Xu(x) for all x G [a, b]. (32) / J a Suppose that N K(%, V) -= Yl fj(x)9j(y) for all x,y G [a, 6], .7=1 where the nonzero functions fj,gji [a, 6] —» R are continuous for all j. Compute the eigenvalues and eigenfunctions of (32). Consider first the special case where N = 1. 4.2. The Green function. We want to study the following boundary-value problem: -(p(x)u'(x)Y + q(x)u(x) = f{x) on [a, 6], (33a) along with the boundary conditions au{a) + 0u'{f)=O, (33b) 7u(t»)+£?/(&) =0, (33c) where a, /?, 7, and (5 are fixed real numbers with a2+/?2 ^ 0 and j2+82 ^ 0. We are given the continuous functions p, q, f: [a, b] —> R such that p(x) 7^0 on [a, 6]. Suppose that the homogeneous problem (33) with / = 0 has only the trivial solution u = 0. Let u\ and ui be a solution of (33a,b) and (33a,c) with / = 0, respectively. Define ^^ Ha<x<y, ' y < x < b, {Ul{x)u2{y) .f t*2(aQm(y) -r, p(*) ir - where p(x) := p(x)(u2(x)u[(x) — ui(x)uf2(x)). Show that (i) Q is the Green function to (33), i.e., if we set v(x) := G(x, y) for fixed y G [a, 6], then i; is a solution of (33) with in the sense of generalized functions.

252 4. Eigenvalue Problems for Linear Compact Symmetric Operators (ii) Q is symmetric, i.e., G(x,y) = G(y,x) for all x,y G [a, 6]. (iii) Problem (33) has the solution u(x) = / G{x,y)f{y)dy for all x G [a, 6]. (iv) Reduce the boundary-eigenvalue problem —(p(x)uf(x))f + q(x)u(x) = Xu(x) on [a, 6] along with the boundary conditions (33b,c) to an integral equation and prove the results for (34), which are similar to Section 4.5. 4.3. A special problem. Compute the Green function to the following boundary-value problem: -u" = f on [0,1], u(0) = u'(l) = 0.

5 Self-Adjoint Operators, the Friedrichs Extension, and the Partial Differential Equations of Mathematical Physics In the fall of 1926, the young John von Neumann (1903-1957) arrived at Gottingen to take up his duties as Hilbert's assistant. These were the hectic years during which quantum mechanics was developing at breakneck speed, with a new idea popping up every few weeks from all over the horizon. Jean Dieudonne History of Functional Analysis, 1981 Stimulated by an interest in quantum mechanics, John von Neumann (1903-1957) began the work in operator theory The result was a paper von Neumann submitted for publication to the Mathemati- sche Zeitschrift but later withdrew. The reason for this withdrawal was that in 1928 Erhard Schmidt and myself, independently, saw the role which could be played in the theory by the concept of the adjoint operator, and the importance which should be attached to self-adjoint operators. When von Neumann learned from Professor Schmidt of this observation, he was able to rewrite his paper in a much more satisfactory and complete form Incidentally, for permission to withdraw the paper, the publisher exacted from Professor von Neumann a promise to write a book on quantum mechanics. The book soon appeared and has become one of the classics of modern physics (Foundations of Quantum Mechanics, Springer-Verlag, 1932). Marshall Harvey Stone, 1970

254 5. Self-Adjoint Operators, the Priedrichs Extension, etc. At the very beginning the given elliptic differential operator is only defined on a space of C2-functions. This operator is extended to an abstractly defined operator by using a formal closure. The main task is to show that the extended operator is self-adjoint. In this case it is possible to apply the methods of John von Neumann. Kurt Otto Priedrichs, 1934 The fundamental quality required of operators representing physical quantities in quantum mechanics is that they be self-adjoint which is equivalent to saying that the eigenvalue problem is completely solvable for them, that is, there exists a complete set (discrete or continuous) of eigenfunctions. The problem has, of course, been solved in the case of operators for which the eigenvalue problem is explicitly solved by separation of variables or other methods, but it seems not have been settled in the general case of many-particle systems. The main purpose of the present paper is to show that the Schrodinger Hamiltonian operator of every atom, molecule, or ion, in short, of every system composed of a finite number of particles interacting with each other through a potential energy, for instance, of Coulomb type, is essentially self-adjoint (i.e., this operator possesses a unique self-adjoint extension). Thus, our result serves as a mathematical basis for all theoretical works concerning nonrelativis- tic quantum mechanics. Tosio Kato, 1951 The interaction between physics and mathematics has always played an important role. The physicist who does not have the latest mathematical knowledge available to him is at a distinct disadvantage. The mathematician who shies away from physical applications will most likely miss important insights and motivations. Martin Schechter, 1981 In this chapter we want to study the following problems in a Hilbert space X, where A: D(A) C X —* X is a linear symmetric operator that has additional properties to be discussed ahead (cf. also Figure 5.1). (i) Abstract boundary-value problem: Au = f, u<E D(A). (1) (ii) Abstract Dirichlet problem: 2~\Au \u)-(f\u) = min!, ue D(A), (1*)

5. Self-Adjoint Operators, the Priedrichs Extension, etc. 255 energetic space (abstract Sobolev space) I abstract Dirichlet problem I elasticity abstract wave equation \ semigroup \ irreversible process in nature quadratic variational problem (Chapter 2) orthogonal projection Riesz theorem l duality map I energetic extension I self-adjoint Priedrichs extension functional calculus for self-adjoint operators abstract wave equation \ one-parameter group l reversible process in nature FIGURE 5.1. Hilbert-Schmidt theory for symmetric compact operators (Chapter 4) eigenvalue problem and Fredholm alternative complete ortthonormal system of eigenvectors (abstract Fourier series) abstract Schrodinger equation one-parameter unitary group quantum physics which is equivalent to (1). This minimum problem is also equivalent to 2^{u | u)E - (/ | u) = min !, u G XE, (1**) where Xe denotes the so-called energetic space of the operator A. (iii) Abstract boundary-eigenvalue problem: Au-fjLU-f, ue D(A), ii G R. (iv) Abstract heat equation: uf(t) + Au(t) = 0, t>0, u(0) = uq, with the solution u{t) = e~Atu0 for all t > 0, (2) (3)

256 5. Self-Adjoint Operators, the Friedrichs Extension, etc. which corresponds to the semigroup {e At}t>o- (v) Abstract wave equation: u"(t)+Au(t) = 0, teR, (4) u(0) = uo, uf(0) = ui, with the solution u(t) = (costA?)u0 + A~2(sintA1/2)ui. (vi) Abstract Schrodinger equation: iu(t) = Au(t), teR, (5) u(0) = uq, with the solution u(t) = e~iAtu0, which corresponds to the one-parameter semigroup {e~tAt}teR. Problems (i)-(vi) allow important applications to the partial differential equations of mathematical physics. For example, this concerns boundary-value problems and boundary-eigenvalue problems for the Laplace equation or the Poission equation, the heat equation, the wave equation, and the Schrodinger equation in quantum mechanics. Such applications of the abstract theory will be considered in this chapter. In particular, in applications to elasticity the "energetic space" Xe corresponds to "states" u of the elastic body which have finite energy, and 2~l{u | u)e = elastic energy in the state u, (f \ u) = work of outer forces. Thus, the variational problem (1**) corresponds to the principle of minimal potential energy. Observe that the "solutions" in (iv)-(vi) correspond to classic solutions of ordinary differential equations if we assume that A is a real number and u = u(t) is a real or complex function. The beauty of functional analysis consists in the fact that The classic formulas remain true for operator equations if we define operator functions A ^ F{A) in an appropriate way.

5. Self-Adjoint Operators, the Friedrichs Extension, etc. 257 The simplest method for constructing such operator functions is the following. Suppose that the operator A has a complete orthonormal system {u\, U2,...} of eigenvectors with the corresponding eigenvalues {Ai, A2,...}, i.e., Aun = \nun for all n. Then 00 u = Vj(^n I u)un for all u e X. n=l This yields 00 00 An — 2_^{un I Au)un = 22, ^ri{un I u)un for all u e D(A), (6) n=l n=l since the symmetry of A implies (un I Au) = (Aun I u) = An(un I it). Formula (6) motivates the following definition: 00 F(A)u:=^2F(\n)(un\u)un. (7) n=l It is quite natural to define the domain of definition D(F(A)) of the operator F(A) as follows: it € jD(F(A)) iff the series (7) converges. By the convergence criterion for abstract Fourier series (Proposition 5 in Section 3.1), we get 00 u G D(F(A)) iff ]T \F(Xn)(un I n)\2 < 00. n=l If we apply this to (6), then we obtain 00 u £ D{A) iff £ |AnK I u)|2 < 00. (8) n=l It turns out that 7/ £fte linear operator A: D(A) C X -* X is self-adjoint, then condition (8) is satisfied. This way the functional calculus leads to self-adjoint operators in a natural way.

258 5. Self-Adjoint Operators, the Friedrichs Extension, etc. In applications to mathematical physics one encounters the following situation. For example, the classic boundary-value problem for the Poisson equation ~Au = / on G, (9) u = 0 on dG can be written in the following form: Bu = f, ueD(B), (9*) where we set Bu := —Au and D{B) := {u G C2(G): u = 0 on dG}. Here, G is a nonempty bounded open set in R^. Letting X := L2(G), we get the linear symmetric operator B: D(B) C X -+ X However, if iV > 2, then the operator B is not surjective. More precisely: There are functions f G C(G) for which equation (9*) has no solution. This is identical to the fact that problem (9) has not always a classic solution if/ € C(<3). The idea of Friedrichs was to extend the operator B to a self-adjoint operator A:D(A)CX^X, i.e., we have Bu = .Au for all w G D(B) and £>(£) C D(A). The operator A is called the Friedrichs extension of the original operator B. It turns out that the equation Au = f, u£ D{A) (9**) has a unique solution u for each given / G X. This solution u of (9**) can be regarded as a generalized solution to the classic problem (9). In terms of the expansion (6), the situation is as follows. If G has a sufficiently smooth boundary, then the symmetric operator B: D(B) CI-^I has an orthonormal system {^i, U2,...} of classic eigenfunctions, which is complete in the Hilbert space X = L,2(G). These eigenfunctions un correspond to eigensolutions of the following classic eigenvalue problem: -Aun = \nun on G, un = 0 on dG.

5. Self-Adjoint Operators, the Priedrichs Extension, etc. 259 Using the same argument as for (6), we obtain that oo Bu = ^T \n(un I u)un for all u G D(B). (10) n=l However, this series also converges for points u G X that do not live in D(B). Naturally enough, the Priedrichs extension A of the original operator B is given through formulas (6) and (8), i.e., oo Au = ]P \n(un I u)un for all u G D(A), (10*) n=l where oo oo u G D(A) «=> 22 ^n(Un I u)un is convergent «=> ^ \^n(un \ u)\2 < oo. n=l n=l The preceding considerations motivate the appearance of the Priedrichs extension in a quite natural way. However, the general theory of the Priedrichs extension is independent of Fourier series expansions. The basic idea is the following. We are given a linear, symmetric, strongly monotone operator B: D(B) C X —> X on the real Hilbert space X, i.e., (Bu | u) > c\\u\\2 for all u G D(B) and fixed c > 0. We first construct the so-called energetic extension Be-Xe —> Xg of the operator B, where D(B) QXEQXQX*E, (11) and Be is the duality map of X#. Then, the Priedrichs extension A:D(A)CX^X of B is an appropriate restriction of £#, namely, we set Au := £#?/ for all it G D(A), where D(A) := {it G X#: £#?/ G X}. This construction guarantees automatically that the Priedrichs extension A: D(A) C X —> X is bijective, since the duality map Be is bijective. That is, the equation Au = /, u G jD(-A), has a unique solution for each given / G X. For brevity we write BCACBe, (11*)

260 5. Self-Adjoint Operators, the Priedrichs Extension, etc. i.e., A is an extension of B. In turn, Be is an extension of A. In terms of Fourier series, the energetic space Xe of the symmetric operator B: D(B) CI-^I from (10) is given through XE = <u e X: ^ Xn\(un \ u)\2 \ < oc. If the operator B corresponds to the classic boundary-value problem (9) for the Poisson equation, i.e., B is given through (9*), then XE =Wl(G), i.e., the energetic space is a Sobolev space. As we shall show in this chapter, the compactness of the embedding W\{G) C L2(G) plays a fundamental role. This is the famous Rellich compactness theorem. In fact, this compact embedding guarantees that the Laplacian with zero boundary conditions possesses a complete orthonormal system of eigen- functions in the Hilbert space L,2(G), i.e., series (10) is convergent. This result will be critically used in order to solve the heat and wave equations. Our approach justifies the classic Fourier method of physicists. The Friedrichs extension represents the functional analytic core of mathematical physics. This approach is closely related to the fundamental physical concept of energy. In quantum physics, physical states correspond to unit vectors in a Hilbert space and the physical quantities (e.g., energy, momentum, and so on) correspond to self-adjoint operators. This will be discussed in Section 5.14. In Section 5.2 we shall show that Self-adjoint operators are closely related to both orthogonality and generalized derivatives. 5.1 Extensions and Embeddings Definition 1. Let A: D(A) C X -+ Y and B: D(B) C X -+ Y be operators, where X and Y are linear spaces over K. We write BCA iff An = Bu for all u G D{B) and D(B) C D(A).

5.1 Extensions and Embeddings 261 A B C A FIGURE 5.2. In this case, we say that the operator A is an extension of the operator B. Obviously, A = B iff B C A and A C £. Example 2. Set X = y := R and £>(B) = [0,1] as well as D(A) := [-1,1]. Moreover, let Bu := u on D(B) and Ait := |w| on D(A). Then, BCi (cf. Figure 5.2). Definition 3. Let X and Y be normed spaces over K. (i) We say that the embedding "X C y" is continuous iff there exists an operator j:X^F (12) that is linear, continuous, and injective. (ii) The embedding "X C y" is called compact iff the operator j from (12) is linear, compact, and injective. Let X be a subset of Y, i.e., X C.Y. Then we set j(^) :— u f°r all ^ £ -X"- In terms of sequences, we have the following: (a) The embedding XC7 is continuous iff, as n —> oo, un^> u in X implies un^> u in y (b) The embedding X C Y is compact iff it is continuous and each bounded sequence (un) has a subsequence that converges in Y, i.e., as n —> oo, wn/ —> i; in y

262 5. Self-Adjoint Operators, the Priedrichs Extension, etc. In the general case of Definition 3, we may identify u with j(u). This makes sense since j:X —+ Y is injective. In this sense, we may regard the space X as a subset of Y, and we may write X C Y instead of "X C Y" for brevity. Standard Example 4. Let G be a nonempty bounded open set in RN, N > 1. Then, the following hold true: (i) The embedding C(G) C L2(G) is continuous. o (ii) The embedding W2(G) Q L2(G) is compact. This is the prototype for embedding theorems, which play a fundamental role in modern analysis. Proof. Ad (i). We are given u G C(G). Let j(u) denote the element of L2(G?) that corresponds to u. Then \\3{u)\\l2{G) = ( / \u(x)\2dx) <( dx) max\u(x)\ < COIlSt\\u\\C(Qy Thus, the operator j: C(G) —> L2(G) is linear and continuous. Moreover, j is also injective. In fact, if j(u) = j(v) and u, v G C(G), then u(x) = v(x) for almost all x G G. Since u and ^ are continuous, this implies u(x) = v(x) for all x G G. Ad (ii). This is the famous Rellich embedding theorem, which will be proved in Section 5.7. □ Definition 5. Let X, Y, and Z be linear spaces over K, and let A: D(A) C X —> y and J5: jD(B) C I -> 7 be linear operators. For each a G K, we define the operators (aA)it := olAu for all it G D(A), and (A + S)w := An + B^x for all u G D(A) H D(B), i.e., jD(aA) := jD(A) and D(A + B) := D(A) D D(B). Let C: D(C) C y —> Z be a linear operator. We set {CA)u := C(Am) for all u G jD(A) with An G jD(C). Obviously, the operators a A, A + B, and CA are also linear.

5.2 Self-Adjoint Operators 263 5.2 Self-Adjoint Operators While working with operators A: D(A) C X —> X that are not defined on the total space, observe carefully the specific form of the domain of definition D(A) of A. The following examples show that, roughly speaking, we have the following situation: (i) Linear integral operators correspond to linear operators A: X —> X that are defined on the total space X. (ii) Linear differential operators A: D(A) Cl->I correspond to linear operators that are not defined on the total space X. The definition of the adjoint operator A* is based on the following formula: (Au \v) = (u\ A*v) for all u G D(A), v G D(A*). (13) Definition 1. Let A: D(A) C X —> X be a linear operator, where D(A) is dense in the Hilbert space X over K. By definition, v G D(A*) iff there exists an element w G X such that (Au \v) = (u\w) for all u G D(A). (14) Furthermore, we set A*v := w. This way we obtain the adjoint operator A*:D(A*)CX ^X. We have to show that this definition makes sense. In fact, suppose that relation (14) also holds if we replace w with w\. Then (u | w - w\) = 0 for all u G D(A). Since D(A) is dense in X, we get w = w\. Proposition 2. Let A: D(A) C X -► X and B: D(B) C X ^ X be linear operators, where D(A) and D(B) are dense in the Hilbert space X over K. Then, the following hold true: (i) The adjoint operator A*: D(A*) C X —* X is linear. (ii) For each a G K, (aA)* - aA\ (15)

264 5. Self-Adjoint Operators, the Priedrichs Extension, etc. (iii) ACB implies B*U*. Consequently, if D(A*) is dense in X, then the operator (A*)* exists. In this case, we set A** := (A*)*. Proof. Ad (i). Let aua2 G K. If (Au \ Vj) = (u \ Wj) for all u G D(A), j = 1,2, then (Au | ol\Vi + 0:2^2) = ol\(Au I v\) + a2(Au \ v2) = (u I aiWi + 0:2^2) for all it G jD(-A). Thus, Vj G jD(A*) for j = 1,2 implies a\V\ + 0:2^2 £ £>(.A*) and A*(ai^i + 0:2^2) = ai-A*^i + a2A*^2. Ad (ii). If a ^ 0, then relation (15) follows from (Au \v) = (u\w) <& (aAu \ v) = (u \ aw). For a = 0, the assertion is trivial. Ad (iii). Let A C B. It follows from (Bu I v) = (u I B*t;) for all u G £>(£), v G £>(£*) that (Ait I v) = (u I B*v) for all u G £>(-A), and hence A*v = B*v for all veD(B*). This implies B*U*. □ Definition 3. Let A: D(A) C X —> X be a linear operator, where £>(.A) is dense in the Hilbert space X over K. (i) A is called symmetric iff A C A*, i.e., (Ait | v) = (u \ Av) for all u,v £ i?(A). (ii) A is called self-adjoint iff A = A*. (iii) .A is called skew-symmetric iff A C —A*, i.e., (Ait | v) = — (u \ Av) for all u,v G I>(A). (iv) A is called skew-adjoint iff A= —A*. Proposition 4. Let the operator A.X^X be linear and continuous on the Hilbert space X over K. Then, the adjoint operator A*:X^X

5.2 Self-Adjoint Operators 265 is also linear and continuous. In addition, \\A\\ = \\A*\\. Moreover, A** = A. Proof. Let v G X. Set f(u) := (v | Au) for all u G X. By the Schwarz inequality, |/(t0| < \\Au\\ \\v\\ < \\A\\ \\u\\ \\v\\ for all ueX. Hence the linear functional f:X —> K is continuous with ||/|| < \\A\\ \\v\\. By the Riesz theorem from Section 2.10, there exists an element w G X such that /(it) = (w \u) for all it G X, and ||i(;|| = ||/||. Hence {Au \v) = (u\w) for all u G X. This implies and ||A*v|| < ||A|| ||v|| for all v G X. Therefore, Ml < \\A\\. (16) It follows from (Au \ v) = (u \ A*v) that (A*v | u) == (v | Ait) for all it, ?; G X Hence (A*)* = A. Replacing A with A* in (169), we get p|| = ||(A*)*|| < P*||. This implies \\A\\ = \\A*\\. D Standard Example 5 (Integral operators). Let A: [a, b] x [a, b] —> R be a continuous function, where —oo<a<6<oo. Define (Au)(x) := / A(x,y)u(y)dy for all x G [a, 6], (17) J a and set X := I/2(a, 6). Then, the following are met: (i) The operator A: X —> X is linear and compact, (ii) The adjoint operator A*:X —> X is given through (A*u)(x)= A(y,x)u(y)dy for all x G [a, 6]. (17*) J a The operator .A*: X —> X is linear and compact. (iii) If ^4 is symmetric, i.e., A(x, y) = A(y, x) for all x, y G [a, 6], then the operator .A: X —> X is self-adjoint

266 5. Self-Adjoint Operators, the Priedrichs Extension, etc. Proof. Ad (i). This follows from Lemma 3 in Section 4.4. Ad (ii). We are given v e X. Set w(x) := / A(y,x)v(y)dy for all x G [a, 6]. J a As in the proof of Lemma 3(c) in Section 4.4, it follows from the Tonelli theorem that (Au | v) = / I / A(x,y)u(y)dy J v(x)dx = I A(x,y)v(x)dx\u(y)dy fb — / w(y)u(y)dy = (u\w) for all u G X. J a Hence A*v = w. This yields (17*). By (17*) and Lemma 3 in Section 4.4, the operator A*:X —> X is linear and compact. Ad (iii). If A is symmetric, then A = A*, by (17) and (17*). □ Proposition 6. Let A: D(A) C X —> X be a linear operator on the Hilbert space X over K such that D(A) is dense in X. Then, the following hold true: (i) A is self-adjoint iff A is symmetric and (Au | v) = (u | w) for all u G D(A) and fixed v, w G X (18) implies that v G D(A) and w = Av. (ii) A is skew-adjoint iff A is skew-symmetric and (Au | v) = — (u | w) for all u G D(A) and fixed v, w G X (19) implies v G D(A) and w = Av. (iii) Let K = C and a G R with a ^ 0. Then, A is skew-symmetric <& aiA is symmetric, A is skew-adjoint <^ aiA is self-adjoint Proof. Ad (i). Observe that A = A* iff A C A* and A* C A. Ad (ii). Use A = -A* iff A C -A* and -A* C A.

5.2 Self-Adjoint Operators 267 Ad (iii). Since (aiA)* = —aiA*, we get A C -A* & aiA C (criA)* and A = -A* &aiA = (aiA)*. D Corollary 7. (i) Each self-adjoint linear operator A: D(A) C X —> X on the Hilbert space X overK is maximally symmetric, i.e., by definition, if we have ACS for any symmetric operator S: D(S) C X —> X, then A = S. (ii) Each skew-adjoint linear operator A: D(A) C X —> X zs maximally skew-symmetric. Proof. Ad (i). It follows from iCS that 5* C A*. Since A = A* and 5 C 5*, we get S C A. Thus, 5 = A. Ad (ii). If A C 5 with A = -A* and SC-S*, then 5* C A*, and hence SCA Thus, A = 5. □ Standard Example 8 (Differential operator). Let X := L%(R). Define (Au)(rc) := u'(rc) for all x G R, where jD(A) := {it G X: t/ G X}. Here, the derivative v! is to be understood in the generalized sense. Then, the following hold true: (i) The operator A: D{A) C X —> X is skew-adjoint. (ii) For each a G R, the operator aiA is self-adjoint. Proof. Ad (i). We have u G D(A) iff it G X and there is a function w G X such that / u(x)v'{x)dx = - J w(x)v(x)dx for all v G Cg°(R)c (20) Jr Jr If this holds true, then we get w = v! in the generalized sense. Step 1: Approximation. Let v G D(A). We want to show that there exists a sequence (vn) in Co°(R)c such that, as n —> oo, vn -+ v in l£(R) and < -+ v' in l£(R).

268 5. Self-Adjoint Operators, the Friedrichs Extension, etc. To prove this we will use the same arguments as in the proof of Proposition 7 from Section 2.2. To this end, we set vn(x) := / 0jl (x - y)v(y)dy, n = 1,2,... Jr n (see (15) in Chapter 2). Then, vn G Co°(R)c for all n, and we get vn —> v in 1/2 (R) as n —> oo. Since for fixed x G R, the function y h-> 0i(sc — y) belongs to the space n Cq° (R), it follows from the definition of the generalized derivative that vn(x)= / -j-(t>±(x-y)v(y)dy = - / —cj)i{x-y)v{y)dy Jr ax » JR dy » = / (j)i{x-y)v'{y)dy. Jr n Hence v'n —> v' in L%(R) as n —> oo. Since the function 0j, is real, we also get vn -x v and «) -► (v;) in Lj(R) as n -► oo. (21) 5£ep #: We want to show that the operator A is skew-symmetric. Replacing v with vn in (20) and letting n —> oo, we get / ujifjdx = - u'vdx for all u,v G £>(A). (22) Ve Jr Hence (Av | w) = -(v | Ait) for all u, v G jD(A). Step 5: We prove that the operator A is skew-adjoint In fact, it follows from (Av \u) = —(v \w) for all v G jD(A) and fixed u, w G X that / (V)uGfo = - / vwdx for all v G C£°(R)C. Jr Jr Thus, we get v! = w in the generalized sense, i.e., w = Au. By Proposition 6, A is skew-adjoint. Ad (ii). This follows from Proposition 6(iii). □ The following example shows that the notion of the generalized derivative is quite natural from the operator theory viewpoint. Example 9. Let X := L^(R). In contrast to the preceding Standard Example 8, let us consider the classic differential operator (Bu)(x) := u'{x) for all x G R, where D(B) = X n C\R). Then

5.2 Self-Adjoint Operators 269 (i) The operator B: D(B) C X —> X is skew-symmetric but no£ skew- adjoint. (ii) For each a G R with a ^ 0, the operator m\B is symmetric but not self-adjoint. Proof. Ad (i). By Step 2 of the preceding proof, B is skew-symmetric. S'et u(x) := \x\ and w(x) := ?/(#) if # ^ 0, w(0) := 0. As in Example 5 from Section 2.5.2, we obtain that w = uf in the generalized sense. However, u $■ D(B), since u is not C1. By Standard Example 8, An = w. Hence the operator A is a proper skew-symmetric extension of B. Thus, B is not skew-adjoint, by Corollary 7. Ad (ii). This follows from (i) and Proposition 6(iii). D Standard Example 10 (The multiplication operator). Let X := L%(R). Define (Mu)(x) := xu{x) for all x G R, where D(M) := {u eI:MwG X}. Then, the operator M: £>(M) CI^ X is self-adjoint The self-adjoint operators z.A and M from Standard Examples 8 and 10 correspond to momentum and position in quantum mechanics, respectively (cf. Section 5.14). Proof. For all u, v G D(M), (Mu | v) = / (xu(x))v(x)dx = / u{x) (xv(x))dx = (u \ Mv). Jr Jr Hence M is symmetric. Moreover, it follows from (Mu | v) = (u | w) for all u G D(M) and fixed v, w G X that / it(x)(x^(x))dx = / u(x)w(x)dx for all it G Cq Jr Jr °°^)c.

270 5. Self-Adjoint Operators, the Friedrichs Extension, etc. u M 0 Pu FIGURE 5.3. Hence w(x) = xv(x) for almost all x G R, i.e., w = Mu. Thus, M is self-adjoint, by Proposition 6. □ Next we want to show that orthogonal projections are closely related to a special class of self-adjoint operators. Let M be a closed linear subspace of the Hilbert space X over K. By Section 2.9, for each u G X, there exists the unique decomposition u = v + w, where v G M and w G ML. (23) Definition 11. The operator Pu := v is called the orthogonal projection from X onto M (cf. Figure 5.3). Proposition 12. Let X be a Hilbert space over K. Then (i) The orthogonal projection P:X —> M from X onto the closed linear subspace M of X is linear, continuous, and self-adjoint and P2 = P. IfM^{0}, then \\P\\ = 1. (ii) Conversely, let P: X —» X be a linear continuous self-adjoint operator with P2 = P. Then, P is the orthogonal projection from X onto the closed linear subspace P(X). Proof. Ad (i). By the Pythagorean theorem, it follows from (23) that II 110 II II9 || || 9 INI = lrII + IMI • Hence \\Pu\\ < \\u\\ for all u G X. Moreover, if u G M, then Pu = u. Hence ll^ll = 1- Let Uj = Vj + Wj, where Vj G M and Wj G M-1, j = 1,2. Then, (vj \ Wk) = 0 for j, k = 1,2. Hence (vi I u2) = {ui | v2) = (vi | v2).

5.2 Self-Adjoint Operators 271 This implies (Pui | U2) = (ui I PU2) for all u\,U2 E X. Hence P = P*, i.e., P is self-adjoint. UveM, then v = v + 0, where v G M, 0 G M-1. Hence Pv = v. By (23), P2u =^Pv = v = Pu for all w E X Therefore, P2 = P. Ad (ii). Set M := P(X). Since P is linear, M is a linear subspace of X. It follows from P2 = P that M is closed. In fact, let (un) be a sequence in M such that un —> it as n —> 00. Then, itn = Pvn for some vn and Pitn = P2vn = Pvn = un. Hence u = lim un = lim Pitn = Pit, n—► oo n—>oo i.e., u E M. Furthermore, since P is self-adjoint and P2 = P, we get (Pu I (I - P)v) = (Pu I v) - (Pu I Pv) = (Pit I v) - (P2it I v) = 0 for all u,v E M. Hence (J - P)v E M"1 for all v E X. Thus, it follows from w = Pu + (J - P)w, Pu eM, (I - P^u E M, that P is the orthogonal projection from X onto M. □ Proposition 13. Le£ A: D(A) C X —> X fee a linear symmetric operator on the Hilbert space X, where R(A) is dense in X. Then, (A-1)* = (A*)-\ where all the appearing inverse and adjoint operators exist. If, in addition, A is self-adjoint, then so is A"1. Proof. The operator A is injective. In fact, An = 0 implies (u I Av) = (Au I v) = 0 for all v E D(A). Since R(A) is dense in X, u = 0. The operator A* is also injective. In fact, if A*u = 0, then it follows from (Av \u) = (v\ A*u) = 0 for all v E D(A) that u = 0. Consequently, the inverse operators A-1 and (A*)"1 exist. Since D(A~1) = JR(A) and ii(A) is dense in X, the adjoint operator (A"1)* exists.

272 5. Self-Adjoint Operators, the Priedrichs Extension, etc. Set B := (A~l)\ We have (u\v) = {A~lAu | v) = (Au | Bv) for all u G D{A), v G D{B), (24) and (z\w) = (AA~xz | w) = (A~lz | A*w) for all w G D{A% z G ^-A"1). (25) By (24), Bv G £>(A*), and (u\v) = (u\ A*Bv) for all w G jD(A), v G £>(£). Since D(A) is dense in X, this implies i4*Bv = v for all v G D(B). (26) Analogously, it follows from (25) that BA*w = w, for all w G D(A*). (27) Hence B = (A*)"1, by Proposition 6 in Section 1.20, i.e., (A-1)* = (A*)"1. If A is self-adjoint, then A = A*, and hence (A"1)* = (A*)"1 = A"1, i.e., A-1 is also self-adjoint. □ Proposition 14. Le£ A: D(A) C X ^> X be a linear operator where D(A) is dense in the Hilbert space X over K. Suppose that there exists a sequence (un) in D(A*) such that un —> u and A*un —> v in X as n —> oo. Then, u G D(A*) and A*u = v. This implies that if A is self-adjoint or skew-adjoint, then un —> u and Awn —> ?; in X as n —> oo imply u G jD(-A) and Au = v. Proof. Letting n —> oo, it follows from (Aw | un) = (w I -A*un) for all w G D(A) that (Ait; | u) = (w | v) for all w G D(A). Hence ,A*w = v. D Proposition 15. For a linear operator U: X —> X on the Hilbert space X over K, the following four conditions are mutually equivalent: (i) U is unitary, i.e., U is surjective and (Uv | Uw) = (v \ w) for all v,w G X.

5.3 The Energetic Space 273 (ii) UU* = U*U = L (iii) U is bijective and U~1 = U*. (iv) U is surjective and \\Uv\\ = \\v\\ for all v G X. Proof, (i) => (ii). It follows from (Uv \ Uw) = (y \w) for all v, w G X that U*(Uw)=w for all w e X, i.e., U*U = I. Hence UU*UU~xw = w for all w G X, i.e., £/£/* = /. (ii) <& (iii). This follows from Proposition 6 in Section 1.20. (ii) => (i). Prom UU* = / it follows that /?(£/*) = X, and U*U = I implies (Uv | Uw) = (v | U*Uw) = (v | w) for all v, n; G X. (i) <^> (iv). Observe that the inner product can be expressed by norms (cf. Problem 2.2). For example, if X is a real Hilbert space, then (u | v) =4~1(||it + ^||2 - ||ia-^||2) for all u,v G X. Hence \\Uv\\ = \\v\\ for all v G X is equivalent to (Uv | Uw) = (v \ w) for &Wv,weX. □ 5.3 The Energetic Space Definition 1. The linear operator B:D(B) CI-^I on the real Hilbert space X is called strongly monotone iff (Bu | it) > c|M|2 for all u G D(B) and fixed c> 0. (28) We assume the following: (H) The operator B: D(B) C X —> X is linear, symmetric, and strongly monotone on the real Hilbert space X. Let (• | •) and || • || denote the inner product and the norm on X, respectively. Let us also introduce the energetic inner product (u | v)e := (Bu | v) for all u,v G D(B), and the energetic norm i \\u\\e := (^ | w)Ji for all it G X.

274 5. Self-Adjoint Operators, the Priedrichs Extension, etc. By (28), (u | u)e = 0 implies u = 0. The symmetry of B yields (u | v)e = (v | u)e for all u, v G D(B). Thus, (• | •)# represents an inner product on the linear space D(B). Definition 2. Let the operator B be given as in (H). Then, the energetic space Xe of the operator B consists precisely of all the u G X that have the following two properties: (i) There exists a sequence (un) in D{B) such that un —> u in X as n —> oo. (ii) The sequence (un) is Cauchy with respect to the energetic norm \\-\\e- Each sequence (wn) having the properties (i) and (ii) is called an admissible sequence for u G Xe- For all u, v G Xe, we set (it | v)E := lim (un | vn)tf, n—>oo where (wn) and (vn) are admissible for it and v, respectively. We shall show below that this limit exists and is independent of the chosen admissible sequences. Proposition 3. Assume (H). Then, the following hold true: (i) The energetic space Xe becomes a real Hilbert space with respect to the energetic inner product (• | •)#. The set D(B) is dense in Xe- (ii) The embedding Xe £ X is continuous, i.e., \\u\\ < c~2 \\u\\e foralluEXE- (iii) There exists a continuous embedding "X C X^ " given by the operator j\X —> Xe, where j(f)(v) := (/ | v) for all v G Xe and each fixed f G X. If we identify / with j{f), then X becomes a subset of X%, i.e., Xe £ X C JC_g, and (/, v)s = (/ | v) for all / G X, t; G X^, (29)

5.3 The Energetic Space 275 where we set (g, v)e := g(v) for all g G XE, v G XE. Proof. Ad (i), (ii). Step 1: Let (un) be an admissible sequence for u = 0. We want to show that lim H^nlU = 0. n—»oo In fact, since | \\Un\\E - IkmlUI < \\un - Mm||tf (30) and (un) is Cauchy with respect to || • \\E, the sequence (||^n|U) is also Cauchy. Thus, the limit A := lim \\un\\E n—>oo exists. Furthermore, observe that \{un | U^e ~ (ur \ Ur)E\ = \{un - Ur | Wm)E + (ur \ U^ - Ur)E\ < \\un — Ur\\E\\Um\\E + H^rlUH^m ~ Ur\\E < £ (31) for all n,m,r > no(e). Since un —> 0 in X, we get (un \ u^e = (un \ Bum) —> 0 as n —> oo. Letting n —> oo and r —> oo in (31) for fixed m, we get | A21 < e for all e > 0. Hence A = 0. Step 2: Let u G XE. Choose an admissible sequence (un) for u. By (30), (H^nlU) is Cauchy. Define \\u\\E := lim \\un\\E. n—kx> Let (^n) be another admissible sequence for it. We want to show that \\u\\E = Hm ||vn|b. n—>oo In fact, since the sequence (itn — vn) is admissible for w = 0, we obtain that I ||wn|b - ||Vn|bl < \\un ~ Vn\\E ~► 0 as 71 —► CO, by Step 1. Step #: For each un,vn G D(B), we have the identity Ki | ^n)£ = 4"1 (||wn + Vn\\2E ~ \Wn ~ Vn\\%) . Let (un) and (^n) be admissible for u G Xe and ?; G X^, respectively. Then, the sequence (un ± t>n) is admissible for u± v. By Step 2, the limit (it | v)E := Hm (un | vn)^ (32) n—»oo exists and is independent of the chosen admissible sequences.

276 5. Self-Adjoint Operators, the Priedrichs Extension, etc. Step 4' Let (un) be an admissible sequence for u £ Xe> Then lim \\u — un\\E = 0. n—»oo In fact, since (un) is Cauchy with respect to || • ||#, \\un — u>m\\E < £ for all n,m > no(e). For each fixed m, the sequence (un — Um) is admissible for u — Um. Letting n —> oo, we get \\u — um\\E < £ for all m > no(e), by Step 2. Step 5: Let (un) be admissible for it G X#. By (28), ll^nlli? > cll^n||2 for all ra. Letting n —> oo, this implies hill > c||w||2 for all w G X^. (33) 5£ep #: XE is a pre-Hilbert space with respect to (• | •)#. In fact, (it | u)e = 0 implies it = 0, by (33). Observe that (• | -)e is an inner product on D(B). Using admissible sequences and the limiting relation (32), it follows that (• | -)e is an inner product on the real linear space X#. Step 7: The set D(B) is dense in Xe- In fact, let u G Xe- Then, there exists a sequence (un) in D(B) that is admissible for u. By Step 4, for each e > 0, there is a itn G D(B) such that ||w — un\\E < £• Step 8: Xe is a Hilbert space. To prove this, let (un) be a Cauchy sequence in Xe- We have to show that there exists a u G Xe such that itn —> u in Xe; as n —> oo. (34) In fact, since jD(B) is dense in Xe, there exists a sequence (vn) in £>(£) such that \Wn — vn\\E < n*1 for all n. It follows from \\vn - Vm\\E < \\vn ~ ^ti\\e + \\um - Vm\\E + \\un - Mm||£ < £, for all n, m > no(e), that (^n) is Cauchy in Xe- By (33), (vn) is also Cauchy in X. Thus, ^n —* it in X as n —> oo. Hence the sequence (vn) is admissible for u. By Step 4, Ikn ~ w||# —► 0 as n —> oo.

5.3 The Energetic Space 277 Therefore, ||«n - u\\e < \\un - vn\\E + \\vn - u\\E —>0 as n -> oo. This is (34). Ad (iii). Let feX. Set j(f)(v) := (/ I v) for all v € XE. By (33), l(/ I v)\ < ll/H ||v|| < c-4 H/ll \\v\\E for all v e XB. Hence j(/) e X'E and ||j(/)IU« < c-i||/||. Thus, the operator j:X^X*E is linear and continuous. In addition, if j(f) = .7(g), then (/ - 0 M = 0 for all v G XE. Since £>(£) C X# and D(B) is dense in X, / = #. Hence j is injective. D Standard Example 4 (The Laplacian). Let G be a nonempty bounded open set in RN, iV > 1. Set X := L2(G) and Bu:=-Au, D{B):=C?(G). Then, the following are met: (i) The operator B: D(B) C X -> X is linear, symmetric, and strongly monotone. (ii) The corresponding energetic space is given through XE =Wl(G), i.e., Xe is a Sobolev space, (iii) For all u, v G Xe, f N (u | v)e = / /_J ^ W ®3V ^x-> where the derivatives djU and djV are to be understood in the generalized sense. (iv) The embedding Xe C X is compact.

278 5. Self-Adjoint Operators, the Priedrichs Extension, etc. Proof. Ad (i). Integration by parts yields (Bu \v) = (-Au)vdx = / u(-Av)dx = (u | Bv) for all u,v G D(B). JG JG Hence B is symmetric. For all u, v G D(B), integration by parts yields (u | v)e ~ {u | Bv) = / (—Av)udx = / 2_\®Ju®ovdx- (^) JG JGj^i By the Poincare-Friedrichs inequality from Section 2.5.6, we get c\\u\\2 < {Bu | u) for all w G £>(£), i.e., B is strongly monotone. Ad (ii). Let u G Xe. Then there exists an admissible sequence (un) for u, i.e., wn G D(B) for all n, ttn —> it in X as n —► oo, (36) and (wn) is Cauchy in Xe- Hence (djUn) is Cauchy in X, since AT | ^n ^ra by (35). Thus, for each j = 1,..., N, there exists a function Vj G X such that (?jWn —> Vj in X as n —> oo. (37) Letting n —> oo, it follows from / undjwdx = — / (djUn)wdx for all w G C^(G) Jg Jg that / udjwdx = — / Vjwdx for all w G Cq°(G). Jg Jg Hence ^- = ^-w in the generalized sense. By (36) and (37), u EW^iG)- o Conversely, let u EWKG)- Then there exists a sequence (un) in D(B) such that (36) and (37) hold true with Vj := djU. Thus, (un) is an admissible sequence for it, and hence u G X#. Ad (iii). Let u, v € X#, and let (wn) and (vn) be an admissible sequence for u and v, respectively. Letting n —> oo, it follows from

5.4 The Energetic Extension 279 N (un | vu)e = / 2_]^Un^Vn(^x ^or a^ n Jg3 = 1 that f N (u | v)e = lim (un | vu)e = / /]djudjvdx. n-*°° Jg j=1 Ad (iv). This will be proved in Section 5.7. D 5.4 The Energetic Extension Definition 1. Let B: D(B) C X —> X be a linear, symmetric, strongly monotone operator on the real Hilbert space X. Then, the duality map Be'-Xe —> X^ of the energetic space Xe is called the energetic extension of the operator B. By Section 2.11, the operator Be is defined through {BEu, v)e = {u | v)e for all it, i; G XE. (38) Moreover, £#: X^ —> X|. is a linear homeomorphism with ||Beu||xs = \\u\\e for a11 w G ^- Proposition 2. Tfte operator Be is an extension of B, i.e., Bu = BEu for all u G D(B). Proof. Let u G D(B) be given. It follows from (29) and (38) that (Beu,v)e = (u | v)e = {Bu | v) = (Bu,v)e for all u,v G D(B). Since jD(B) is dense in Xe, this implies (Beu,v)e = (Bu,v)e for all v G X^. Hence Beu = Bu. O

280 5. Self-Adjoint Operators, the Priedrichs Extension, etc. 5.5 The Friedrichs Extension of Symmetric Operators We assume the following: (H) B: D(B) C X —> X is a linear, symmetric, strongly monotone operator on the real Hilbert space X, i.e., (Bu | u) > c\\u\\2 for all u G D(B) and fixed c > 0. Definition 1. The Priedrichs extension A: D{A) CI-^I of the operator B is defined through An := Beu for all u G D(A), where £>(A) := {u G X#: B^u G X}. By Proposition 3(iii) in Section 5.3, we obtain that u G D(A) iff there exists an / G X such that (Beu, v)e = (f | v) for all v G X#. Observe that £>(B) CI^CICI* and bcacbe. Theorem 5.A. Assume (H). The Priedrichs extension A possesses the following properties: (i) The operator A: D(A) C X —> X is self-adjoint and bijective, and (Au | it) > c||u||2 /or a// u G I>(A). (ii) Tfte inverse operator A~l:X^X is linear, continuous, and self-adjoint. (iii) If, in addition, the embedding Xe Q X is compact, then A-1: X —> X 25 compact. Proof. The operator A is a restriction of Be- Since Be'Xe —* X^ is bijective, so is the operator A. For all u G D(A), (Au | it) = (Au,u)e = (Beu,u)e = (u \ u)e > IM|2?

5.5 The Friedrichs Extension of Symmetric Operators 281 by (33). The operator B^iX^ —> Xe is linear and continuous. Since the embedding X C X^ is continuous, the restriction BE : X —> Xe is also continuous.1 This operator is identical to A~l. Hence the operator A'^.X^Xe (39) is linear and continuous. In turn, since the embedding Xe Q X is continuous, the operator A'hX^X (40) is linear and continuous. Moreover, if the embedding Xe Q X is compact, then it follows from (39) that the operator (40) is compact. Let f,geX. By (29) and (38), (A-1/ | A~1g)E = (BE(A-1f),A-1g)E = (/, A"1*)* = (f\A-1g). Since (A-1/ | A-1g)E = (A^g | A"1/)*, we get (f\A-1g) = (A-1f\g) for all/, ^ e X. Thus, the linear, continuous operator A~l:X —> X is symmetric, and hence A~1 is self-adjoint (cf. Problem 5.1). Finally, by Proposition 13 in Section 5.2, the operator A is self-adjoint. □ 5.5.1 Variational Problem For given / G X, let us consider the following two variational problems: 2~1(u | u)E -{f\u)= min!, ue XE, (41) and 2~1(Au \u)-(f\u)= min!, u G D(A). (41*) Proposition 2. Let A be the Friedrichs extension of the operator B given through (H). Then, the two variational problems (41) and (41*) have the unique solution no = A-1/. 1 Observe that, for all u G X, WB^Ae < constHSj^uHx* < constWB^ul

282 5. Self-Adjoint Operators, the Priedrichs Extension, etc. Proof. Ad (41). By (29) and (38), it follows that, for all u G XE, (f \u) = (f,u)E = {Auq,u)e — {Beuq,u)e = (uo I u)e- Hence 2~x{u -u0\u- u0)E = 2~x{u | u)E - (/ | u) + 2~1{uq \ u0)e- Thus, the minimum problem (41) is equivalent to 2~x{u — Uq I u — Uq)e = mini, u G Xe- Obviously, this problem has the unique solution uq. Ad (41*). Since (u0 \ Au) — (Au0 \ u) = (/ | u), 2~1(Au - Au0 | u - u0) = 2~1(Au \u)-(f\u) + 2~1(Au0 \ u0). Therefore, problem (41*) is equivalent to 2~1(A(u- uo)\u — u0) = mini, u G D(A). Because of (Av \ v) > c\\v\\2 for all v G D(A), this problem has the unique solution u = uq. □ 5.5.2 Operator Equation Suppose we are given the operator equation Bu = f, ueD(B), (42) where the operator B satisfies condition (H). Let A be the Priedrichs extension of B. Note that D(B) C D(A) C X and Bu = Au for all u G D(B). Consequently, each solution of (42) is also a solution to the following equation: Au = f, ueD(A). (43) Let us also consider the problem (u | Bv) = {f\v) for fixed u e XE and all v G D(B), (43*) along with the minimum problem 2~\u | u)E -{f\u)= mini, u G XE. (43**) Theorem 5.B (The abstract Dirichlet problem). Assume (H). Let f € X. Then, each of the three problems (43), (43*), and (43**) has the unique solution uq = A~x f.

5.5 The Priedrichs Extension of Symmetric Operators 283 The solution uq is called a generalized solution to the original "classic problem" (42). We will show in Section 5.6 that (43**) corresponds to the Dirichlet problem and that the solution uq of (43*) is a solution to the Poisson equation, in the sense of the theory of generalized functions. Proof. By Theorem 5.A and Proposition 2, problems (43) and (43**) have the unique solution uq. Equation (43*) is identical to (Av | u) = (/ | v) for fixed u G XE and all v G D(B). Since (Av | u) = (Bev,u)e = (v \ u)e and (v | u)e = (u \ v)e for all u G Xe and v G D(B), equation (43*) is equivalent to (Beu,v)e = (f,v)E for fixed u G Xe and v G D(B). Since D(B) is dense in Xe, this is equivalent to Beu = /, u G XE (cf. Problem 1.10). Finally, since / G X, this is equivalent to An = /, u G D(A). D 5.5.3 Eigenvalue Problem We now consider the operation equation Bu = ^u + f, ueD(B), /iGR, u^O, (44) along with the following two generalized problems: Au = ti,u + f, ueD(A), /xGR, (44*) and (u | Bv) = fi(u | v)+(/ | v) for fixed u G X#, /x G R, and all v G jD(-B). (44**) It follows as in t\ie proof of Theorem 5.B that Problem (44*) is equivalent to (44**). We make the following assumptions: (HI) Let X be a real separable Hilbert space with dim X = oo. We are given the operator B: D(B) C X -* X as in (H), i.e., B is linear and symmetric, along with (Bu \ u) > c\\u\\2 for all u € £>(B) and fixed c> 0. Let A:D(A) CX^X be the Priedrichs extension of £?. In addition, we asume that the embedding Xe Q X is compact.

284 5. Self-Adjoint Operators, the Priedrichs Extension, etc. Theorem 5.C (Eigenvalue problem). Assume (HI) and set f = 0. Then, the following hold true: (i) The operator A has a countable system {un,/j,n} of eigensolutions that contain all the eigensolutions of A. (ii) The eigenvectors {un} form a complete orthonormal system in the Hilbert space X. In addition, un G Xe for all n. (iii) All the eigenvalues \in have finite multiplicity. Furthermore, we have 0 < c < \i\ < \ii < • - - and \in —» +oo as n —> oo. Proof. According to Theorem 5.A, (Au | u) > c\\u\\2 for all u G D(A). If it is a solution of (44*) with / = 0 and u ^ 0, then /j,(u | u) = {Au | u) > c(u | u), and hence \i > c > 0. Again by Theorem 5.A, the operator A~X:X —> X is symmetric and compact. Let A := /x_1. Then the eigenvalue problem (44**) with / = 0 is equivalent to A~xu = \u, ueX, A G R, u ^ 0. (45) Observe that A = 0 is not an eigenvalue of A-1, since A~xu = 0 implies u = 0. The assertion follows now from Theorem 4.A in Section 4.2 applied to the inverse operator A-1. D 5.5.4 The Fredholm Alternative Theorem 5.D. Assume (HI). We are given f G X and \i G R. Then, the following hold true: (i) // \i is not an eigenvalue of the operator A, then equation (44*) has a unique solution u. (ii) If 11 is an eigenvalue of A, then equation (44*) has a solution u iff (f \v) = 0 for all eigenvectors v of A corresponding to fi. Proof. By Theorem 5.A, the operator A~l:X —+ X is symmetric and compact. Case 1: Let \i ^ 0. Equation (44*) is equivalent to \u-A-1u = \A~1f, ueX, (46)

5.6. Applications to Boundary-Eigenvalue Problems 285 where A := /i_1. Ad (i). This follows from Theorem 4.B in Section 4.3 applied to A~1. Ad (ii). By Theorem 4.B, equation (46) has a solution u iff (XA-1/ \v) = 0 (47) for all eigenvectors v of A~l corresponding to A. Observe that {A'1! I v) = (f | A-\) = \{f | v). Thus, condition (47) is equivalent to (/ | v) = 0 for all eigenvectors v of A~1 corresponding to A. In turn, this is equivalent to assertion (ii). Case 2: If \x = 0, then fi is not an eigenvalue of A. By Theorem 5.B, equation (44*) has a unique solution u. □ 5.6 Applications to Boundary-Eigenvalue Problems for the Laplace Equation Let us consider the following fundamental classical boundary-eigenvalue problem for the Laplacian: —Au — nu = f on G, /i E R, (48) u = 0 on dG. Let G be a nonempty bounded open set in RN. For /x = 0, (48) is called the Poisson equation. Definition 1. The generalized problem to (48) reads as follows. We are o looking for a function u EWliG) sucn that / u(-Av)dx -\i [ uvdx= [ fvdx for all v E Cf?(G). (48*) Jg Jg Jg This means that equation (48) is satisfied in the sense of the theory of generalized functions. Formally, we obtain (48*) by multiplying (48) with v and subsequent integration by parts. Proposition 2 (Eigenvalue problem). Let / = 0. Then, the following are met: (i) The generalized problem (48*) has a countable system {un,nn} of eigensolutions that contain all the possible eigenvalues.

286 5. Self-Adjoint Operators, the Priedrichs Extension, etc. (ii) The system {un} of eigenfunctions forms a complete orthonormal o system in the Hilbert space L,2(G). In addition, un €W\{G) for all n. (iii) All the eigenvalues fin have finite multiplicity. Furthermore, we have 0 < /^i < /^2 < • • • and fjLn —> +oo as n —> oo. Proof. We set X := L2(G) and Bu := -Am, where I>(B) := C?(G). By Standard Example 4 in Section 5.3, the operator B: D(B) C X —> X is linear, symmetric, and strongly monotone. The corresponding energy space is given through XE =W\(G\ and the embedding Xe Q X is compact. Moreover, f N (u\v)e= I 2_^djudjvdx for all u,v G XE. Let A be the Priedrichs extension oiB. Then, problem (48*) corresponds to (44**). The assertion follows now from Theorem 5.C. □ Denote the energetic extension Be of B = — A by —A#. Then, -AE:Wl(G) ^wl(G)* is a linear homeomorphism, by Section 5.3. More precisely, the operator o —Ae is identical to the duality map of the Sobolev space W\{G)> Explicitly, (—Aeu)(v) = (u | v)e for all u,v E Xe- This means r N (-AE)(v)= / ^djudjvdx for all u,v ewl{G). Jg j=i Recall that the Priedrichs extension A of B = —A is given through An = -AEu for all u E D(A), o where D(A) consists precisely of all those functions u EW\{G) for which o -AEu E L2(G). This means that u E D(A) iff u EW\(G) and that there exists a function / G L2(G) such that r N r / \^djudjvdx— I fvdx for all v EW\{G). JgJ^[ jg

5.7. The Poincare Inequality 287 Proposition 3 (The Fredholm alternative). We are given f G L2(G) and H G R. Then, the following hold true: (i) If ii is not an eigenvalue, then problem (48*) has a unique solution. (ii) If ji is an eigenvalue, then (48*) has a solution iff L fvdx = 0 for all eigenfunctions v £W\{G) corresponding to fi. Proof. Problem (48*) corresponds to (44**), which is equivalent to (44*). Thus, the assertion follows from Theorem 5.D. □ Proposition 4 (The variational problem). Let \i = 0, and let f £ I/2(G). Then, the unique solution u of problem (48*) is equal to the unique solution of the following minimum problem: r N r 2~l / J2dJudJvdx~ / fudx = mm\, u£W\(G). (49) Proof. Problems (48*) and (49) correspond to (43*) and (43**), respectively. Thus, the assertion follows immediately from Theorem 5.B. □ Problem (49) is identical to the Dirichlet problem from Section 2.5. 5.7 The Poincare Inequality and Rellich's Compactness Theorem Proposition 1. Let G be a nonempty bounded open set in RN, N > 1. Then, the embedding W\(G) C L2(G) is compact This proposition was proved by Rellich in 1930. Our proof will be based on the following special Poincare inequality: fu2dx<2R2N fj2(dju)2dx + (2R)-N ( f udxj . (50)

288 5. Self-Adjoint Operators, the Priedrichs Extension, etc. Lemma 2. Let C be a closed cube in RN, N > 1, with edge length 2R > 0. Then, relation (50) holds for all u G Cl{C). Proof. We will use the well-known inequality (N \ 2 N ^anj <NY^al for allai,...,aiv GR, (51) n=l / n=l which follows from 2ab < a2 + b2 for all a, b G M. Observe that (50) is invariant under a translation. Moreover, using the transformation x i—► 2Rx, it is sufficient to prove (50) for R = |, i.e., ^ == l~2' 2J X ' ' ' X L 2' 2J* Step 1: Let JV = 1 and C = [-±, \). We are given w G C^C). Then, it(y) — u{x) = / uf(t)dt for all x,y eC. J x By the Schwarz inequality, (u(x) - u{y))2 = u{x)2 + it(y)2 - 2u{x)u{y) < I l-\u\t)\db< J u'(t)2dt, since Jc dt = 1. Applying the integral JCxC ... dxdy, we get / u{x)2dx + / u(y)2dy < / u'(t)2dt JC JC JC + 21 / u(x)dx J ( / u(y)dy J . Hence 2 / it(x)2dx < / u'{x)2dx + 2 ( / it(x)dx J . This is (50) with R = |. Step jg: Let AT = 2 and C = [-±, \] x [-|, ±]. We are given u G C1^). Let x = (£, 77) and y = (a, /?). Then uz(t,0)dt + By (51), (a + 6)2 < A^a2 + Nb2 for all a, b G R. Hence (u(rc) - u{y))2 = u{x)2 + u{y)2 - 2u{x)u{y) u{x) — u{y) = / u$(t, (3)dt + / ^(^ t)dt for all x,y eC. Jot JB < n(J ut(t,0)dt\ +N(J*uri&t)dt\ .

5.7. The Poincare Inequality 289 By the Schwarz inequality, u{xf + u{yf <N [ut(t, /?)2 + Ur,(£, tf]dt + 2u{x)u{y). Applying the integral JCxC •.. dxdy and observing that JCxC dxdy = 1, we get 2 / u2dx <N I (u\ + ufydx + 2 ( / udx J . This is (51). 5fep #: The proof proceeds analogously for RN with N > 3. D Proof of Proposition 1. We set j(u) := it. Then, the linear operator j:^(G)->L2(<?) is continuous because lb'Wlll2(G) =JGU2dx<jGL2 + f^(djU)2 I <fe = ||«||^(g) Since the set Cq°(G) is dense in W\{G)i the compactness of j follows from the compactness of the operator j:C^(G)CW12(G)^L2(G), (52) by the extension principle from Section 3.6. Let B be an open ball in RN such that G C B. Since each function u G Cq°(G) vanishes on a compact subset of (7, we get Cq°(G) C C§°(B), and the compactness of the operator j:C^(B)Cwl(B)^L2(B) (53) implies the compactness of j from (52). Consequently, Proposition 1 follows from Lemma 3, given next. □ Lemma 3. Let B be a closed ball in RN. Set j(u) := u. Then, the operator j from (53) is compact Proof of Lemma 3 for N = 1. Let B := ]a, 6[, where —oo < a < b < oo. We are given it G Cg°(B). Then, a(sc) = / uf{t)dt for all x G [a, 6]. ♦/a

290 5. Self-Adjoint Operators, the Priedrichs Extension, etc. By the Schwarz inequality, i(x)\ < / 1 • \u'(t)\dt <( f dt) If u'(t)*\ < (6-a)*||M||if2, (54) \u(x) - u(y)\ = \ u'(i)dt\ max \u[ a<x<b and < |a?-2/|*||w||ij2, (55) where i 2 / {u2+u,2)dx INIl,2 •= I / \U +U )CLX I . o Let M be a bounded set in Wl(B) H Cg°{B). By the Arzela-Ascoli theorem from Section 1.11, it follows from (54) and (55) that the set j(M) is relatively compact in C(B). Since L J a b 2 (vn(x) — v(x)) dx ] <(b — a)2 max \vn(x) — v(x)\, J a<x<b it follows from vn —> v in C(#) as n —> oo that vn ^ v in L2OB) as n —> 00. Consequently, the set j(M) is also relatively compact in Li2{B). D Proof of Lemma 3 for N = 2. Let x = (£, 77) and set 1 2 Hli,2 •= f / (^2 + ^| + ^)dxj Define M:={ueCnS):IM|if2<l}. We want to prove that (A) The set j(M) is relatively compact in 1/2(0)- Since the operator j is linear, j(aM) = aj(M) for each a > 0. Thus, o assertion (A) implies that j sends bounded sets from W\{&) ^ Cq°(B) to relatively compact sets in L2(B). Consequently, (A) implies the assertion of Lemma 3. By Proposition 10 in Section 1.11, the set j(M) is relatively compact iff, for each e > 0, the set j(M) has a finite e-net. Therefore, it remains to prove the following:

5.7. The Poincare Inequality 291 C = 0(M) (b) FIGURE 5.4. (B) For each e > 0, there exist functions u\,..., ur G M such that miri \\j(u)-j(uk)\\2 = nun / (u - uk)2dx < e. (56) Step 1: Boundary strip. We first prove the following: For each 6 > 0, there exists an open subset H of the open ball B such that H C B and / u2dx < 6 (57) JB-H (cf. Figure 5.4(a)). To this end, we choose a local (/i, ^-coordinate system as pictured in Figure 5.4(b). More precisely, we assume that the boundary OB has a local representation of the form C = g(v), /* e J, where g: J —> R is a (^-function on the interval J :=] — a, a[ with a > 0. For sufficiently small (3 > 0, the local boundary strip Bp := {(/i, C): ^J, <?(/i) - 0 < C < g(li)} is a subset of B. Let u G M. For all points (//, C) and (//, r) in S^, K U>(fJ»t)= U{([JL,t)dt + u(tl,,T). From (a + 6)2 < 2a2 + 262 for all a, 6 G R and the Schwarz inequality, it follows that ^(/i,C)2<2( / l.\uc(ii,t)\dt) +2^(/i,r)2 < 2/3 / wc(/x, *)2d* + 2w(/x,r)2. Integration over r yields gW-P Pufa C)2 < / W2H(», t)2 + 2u(ji, t)2]dt, Ja(u)-0

292 5. Self-Adjoint Operators, the Friedrichs Extension, etc. and integration over B£ yields (3 j u2dx <e I {2(32u2c + 2u2)dx < e - const111*11f 2- There exists a number n independent of e such that n local boundary strips cover a boundary strip of the ball B. Thus, choosing the number e sufficiently small, we obtain (57). Step 2: The inequality of Poincare on H. We choose closed cubes Ci, ..., Cs of edge length 2R that cover the set H such that Cj C B for all j. Let V:=(jCd. 3 = 1 By the special Poincare inequality (50) with N = 2, / u2dx < I u2dx < 4R2 / (uj + u2)dx Jn Jv Jv +(2i?)-2E 3 We set Jcn udx 2 forallueCo°°(£). (58) F(u) := (2/1,..., ys), where %• := / ucfo. Relation (58) yields / u2dx < m2\\u\\\ 2 + (2R)-2\F(u)\2 for all u G C^B), (59) where li7^)!2 = J2j \yj\2' By the Schwarz inequality, \yj\2 < meas(Cj) / u2dx < meas(S) / u2dx, JCj JB for all j. Thus, the set F(M) is bounded in Rs and is hence relatively compact Consequently, for each 77 > 0, the set F(M) has a finite 77-set, i.e., there exist functions ui,...,MrEM such that min \F(u) - F(uk)\ < rj for all ue M. l<k<r Prom (57) and (59) along with (u — Uk)2 < 2u2 + 2u\, we get the key formula (u — Uk)2dx = (u — Uk)2dx + I (u — Uk)2dx JB JB-H J?i <46 + 4R2\\u - uk\\l2 + {2R)~2\F{u) - F(uk)\2 <46+WR2 + {2R)-2ri,

5.8 Functions of Self-Adjoint Operators 293 for all u G M and k = 1,..., r. Finally, if we choose the positive numbers 6, i?, and rj sufficiently small, then we get the desired estimate (56). □ For N > 3, the proof proceeds completely analogously. 5.8 Functions of Self-Adjoint Operators Our objective is to construct a simple functional calculus for an important class of self-adjoint operators. We make the following assumptions: (H) The linear operator A: D(A) C X —> X is self-adjoint on the separable Hilbert space over K, and A possesses a complete orthonormal system {un} of eigenvectors in X, where Aun = Xnun for all n. The system {un} is finite or countable if dim X < oo or dim X = oo, respectively. Proposition 1. Assume (H). Then Au = 2_] Xn(un | u)un for all u G £2(.A). (60) n Furthermore, the following three conditions are mutually equivalent: (i) u e D(A). 00 Hn^n{un | u)un is convergent (iii) ]£n \K(v>n \u)\2 is convergent Proof, (i) => (ii). Since {un} is complete, v = 22(un I v)un for all v e X. (61) n If u e D(A), then (un \ Au) = (Aun \ u) = Xn(un \ u). Hence Au = /_^(^n I Au)un = 2_] Xn(un | u)un for all u G ^(^4). n n (ii) <=> (iii). This equivalence represents the convergence criterion for Fourier series (Proposition 5 in Section 3.1). (ii) => (i). We construct the linear operator C: D(C) C X —> X through Cu := YJ Xn(un | u)un for all ^ G D(C),

294 5. Self-Adjoint Operators, the Priedrichs Extension, etc. where u G D(C) iff this series converges. It follows from (Cu | v) = 2^ Xn(u | un)(un | v) = (u | Cv) for all u, v G D(C) n that C is symmetric. Obviously, C is an extension of A. Since .A is se//- adjoint, there does not exist a proper symmetric extension of A (Corollary 7 in Section 5.2). Hence A = C. D Let the functions F: E —> K be given. We define the operator F(A):D(F(A))CX -> X through the quite natural formula F(A) := £ F(K)(un | u)un for all u G £>(F(A)), (62) n where u G D(F(.A)) iff X^n^(^)(^n I u)un is convergent. By Proposition 5 in Section 3.1 this means that u G D(F(A)) iff ^|F(An)K|^)|2<oo. n In particular, we obtain un G D(F(A)) and F(A)iin = F(An)iin for all n. Since {^n} is a complete orthonormal system in X, the set span {u\,U2,...} is dense in X, and hence D(F(.A)) is dense in X. □ Proposition 2. Assume (H). T/ien, /or eac/i real function f:R-+R, the operator F(A) is self-adjoint Proof. Let C := F(A). As in the proof of Proposition 1, we obtain that C is symmetric, i.e., C C C*. In order to prove that C* C C, let u G C*. Since (ixn | C*ix) = (Ci£n | u) = F{Xn)(un | u) for all n, we obtain C*u = Y^(un I C*u)un = ^2F(Xn)(un | u)un. n n Hence ugD(C). D With a view to applications in the next section, we now consider functions of the form F(A,t)u = Y^F(\n,t)(un | u)un n

5.8 Functions of Self-Adjoint Operators 295 depending on a real parameter t, which will play the role of time. Set x(t) :=F(A,t)u for fixed u G X. We expect that x'{t) = Ft(A,t)u, (63) where Ft(A,t)u :=Y^Ft(K,t)(un | u)un. n In order to justify this we need the following two majorant conditions: y2\cn(un | u)\2 < oo, wheie cn := sup |F(An,£)|, (64) and \2 \dn(un | u)\2 < oo, where dn := sup |Ft(An,t)|. (65) Proposition 3. Suppose that (H) /io/ds. Le£ the function F:Rx J -+R be given, where J is a real interval. Then, the following hold true: (i) Assume (64) for ueX.Ift\-> F(\,t) is continuous on J for each X eR, then t\->x(t) is continuous on J. (ii) Assume (64) and (65) for fixed u e X. Ift\-+ F(\,t) is continuously differentiate on J for each A G R, then t \-> x(t) is continuously differentiable on J and relation (63) holds true for all t G J. Proof. Ad (i). We set Fn(t) := F(\n,t) and an := (un \ u). Since |a + 6|2 < 2|a|2 + 2|6|2 for all a, b e K, we get A := \\F(A,t)u - F(A,s)u\\2 = ]T \(Fn(t) - Fn{s))an\2 n n<N n>N Lets > 0 be given. By (64), there is a number TV such that 2^n>j/v lcn&n|2 < e. Since Fn is continuous on J, this implies A <e + e, provided \t — s\ is sufficiently small. Ad (ii). Use the mean value theorem Fn{s)-fn{t) - <(*) = F'n{t + 4{8 -1)) - Kit), 0 < 4 < 1, s — t

296 5. Self-Adjoint Operators, the Priedrichs Extension, etc. and an analogous argument as in the proof of (i). □ Example 4 (Characterization of the energetic space Xe)> Let B:D(B) C X —> X be a linear, symmetric, strongly monotone operator on the real separable Hilbert space X. Suppose that the embedding xEcx is compact. Let A: D(A) C X -» X denote the Friedrichs extension of B. Then XE = D(A%), (66) and (u | v)e = (A^u | A%v) for all u,v G Xs. (67) Proof. By Theorems 5.A and 5.C, the operator A satisfies condition (H), where An > c > 0 for all n. Then A^u = ^\l(un | u)un for all u G D(A%), (68) n where iz G J5(A2) iff ]PAn|(un | u)|2 < oo. n Observe that it G 2?(-A) iff $Zn ^nlfan I u)\2 < oo. Since 0 < An < const • A2 for all n, D(A)CD(Ai). In addition, note that D(B) C D(.A). Furthermore, it follows from (60) and (68) that (u | v)£? = (Bu | v) = (Au | v) = jP An(ii | ixn)(ix | v) n = (i4iw|i4*v) for all u,?;G £>(£). (69) Moreover, by (68) we get \\Aiu\\2 = £ An|K | u)\2 > c^ |K | u)|2 n n = c||u||2 for allueD(Ai). (69*) Step i: We set Y := D(A^) and (u | v)y := (A5ix | A*v) for all w,-yG7.

5.8 Functions of Self-Adjoint Operators 297 By (69*), \\u\\y > c\\u\\2 for all ueY. (70) We want to show that Y becomes a real Hilbert space equipped with the inner product (• | -)y. In fact, if \\u\\y — 0, then u = 0, by (70). Now let (un) be a Cauchy sequence in Y. Then, (A^un) is Cauchy in X. Hence A*un-*v mX as n —► oo. (71) It follows from (70) that (un) is also Cauchy in X, and hence un -» u in X as n —> oo. (71*) The operator A* is self-adjoint. According to Proposition 14 in Section 5.2, it follows from (71) and (71*) that u G D(A%) and A%u = v, i.e., u eY. Finally, by (71), ll^n — u\\y = \\A*un — A*u\\ —> 0 as n —> oo. In the following we want to show that Xe = Y. Step 2: We prove that the set D(A) is dense in Y. In fact, if dim X < oo, then D(A) —Y = X. Now let dim X = oo. Suppose we are given ix G D(A). Hence u G D(i4i). Set m n=l Then, iim G V for all m, and \\Um - u\(y = \\A*Um - A*u\\2 = ^2 ^n|(^n | ^)|2 -» 0 as 771 -» 00, n>ra by (68). Step 3: We prove that Xe Q Y. Let u G X#. Then there exists an admissible sequence (un) for u. This means that un G £>(#) for all n as well as iin —> ii in X as n —> oo, and (iin) is Cauchy with respect to || • \\e- By (69), (un) is Cauchy in Y. Thus, Step 1 tells us that un —> w in Y as n —* oo, along with c||ixn — w||2 < \\un — w\\y —» 0 as n —* oo.

298 5. Self-Adjoint Operators, the Priedrichs Extension, etc. Hence u = w, i.e., u e Y. Step 4: Let us prove (67), i.e., (u | v)e = (u | v)y for all u, v e XE. (72) If u, v G Xe, then there exist admissible sequences (un) and (vn) for u and v, respectively. Step 3 yields un —> u and vn -+ v mY as n —> oo. By (69), (i/n | vn)£; = (un | t>n)y for all n. Letting n -^ oo, we get (72). Step 5: Since X# is a Hilbert space, X# is closed with respect to the norm || • H^. By Step 4, \\u\\E = \\u\\y for all u e Xe- Thus, XE is a closed linear subspace of Y. Since D(A) C X# C Y and D(.A) is dense in F, we get Xs = Y. □ 5.9 Semigroups, One-Parameter Groups, and Their Physical Relevance The notion of a semigroup is the most important notion for describing time-dependent processes in nature in terms of functional analysis. The key relations are S(t + s) = S(t)S(s) for all M > 0, (73) 5(0) = /, (74) and (75). Definition 1. Let X be a Banach space over K. A semigroup {S(t)}t>o on X consists of a family of operators S(t): X —> X for alH > 0 such that (73) and (74) hold true. The generator A:D(A) C X —> X of the semigroup {#(£)} is defined through Au:= Urn r1^) - J)ia, (75) where ii G D(.A) iff this limit exists. Let S+ = {S(t)} be a semigroup. Set u(i) := S(t)u0 for all t > 0. (76) Then (i) S+ is called strongly continuous iff the function u: [0, oo[ —* K. is continuous for each uq € X.

5.9. Semigroups, One-Parameter Groups, and Their Physical Relevance 299 (ii) <S+ is called nonexpansive iff<S+ is strongly continuous and S(t): X -» X is nonexpansive for all t > 0, i.e., ||5(0^o — 5(0^o|| < ||^o — vo\\ f°r all ^Cb^o £ X and each t>0. (iii) <S+ is called linear iff S(t):X —> X is linear and continuous for all Definition 2. A one-parameter group {5(0}teR on the Banach space X consists of a family of operators S(t):X —> X for all £ G R such that 5(0) = I and S(t + s) = S{t)S(s) for all t, s G R. (77) The generator A: D(A) C I -^ I of the one-parameter group {5(0} is defined through Au :=\im t-\S(t)-I)u, (78) where u G D(A) iff this limit exists. Let S = {5(0} be a one-parameter group on X. Set u(t) := S(t)uo for all t G R. (79) Parallel to semigroups, we introduce the following terminology: (i) S is called strongly continuous iff the function u: R -» R is continuous for each i^o £ -X". (ii) S is called linear iff S(t):X —> X is linear and continuous for all (iii) S is called uniformly continuous iff <S is linear and the function 11—> 5(0 is continuous from R to L(X,X), i.e., ||5(t +ft)-5(t)|| -+0 asft-+0, for all t G R. Each uniformly continuous one-parameter group {5(0} is strongly continuous. This follows from \\S(t + h)u0 - S{t)u0\\ < \\S(t + h)- 5(t)|| ||t£o|| -♦ 0 as h -+ 0, for all uo G X. Example 3. Let X be a Banach space over K, and let i:l -^ 1 bea linear continuous operator. We set 5(t) := etA for all t G R. Then

300 5. Self-Adjoint Operators, the Priedrichs Extension, etc. (i) S = {S(t)} forms a linear one-parameter group on X with the generator A. (ii) S is uniformly continuous and hence strongly continuous. (iii) Let u$ G X be given. If we set u(t) := etAu0 for all tGK, then u = u(t) is the unique solution to the following ordinary differential equation: u'(t) = Au(t), —oo < t < oo, (80) u(0) = u0. This shows that there exists a close connection between one-parameter groups and ordinary differential equations. We shall discuss later that one- parameter groups describe reversible processes in nature. Therefore: If the operator A: X —> X is linear and continuous, then the differential equation (80) cannot describe an irreversible process in nature. This explains why it is necessary to study the differential equation (80) in the case where the linear operator A:D(A)CX-*X cannot be extended to a continuous operator on X. If D(A) is dense in X, then this means that sup H-^H = oo? ||u||<l,u€£>04) by the extension principle from Section 3.6. Such linear operators are called unbounded. Proof. By Example 5 in Section 1.23, e(t+s)A = etAesA fbralU,s€R. This is the group property (77). Statement (iii) follows from Proposition 3 in Section 1.24. In particular, we get u'(0) = Au$ for each u0 G X, i.e., the operator A is the generator of {£(£)}. Furthermore, h2 A2 ehA = I + hA+ _^ + ... for all ft € R.

5.9. Semigroups, One-Parameter Groups, and Their Physical Relevance 301 Hence \\ehA - I\\ < \\hA\\ + J^ll! + • ■ • < c»fcA» - 1. For each t e R, this implies \\S(t + h)- 5(t)|| = ||e(t+^A - etA\\ = \\etA(ehA - I)\\ < ||ctA||||c*iA_/|| _^0 as/i-^o, i.e., {5(0} is uniformly continuous. D Definition 4. Let X be a Hilbert space over K. By a one-parameter unitary group we understand a strongly continuous, one-parameter group {5(0} where each operator S(t):X —> X is unitary, i.e., \\S(t)u\\ = \\u\\ for all u e X, te R. Example 5. Let A: X —> X be a linear continuous self-adjoint operator on the complex Hilbert space X. Set 5(0 := eiAt for all t e R. Then, {5(0} is a one-parameter unitary group with the generator zA Proof. By Example 3, {5(0} is a uniformly continuous, one-parameter group with the generator iA. Since A is self-adjoint, we get (A2u | v) = (Au | Av) = {u | .A2f) for all u,v e X, i.e., A2 is also self-adjoint. Analogously, An is self-adjoint for n = 0,1,2, — This implies f E ^ i«) = («i E ^«) for a11«.«* *• \n=0 ' / \ n=0 ' / Letting m —> oo, we get (e*Au | v) = (u | e~Mt>) for all v e X. Thus, (e*tA)* = e~*tA, and hence 5(05(0* = 5(0*5(0 = J for all t G R. By Proposition 15 in Section 5.2, 5(0 is unitary. □ Remark 6 (Physical interpretation). Let X be a Banach space. We regard the elements u of X as "states" of a physical system. Furthermore, let t = time.

302 5. Self-Adjoint Operators, the Priedrichs Extension, etc. (i) One-parameter groups and reversible processes in nature. Let S = {£(£)}teR be a one-parameter group. Each function u — u(t) given through u(t) := S(t)u0 for silt G R and fixed u0 (81) is called a possible process of the system. We say that the system is in the state u(t) at time t. In particular, since 5(0) = /, the state u$ corresponds to the "initial state" of the system at time t = 0. It follows from the group property S(t + to) = S(t)S(to) for all t, to G R that u(t +10) = S(t)u(t0) for all* € R and fixed t0 G R. (82) This allows the following interpretation: (C) Strong causality. The state of the system u(to) at a fixed time to determines uniquely all the states of the system in the future t > to and in the past t < to- (H) Homogeneity in time. Ift\-+ u(t) is a possible process of the physical system, then the process t i-> u(t + to) is also possible for each fixed to GR. Observe that the transformation 11—> t + to corresponds to a translation of time. To explain this, consider two observers 0\ and 02. Suppose that 0\ and O2 perform two experiments that correspond to possible processes. Furthermore, assume that 0\ measures the state uo at time t = 0, whereas O2 measures the state uo at time t = to- Then, 0\ and O2 observe the same process provided O2 changes his clock by replacing the initial time to with the time t = 0. In addition, the group property tells us that S(t)S(—t) = S(—t)S(t) = 5(0) = / for each t G R. Thus, we obtain that the operator S(t):X -» X is bijective and S(t)-X = S{-t) for all t G R. Hence u(-t) = S{t)-xuo. (83) This allows the following interpretation: (R) Reversibility. Ift\-+u(t) is a possible process of the system, then the reverse process t \-> u(—t) is also possible. Consequently, one-parameter groups describe reversible processes in nature, e.g., wave processes without friction (energy dissipation). In fact, if u = u(t) corresponds to such a wave process, then the reverse process u — u{—t) is also possible (cf. Figure 5.5). If the one-parameter group S is strongly continuous, then each possible process u = u(t) depends continuously on time t for all teR.

5.9. Semigroups, One-Parameter Groups, and Their Physical Relevance 303 (a) (b) FIGURE 5.5. One-parameter groups are also called dynamical systems. For example, many systems in mechanics are dynamical systems (e.g., the motion of planets). Suppose that the gravitational field of a star changes in time. Then the motion of its planets is not homogeneous in time, i.e., this motion cannot be described by a one-parameter group. We will show in Sections 5.11 and 5.14 that One-parameter unitary groups reflect energy conservation of wave processes or probability conservation of quantum processes. (ii) Semigroups and irreversible processes in nature. Let <S+ = {S(t)}t>o be a semigroup. Each function u = u(t) defined through u(t) := S(t)u0 for all* > 0 and fixed u0 e X (84) is called a possible process of the physical system. In contrast to one- parameter groups, such a process is only defined for time points t > 0, and the reversibility condition (R) is not satisfied. Proper semigroups describe irreversible processes. For example, the growth of a human being is irreversible. The semigroup property S(t + to) = S(t)S(t0) for all *, t0 > 0 yields u(t +10) = S(t)u(t0) for all* > 0 and fixed t0 > 0. (85) This allows the following interpretation: (C) Causality. The state of the system at time to > 0 uniquely determines all the states u(t) of the system in the future t > to. (H) Homogeneity in time. If t \-> u(t) is a possible process for all t > 0, then so is the process 11-> u(t +10) for allt>0 and fixed to > 0. Example 7 (The harmonic oscillator). The motion x = x(*) of a mass point of mass m > 0 on M1 is described by the basic equation of classic mechanics, "force equals mass times acceleration," i.e., K = rax", (86)

304 5. Self-Adjoint Operators, the Friedrichs Extension, etc. x e OK K (a) (b) FIGURE 5.6. where x = ue and K = Ke. Here, e denotes a unit vector (cf. Figure 5.6). For small \x\, the Taylor expansion yields K(u) = K(0) + K'(0)u + --- . We assume that K(0) = 0 and K'(0) < 0. From (86), we get u"(t) + Au = 0, -oo < t < oo, u(0) = u0, u'(0) = v0, /^-... (87) where .A = ^-^. For example, equation (87) describes the motion of a spring (cf. Figure 5.6(b)). Introducing the new variable v := u', equation (87) is equivalent to the following first-order system: U' = V' (87*) v' — -Au, u(0) = u0, v(0) =vo. y J This can be written as w' = Aw, w(0) = wo, (87**) where we set w := (u,v) and .Aw := (v, —Au). The space X := M2 with the inner product (wi | w2) := AuiU2 + ^1^2 becomes a Hilbert space, and the operator A: X —> X is linear, continuous, and skew-symmetric, i.e., (Ait; | 2) = —(w I Az) for all w,z e X. According to Example 3, for each given wo e X, problem (87**) has the unique solution w(t) = etAw0 for all * e R, (88) where {e**'4} represents a one-parameter group on X. On the other hand, one checks easily that u(t) = (costC)uo + C~1(sintC)vo1 v(t) = -C(sin£C>o + (cos*C)v0, C := A%,

5.10 Applications to the Heat Equation 305 is the solution of (87*). In matrix notation, this means that fu(t)\ _ ( costC C^sintCN f u0\ _ tA (uo\ \v(t)J \-CsintC costC J\v0J 6 \v0 J ' The group property e^t+s^Auo = etAesAwo for all t,sGlis equivalent to / cos(* + s)C C^sin^ + ^CA V-Csin(£ + s)C cos{t + s)C J _( costC C-t-smtC^f cossC C^sinsCN ~ \-CsintC costC J \-CsinsC cossC /' for all i,5Gl. In turn, this is equivalent to cos(£ + s)C = (cos tC) cos sC — (sin tC) sin sC, sin(t + s)C = (sin tC) cos sC + (cos tC) sin sC for all t,s6R, which represents the addition theorems for the trigonometric functions. In summary, The addition theorems for the trigonometric functions are equivalent to the fact that the harmonic oscillator corresponds to a dynamical system. Finally, let us study the energy E := 2~1mv2 + 2~1mAu2 of the harmonic oscillator. We want to show that The energy E is constant along each possible motion of the harmonic oscillator. In fact, it follows from (87*) that —r+ = mv(t)v'(t) + mAu(t)u'(t) = 0 for all t e R. at In Section 5.11 we shall show that the same argument applies to the wave equation. Then, the original equation (87) represents the wave equation, where the self-adjoint operator A is the Friedrichs extension of —A. 5.10 Applications to the Heat Equation We consider the following initial-value problem: u'{t) + Au(t) = 0, 0 < t < oo, (8Q, We assume the following:

306 5. Self-Adjoint Operators, the Friedrichs Extension, etc. (HI) The operator B: D(B) C X —> X is linear, symmetric, and strongly monotone on the real separable Hilbert space X with dim X — oo. (H2) A: D(A) C X —> X is the Friedrichs extension of I? with the energetic space X#. Suppose that the embedding Xe £ X is compact Theorem 5.E (The abstract heat equation). For each given initial value uo G D(A), the original equation (89) has a unique C1-solution u: [0, oo[ —► X. This solution is given by u(t) = e~tAu0 for all t > 0. (90) The operator —A is the generator of the linear, strongly continuous, non- expansive semigroup {e~tA}. We regard (89) as a generalized problem to the following classic problem: u'(t) + Bu(t) =0, 0 < t < oo, ( . The function u = ix(t) from (90) is defined for each uo e X. This function is called a mi/d (generalized) solution of both (89) and (91). Standard Example 1 (The classic heat equation). Let G be a nonempty bounded open set in RN, N > 1. Set M+ =: [0, oo[. We consider the following initial boundary-value problem for the heat equation: ut-Au = 0 onGxl+, u(x, t) = 0 on dG x M+ (boundary condition), (92) u(x,t) = uo(x) on G (initial condition). For example, this problem allows the following physical interpretation. Set u(x,t) = temperature at the point x at time t. Then equation (92) describes the distribution of temperature in the "body" G without any outer heat sources.2 The given function u$ corresponds to the initial temperature of the body at time t = 0. We set X := L2(G) and B := -A with D(B) := C?(G). By Standard Example 4 in Section 5.3, conditions (HI) and (H2) are satisfied with the energetic space XE =Wl(G). 2 A detailed physical motivation can be found in Zeidler (1986), Vol. 4, Section 69.2.

5.10 Applications to the Heat Equation 307 Theorem 5.E tells us that for each given initial temperature u$ G £2 (GO the original classic problem (92) has a uniquely determined mild (generalized) solution in the sense of (90). The semigroup property of {e~~tA} reflects the fact that Heat conduction represents an irreversible process in nature. Proof of Theorem 5.E. Uniqueness. Let i/,v:R+ —> X be two C1- solutions of (89). Define w(t) := u(t) - v(t). Then, w(0) = 0. By the product rule (Example 5 in Section 2.1), it follows from 4m*) I ™W) = 2(u/(t) I w(t)) = -2(Aw(t) I w(t)) < 0 at for all t G R+ that (w(t) \w(t)) = 0 for all t G R+, and hence w(*) = 0 for all t G R+. T/ie semigroup. By Theorem 5.C, the operator .A has a complete or- thonormal system {u}n>i of eigenvectors with Aun = \nun and An > c> 0 for all n = 1,2,... . We now use the functional calculus from Section 5.8. Then, Au = ^Xn(un I u)un, n and u G D(-A) iff J2n \^n{un \ u)\2 < 00. By definition, e~tAu := ^^ e~tAn (iin I ^)^n for each £ G R+, n where u G D(e~tA) iff £)n |e~*An(i/n I ^)|2 < °°. Observe that An > 0. Thus, for all u G X and £ G R+, ||e-^||2 = Y, |e-tA"(«„ I u)\2 < Y IK I u)\2 = H|2. n n Hence D(e~tA) = X and ||e-*A|| < 1. From e-(i+s)An = e-tAne-sAn we get the semigroup property e-{t+s)A = e-tAe-aA for aU ^ s e R+> (93) In fact, let u G X and t,5G R+. Since the operator e~sA is self-adjoint, (un I e~sAu) = (e'sAun I u) = e'sXn(un \ u).

308 5. Self-Adjoint Operators, the Friedrichs Extension, etc. Hence e-tA(e^sAu) = J2e~tXn(un I e~S^K n = £e-(t+s>A"K | u)un = e-^+^u. n We now set u(t) := e~tAu0 for all t G R+. For all i^o G X and teE+,we have the decisive majorant condition ]T \e-tXn(un I u0)|2 < 53 IK I u0)\2 < oo. n n Thus, by Proposition 3 in Section 5.8, the function 11~> u(t) is continuous on R+, and ^(0) = u$. Let i^o G D(A) and teR+. Then, we have the majorant condition 53 \Xne-tXn(un | u0)|2 < J] lAnK | ^o)|2 < oo. n n Again by Proposition 3 in Section 5.8, this implies that the derivative n exists for each teR+, and the function v! is continuous on R+. Furthermore, 53 lAnKi | e~tAu0)\2 = 53 \Xne~tXn(un I u0)\2 < oo for all t G R+. n n Hence ?/(£) G D(A). In addition, Au(£) = 53-Mw* I e~tAii0)^n = 5Z^ne_tAn^n I wo)^n = -w'C*)? n n for all teR+. Therefore, u = ?/(£) represents a solution of the original problem (89). Generator of the semigroup. Let us define the operator C: D(C) CI-> X through Cw:= lim hT1 {eThA - I)w, h-++0 where w G D(C) iff this limit exists. We want to show that C = —A. In fact, differentiation of the relation (e~tAu \v) = (u\ e~tAv) for all u,v G D(C) and all t € R+

5.11 Applications to the Wave Equation 309 with respect to t at t = 0 yields (Cu | v) = (u | Cv) for all u, v e D(C), i.e., C is symmetric. Let u(t) := e~tAuo for all t e R+. For each u0 € D(A), u'(0) = — Auo. Hence —A C C. That is, the symmetric operator C is an extension of the self-adjoint operator —A. By Corollary 7 in Section 5.2, C = -A. □ 5.11 Applications to the Wave Equation We consider the initial-value problem u"(t) + Au(t) = 0, -oo < t < oo, u(0) = u0, u'(0) = v0, and we make the following assumptions: (94) (HI) The linear operator B:D(B) C X —+ X is symmetric and strongly monotone on the real separable Hilbert space with dim X = oo. (H2) A: D(A) C X -» X is the Friedrichs extension of B with the energetic space Xe> Suppose that the embedding Xe Q X is compact. We also set C \— A*. In the trivial case where X = R and A > 0, equation (94) describes the motion of a harmonic oscillator with the classic solution u(t) = (costC)uo + C~1(sintC)vo, /QC,x u'(t) = -(smtC)Cuo + (costC)vo- [ ' Up to a constant, the energy of the harmonic oscillator at time t is given through E(t) = 2-\\u'(t)\2 + \Cu(t)\2) (96) (cf. Example 7 in Section 5.9). Definition 1. By a classic solution to the original problem (94), we understand a C2-function u: E -» D(.A) such that (94) holds true and 11-> Cii(t) is C1 from R into X. By Example 4 in Section 5.8, I>(i4) C D(C), Xs = D(C), and \\u\\E = \\Cu\\ for all ueX. (97) Generalizing the classic expression (96) for the energy of a harmonic oscillator, we define the energy of a classic solution u = u(t) of (94) at time t through E(t):=2-1{\\u'(t)\\2 + \\Cu(t)f}.

310 5. Self-Adjoint Operators, the Friedrichs Extension, etc. This is equal to ^(t) = 2-1{ll«,WII2 + ll«(t)lll}- (98) Theorem 5,F (The abstract wave equation). Assume (HI) and (H2). Then, for given initial values u0 G D(A) and v0 e D(C), the original problem (94) has a unique classic solution. This solution is given by (95). The energy E(t) is constant along this solution. The proof will be given ahead. We regard (94) as a generalized problem to the following classic problem: u"(t) + Bu(t) = 0, -oo < t < oo, ( v u(0)=u0, u'{<S)=v*. (yyj Let us introduce the product space XE x X, which consists of all ordered pairs3 w := (u,v), where u G Xe, v G X. Then, Xe x X becomes a real Hilbert space equipped with the inner product (wi I W2)xExx := (ui | u2)e + (vi | v2). Let us define the operator S(t)(«o,t;o) = Wt),«,(t)) (100) through (95). If u0 G i?(i4) and v0 G D(C), then w = u(t) from (100) is the classic solution to the original problem (94). Observe that the energy at time t is given by E(t) = 2-1\\S(t)(u0,v0)fXEXX. This shows that the use of the space Xe x X is quite natural. Energy conservation means that \\S(t)(uo,v0)\\xBxx = \\(uo,v0)\\XbXx for all t G R. (100*) However, we will show that the operator S(t) is still defined for all u0 G XE and ^ G X, (100**) 3General product spaces will be studied in Section 3.6 of AMS Vol. 109.

5.11 Applications to the Wave Equation 311 and the relation (100*) remains valid. Definition 2. Assume (100**). Then the function u = u(t) from (100) is called a mild (generalized) solution of both (94) and (99). Corollary 3 (One-parameter group). Assume (HI) and (H2). Then the operator family {S(t)} defined in (100) represents a one-parameter unitary group on the product space XE x X. The proof will be given later. The original equation (94) describes wave processes. Corollary 3 reflects the fact that these wave processes are reversible. Standard Example 4 (The classic wave equation). Let us consider the following initial boundary-value problem for the wave equation: utt- Au = 0 on G x R, u(x, t) = 0 on dG x K. (boundary condition), . , u(x, 0) = uq(x) on G (initial position), ^ ' ut(x, 0) = vo(x) on G (initial velocity). Here, G is assumed to be a nonempty bounded open set in RN, N > 1. We set X := L2(G) and Bu := -Au with D(B) := C%°(G). By Standard Example 4 in Section 5.3, conditions (HI) and (H2) are satisfied, where XE =W\{G) and \\u\\E Thus, we may apply the results to (101). In particular, for given initial values o ^o e^(G) and ^o g L2(G), we get a uniquely determined mild (generalized) solution for (101). An application to the vibrating string will be considered in the next section. According to (98), the energy of the wave process at time t is given through N E(t) = 2'1 [ JG \ut{x,t)2 + ]T(d^(M))2 This is the well-known classic energy formula. dx. Proof of Theorem 5.F. We will use the functional calculus from Section 5.8. In fact, the assertions of Theorem 5.F follow by means of simple formal

312 5. Self-Adjoint Operators, the Priedrichs Extension, etc. computations. The point is that we have to justify these formal computations by respecting the domains of definition of the operators A, C, and so forth, and by using majorant conditions in the sense of Proposition 3 in Section 5.8. By Theorem 5.C, the operator A has a complete orthonormal system {un} of eigenvectors with Aun = Xnun for all n = 1,2,... , where 0<c<Ai<...<An<...—>oo as n —> oo. By Section 5.8, we get Aau := ^2^n(.un I w)wn, a > 0, n where u G D(^a) iff £n lAnK I u)|2 < oo. Hence D(A) C D(C), where C := A2. Moreover, C(Cw) - An for all u G D(A). i In fact, since C is self-adjoint, we get (un \ Cu) — (Cun \ u) = \n{un \ u). Hence, for all u G D(A), C{Cu) = ^2^n (Un | Cu)un = ^2 ^n{un \ u)un = Au. n n This implies {Cu | Cv) = {u | C(Cv)) = (u | Av) for all u, v G I>(j4). Uniqueness via energy conservation. Let u = u(t) and v = v(£) be two classic solutions to (94). Set w(t) := ii(t) — ?;(£). Then, w(-) is a classic solution of (94) with w[o) = w'(0) = 0. Observe that w(t) G D(A) for all t G R, and hence w(t) G D(C) for all t G R. We shall show that (Cw(t))' = Cw'(t) for all t G R. (102) By the product rule (Example 5 in Section 2.1), differentiation of the energy function E(t) = 2-\w'{t) | w'(t)) + 2-\Cw{t) | Cw(t)) yields E'(t) = (w"(t) | w'(t)) + [Cw\t) | Cti;(t)) == ~(Aw(t) | w'(t)) + (w'(t) | Aw(t)) = 0 for all t G R.

5.11 Applications to the Wave Equation 313 Since £"(0) = 0, this implies E(t) = 0. Therefore, w'(t) = 0, and hence w(t) = 0. Energy conservation. Similarly, we obtain that E'(t) = 0 along each classic solution u = u(t) of (94). Proof of (102). Let wh(t) := h'1{w{t + h) - w(t)). Then Cwh(t) = h-\Cw{t + h)~ Cw(t)). By Definition 1, the derivatives w'(t) and (Cw(t))' exist. Letting h -» 0, we get wh(t) -♦ w\t) and Cwh(t) -+ (Cw(t))'. Since C is self-adjoint, this yields (Cw(t))' = Cw'(t), by Proposition 14 in Section 5.2. This proves (102). i Let uo G D(A) and vo G D(C) be given. Set /xn := An- Then, ]T |/4 (un I ^o)|2 < oo and Y^ \^n{un I v0)\2 < oo. (103) n n For all t G R, define4 u(t) := (costC)iio + C~1(sintC)v0 sin/xn£ = ]PcOS/Xn£(un I li0)^n + J^ ~ (^n | V0)un. (104) n ^n Formal differentiation yields u'(t) = ^2-/j,n sin/j,nt(un | u0)un + ^2 cos/jint(un \ Vo)un, u"^) = -^^ncosM(^n I u0)un - ]P/xnsin/xn£(un I v0)un. n n For alH G R and all n = 1,2, ..., we get in //.«.£ I < const, (105) |cos/xn£|2, |sin/xn£|2, , ,2 sin /xn£ Mn and |/xn| < const |/xn|2. Thus, using (103), the necessary majorant conditions from Proposition 3 in Section 5.8 are satisfied. This justifies the preceding formal differentiation. Hence the function u:R -» X from (104) is C2 and u(0) = u0, u'(0) = v0. 4For brevity of notation, we write cos ^nt{un \ uq) instead of (cos ^nt){un \ uq), and so on.

314 5. Self-Adjoint Operators, the Friedrichs Extension, etc. Similar arguments5 show that Au(t) = ]P /4 COS fJLnt(un | U0)un + ]P fJLn sin /jLnt(un I Vo)ttn, n n and u(t) G D(A) for all t G R. Hence u"(t) = -4u(t) for all t <E R. In addition, we get Cu(t) = ^2^n cosHnt{un \ u0)un + ^ sinnnt(un \ v0)un, (106) n n and u(t) G D(C) for all t G R, along with (Cu(t))' = ]P -/4sin/xn£(un | u0)un + ^/xncos/xn£(un | v0)un, n n for all t G R, i.e., the function £ i-> Cit(t) is C1 on R. D Proof of Corollary 3. Let u0 G D(C) and v0 G X. Then, ]P |/xn(^n I ^o)|2 < oo and ]P |(un | v0)\2 < oo. n n By the majorant criterion from Proposition 3 in Section 5.8, it follows from (104) and (105) that u = u(t) is C1 on R. Set S(t)(uo,v0):=(u(t),u'(t)). Since D(A) and D(C) are dense in X# and X, respectively, it follows from the extension principle in Section 3.6 that relation (100*) remains valid for all uo G Xe and vo G X, i.e., ||S(t)(u0, vo)||xBxx = IK^o,vo)\\xExx for all (u0,vo) G XE x X (107) 5For example, observe that ^(t) = y^anUn, n where an := (un | it(t)) = (itn I uo) cos/int + («n | vo)^'1 sin/in£, by (104). To prove that u(t) G D(-A), we need 2^|/i^an|2 < oo. n However, this follows from (103) along with \a + b\2 < \a\2 + |&|2 for all a, b G C.

5.12. Applications to the Vibrating String 315 Hence the operator S(t) is linear and continuous on Xe x X. The group property S(t + s) = S(t)S(s) for all t, s e R follows as in Example 7 from Section 5.7 by using the addition theorems for sin/xt and cos/xt along with the functional calculus. Thus, the operator S(t) is bijective by Section 5.9. Finally, it follows from (107) that S(t) is unitary for each t. □ 5.12 Applications to the Vibrating String and the Fourier Method We will show that the motion of a vibrating string6 is governed by the following equations: Utt — Uxx =0, 0 < X < 7T, —00 < X < 00, u(x, i) = 0, x = 0,7r, —oo < t < oo (boundary condition), u(x, 0) =uo(x), 0 < x < n (initial position), ut(x, 0) — vo(x) 0 < x < it (initial velocity). (108) Here, u(x, t) denotes the deflection of the string at the point x at time t (cf. Figure 5.7). Set un{x) := 7T~2 sinnx, An := n2, (u \ v) := / u{x)v{x)dx. Jo By a classic solution to (108), we understand a solution u such that u,ux,uxx,uuutt e C([0,7r] x R). Proposition 1. (i) Classic solution. We are given the functions u0,vo€ ^[0,^] with the boundary conditions u^ ' (0) = Uq ' (n) = Vq ' (0) = Vq \it) = 0 for Jfc = 0,2. Then, the original problem (108) has a unique classic solution u, where oo u{x,t) = /_^{(^n I ^o)cosnt+ (ixn I vo)^1 smnt}un(x). (109) n=l 6For simplicity of notation we set c = 1, where c denotes the wave velocity (cf. Remark 4 ahead). Moreover, we choose the string length £ = tc.

316 5. Self-Adjoint Operators, the Friedrichs Extension, etc. ,u = u(x, t) FIGURE 5.7. This series converges uniformly for all x G [0, it] and t G R. (ii) Generalized solution. Suppose that the initial data satisfy the weaker conditions o u0 €^2(0,71-) and v0 G L2(0,7r). Then, for each t G R, series (109) converges in the sense of the Hilbert space 1^(0,7r). The proof ahead shows that, under the assumption (ii), series (109) represents the mild (generalized) solution of the original problem (108), in the sense of Theorem 5.F in Section 5.11. Remark 2 (The classic Fourier method). Let us recall the famous classic motivation of the ansatz (109). First we are looking for special solutions of the string equation of the form u(x,t) = <p(x)ip(t). If u satisfies the equation utt — uxx = 0, then (f){x)^"(t) - <i>"{x)il){t) = 0. (110) Suppose that there exist points xo and to such that (j)(xo) ^ 0 and ^(to) ¥" 0. Then, the boundary condition "u(Q, t) = u(ix, t) = 0 for all times t G R" implies 0(0) — 0(7r) = 0. Letting either t = to or x = xo in (110), we obtain the following boundary-eigenvalue problem: 4f'{x) = -\4>{x), 0(0) = 0(tt) = 0, along with the differential equation </>"(*) = -A^(t), 0 < x < n, A G -00 < t < 00, (111) (112) where A = -^^ = ~^p Problem (111) has been studied in Section 4.5. The eigensolutions of (111) are cj) = un, A = An = n, n=l,2,....

5.12. Applications to the Vibrating String 317 Moreover, each solution of (112) with A = An is a linear combination of the two special solutions i/>i(t) = cosnt and i/>2(t) = sinnt, n=l,2,.... The special solutions u(x,t) := un(x)ipi(t) and u(x,t) := un(x)ip2(t) of ^tt - wxx = 0, i.e., un(x) cosnt and un(x) sinnt, n = 1,2,... , are called the eigenosdilations of the string. Now to the point of the classic Fourier method. We assume that Each motion of the string is a superposition of eigenoscillations. Therefore, we make the following ansatz:7 oo u(x,t) = Y^(an cosnt + /?n sin nt)un(x). (113) 71=1 To determine the unknown coefficients an and /?n, we use the following formal argument. Let t = 0. Then, oo U0(x) = u(x, 0) = ^2 <XnU>n(x)- (113*) n=l Observe the orthogonality relation (wn I %) = / wn(a;)wm(a;)da; = 8nm for all n, m = 1,2,... . Jo Thus, multiplying equation (113*) by um(x) and integrating over the interval [0,7r], we get otm = Kn I uo), rn = 1,2,... . Furthermore, formal differentiation of (113) yields oo v0(x) = ut(x, 0) = ]T n/3nun(x), (113**) n=l and hence An = Kn | ^o)ra~\ m = 1, 2, . . . . 7This general superposition principle dates back to a famous paper by Daniel Bernoulli in 1753. Interestingly enough, Euler did not believe that the ansatz (113) describes the most general form of the string vibration. Obviously, the time was not ripe for a general theory of Fourier series. A detailed discussion can be found in Szabo (1987), Chapter 4.

318 5. Self-Adjoint Operators, the Priedrichs Extension, etc. This way we obtain the expression (109). In the following proof we have to justify these formal considerations. Here, our earlier investigations about the boundary-eigenvalue problem (111) will play a decisive role (cf. Section 4.5). Proof. Ad (i). Uniqueness. Let u be a classic solution to (108). Then, for each tGR, u(>, t) e C2[0,tt] and u(0, t) = u(tt, t) = 0. By Proposition 2(iii) in Section 4.5, oo u(x,t) = \] bn(t)un(x) for all x G [0,7r], teR, n=l where M*) := (iin | u(-,t)) = / un(x)u(x,t)dx, Jo n=l,2,.... Hence /»7T &n(*) = / Un(x)ut(x,t)dx. JO to Let n = 1,2, Integration by parts yields pTT PIT b'n(t)= I un(x)utt(x,t)dx = / un(x)uxx(x,t)dx Jo Jo = / Un(x)u(x,t)dx = -\nbn(t), Jo since u^ = —\nun. This implies the following initial-value problem 6n(*) = -An6(t), -oo < t < oo, MO) = Ki | ^o), &n(°) = Ki I vo), n = 1,2,... , which has the following unique solution: bn(t) = (un | u0)cosnt + (un | v^jn"1 smnt for all tGl. Recall that An = n2. This way we obtain (109). Existence. Formal differentiation of (109) yields the following formulas: oo Ut(x,t) = 2_]{~(Un I ^o)^sm^+ {un | Vo) COS Tlt}un(x), n=l oo v>tt(x,t) = /_^{—(^n | uo)n2cosnt — (un \ vo)nsinnt}un(x), n=l

5.12. Applications to the Vibrating String 319 ux(x,t) = 2j{(^n | uo)cosnt + (un | vq)™"1 sinnt}ufn(x), oo n=l oo uxx(x,t) = ^{(^n | ^o)cosnt + (un | v0)n 1sinnt}wJ((a;). (114) 71=1 To justify this, observe the following. By Proposition 2 in Section 4.5, each of the series oo J2\{un\w)u^\x)l k = 0,1,2, ^ = ^o,^o (115) 71=1 converges uniformly on [0, n]. For alH e K. and n = 1,2, ..., |cosn£|, |sinn£|<l and n<n2. In addition, n2un = —u^. Therefore, each of the series from (114) can be majorized by (115) and is hence uniformly convergent on [0, n] x R. This justifies the formulas (114). In addition, we obtain that the functions u, ut, v>tti Ux, uxx are continuous on [0, tt] x R. One checks easily that the function u from (109) represents a solution of the original problem (108). In fact, from un(0) = un(7r) =0, n = 1,2,... , we obtain the boundary relation ^(0, t) = u(ir, t) = 0 for all tGR. Since u„ = -n2un, n= 1,2,... , it follows from (114) that the differential equation u>tt - uxx = 0 is satisfied on [0, tt] x R. Finally, using Proposition 2 in Section 4.5, we obtain the following two initial conditions: u(z,o) = 5^Ki I ^o)Mz) = M^O, 71=1 oo ut(x,0) = 5^(^71 I Vo)un(x) = v0(x) for all x e [0, tt], 71=1 Ad (ii). We set X := L2(0,tt) and Bu:=-u" with £>(£):= C0°°(0,7r).

320 5. Self-Adjoint Operators, the Priedrichs Extension, etc. Then, problem (108) is a special case of Standard Example 4 in Section 5.11. Let A denote the Priedrichs extension of the operator B. The corresponding energetic space is given through Since un(0) = un(n) = 0, it follows from Standard Example 11 in Section 2.5 that Un G Xe, n- 1,2,... . Integration by parts yields / unv"dx = I u'^ydx = —\n I unvdx for all v G C£°(0,7r). Jo Jo Jo That is, (un | Bv) = \n{un | v) for all v e D(B). By (44**), this means that Aun = Aniin, n = 1,2,... . By Proposition 2 in Section 4.5, the orthonormal system {un} is complete in £2(0,7r), by Section 4.1. Thus, the self-adjoint operator A has no eigenvalues different from {An}. According to (104), the mild (generalized) solution to the original problem (108) is given through 00 u^) = y%2i(Un I wo) cos/xnt + (iin I vq)^1 sinHnt}un, 71=1 I where /xn := An = n. This is precisely the series from (109). □ Example 3 (Physical motivation of the string energy). Let p denote the constant density of the string, where p > 0. Then, a small piece V of the string has the mass Am = pAs, where As denotes the arclength of V (cf. Figure 5.7). Let x0 denote the position of V. Then, the velocity v of V at time t is given through v = ut(x0,t). Thus, we obtain £kin(P) = 2-\Am)v2 = 2-1p(As)ut(x0,t)2 for the kinetic energy of V. Finally, we assume that the potential energy Epot(V) of V is proportional to the extension of V, i.e., Spot(V) = a(As — Ax), where a = const > 0.

5.12. Applications to the Vibrating String 321 Then, the total kinetic energy E^n and the total potential energy Epot of the string are given by summing over all small pieces of the string, i.e., Ekin = J2 2-V(As)u? and Epot = ]T a(As - Ax), v v Observe that As = ^l+ux(x,t)2Ax. More precisely, the final definition of E^n and Epot at time t will be based on the following integrals: £kin(*):= / 2-1put{x,t)2dx, Jo Epot{t): = a {^l + ux{x,t)2-l)dx. Jo Suppose that the deflection of the string is small, i.e., \ux\ is small. By Taylor series expansion, y/l+U* = l + 2-1U2x + .-.. Hence the total energy E(t) of the string at time t is 5, given approximately through /»7T E(t) = SkinW + EPot(t) = 2"V / (utfa *)2 + c2ux(x, t)2)dt, Jo where c := (-) . Let (j>: K. -* K. be a C1-function. Then, / is said to be stationary at the point x iff /'(») = o, i.e., the tangent line at the point x is horizontal (cf. Figure 5.8). Thus, / is stationary at local minima, local maxima, and horizontal inflection points. If f'(x) — 0, then we also say that x is a critical point of /, and f(x) is called a critical value of /. Remark 4 (Physical motivation of the string equation). The fundamental principle of stationary action in mechanics says that action A(u) := / (E^[n(t) — Epot(t))dt = stationary!. J to Here, we vary over those states of the system that are fixed at both the initial time to and the terminal time t\ and those that satisfy the boundary

322 5. Self-Adjoint Operators, the Friedrichs Extension, etc. I ^x FIGURE 5.8. conditions. For the string with fixed end points, this means the following for the action A(u)\ A(u) := 2~1p / ( / (u2 — c2u2x)dx 1 dt = stationary!, (116a) where u(0,t) = u(n,t) = 0 for all t e [to,ti] (boundary condition), u(x,to) = fixed for all x G [0, n] (initial condition), u(x,ti) = fixed for all x e [0, it] (terminal condition). (116b) The meaning of "stationary!" will be explained in (117*). Define n:=]0,7r[x]t0,ti[. Let u = u(x,t) be a sufficiently smooth solution of (116). We set w(x, t) := u(x, t) + tv(x, t), where r is a real number. The sufficiently smooth function v is called admissible iff v = 0 on dW. This guarantees that both w and u satisfy the same side conditions (116b). Set <t>v(r) := A(u + rv). By definition, A is stationary at u iff 4>'v(0) = 0 for all admissible functions v. (117*) We want to show that (117*) implies the string equation utt(x, t) - c2uxx(x, t) = 0 on £L (117) In fact, it follows from (117*) that 0^(0) = p I (utvt - c2uxvx)dx dt = 0 for all v e C^(n). Integration by parts yields / {utt - c2uxx)v dxdt = 0 for all v e C^(n). Jq

5.13 Applications to the Schrodinger Equation 323 c -m (a) u = a(x — ct) (b) u = b(x + ct) FIGURE 5.9. By the variational lemma from Section 2.2.3, this implies (117). If the functions a, b: R —> K. are C2, then the function u(z, t) := a(x - ct) + b(x + ct) (118) is a solution of the string equation (117) for all (x,t) G M2. Here, x i-> a(x — ct) corresponds to a wave that moves with velocity c from left to right (cf. Figure 5.9), whereas x h-» b(x + ct) moves with velocity c from right to left. In each textbook on partial differential equations one proves that (118) represents the most general C2-solution of the string equation (117). This justifies the designation "one-dimensional wave" equation for the string equation. 5.13 Applications to the Schrodinger Equation The following so-called abstract Schrodinger equation u'(t) = -iAu(t), -oo < t < oo, ( v u(0) = uo (119) governs the motion of quantum systems. Theorem 5.G. Let A: D(A) C X -» X be a self-adjoint operator on the complex Hilbert space X. Then, the following hold true: (i) There exists a unique one-parameter unitary group generated by the skew-adjoint operator —iA.

324 5. Self-Adjoint Operators, the Priedrichs Extension, etc. (ii) For each uo G D(A), the function u(t) := S(t)u0 for allteR (120) is the unique C1-solution to (119). For each uo G X, the continuous function u: R —> X from (120) is called a mild solution of (119). Proof. Uniqueness. Let {S(t)} be a one-parameter unitary group generated by the operator C := —iA. Then S{t + h)- S(t) = S{t)(S(h) -I) = (S(h) - I)S(t). (121) Let uo G D(A) be given. Then ~(S(t)u0) = lim S(t)h-\S(h) - I)u0 = S(t)Cu0, dt h—>o since {£(£)} is strongly continuous. On the other hand, it follows from (121) that lim h'^Sih) - I)S(t)u0 = S(t)Cu0. h—>0 By the definition of the generator C in Section 5.9, this implies S(t)uo G D(C) and CS(t)u0 = S(t)Cu0 for all t G R. Thus, the function u(t) := S(t)uo satisfies the initial-value problem u'(t) = Cu(t), -oo < t < oo, ,122, u(0) = u0. { } Let v = v(t) be another solution of (122). We shall show below that 4~S(t - s)v{s) = -S(t - s)Cv(s) + S(t - s)v'(s) ds (123) = -S{t - s)Cv{s) + S(t - s)Cv{s) = 0 for all t,sGM. This implies that S(t - s)v(s) = const(t) for all s G R, and hence S(—s)v(s) = v(0) for all s G M. Applying S(s) to this equation, we get v(s) = S(s)uo = u(s) for all s G M. This proves the uniqueness of the solution u = u(t) to (122).

5.13 Applications to the Schrodinger Equation 325 Let {T(t)} be another one-parameter unitary group generated by the operator C. Then, the function w(t) := T(t)uo for alU G R is a solution of (122), and hence w(t) = u(t) for all tGl. This implies (T(t) - S(t))uo = 0 for all uq from the dense subset D(A) of the Hilbert space X. By the extension principle from Section 3.6, T(t) = S(t) for all t e R. This proves that the operator C cannot generate two different one-parameter unitary groups. Proof of (123). Since the derivative v'(s) exists, we get v(s + ft) = v(s) + hv'(s) + he(h), where £(ft)-»0asft-»0. Observe that h'^Sit - s - h)v(s + h)- S(t - s)v{s)] = Ai + A2, where Ax := h-l[S(t-s-h)-S(t-s)]v(s), A2 := h-lS(t-s-h)(v(s+h)-v(s)). Obviously, Ax = S(t - s)h-\S(-h) - I)v(s) -+ -S(t - s)Cv(s) as h -♦ 0. Moreover, A2 = S(t-s- h)v'(s) + r(h), where r(h) := S(t - s - h)e(h). Since ||5(t)|| < 1 for all t, we get ||r(ft)|| < ||e(ft)|| -* 0 as h -* 0, and hence A2 -^ 5(t - s)t/(s) as ft -^ 0. Existence. Case 1: We first make the following additional assumption. (H) The self-adjoint operator A: D(A) C X -+ X possesses a finite or countable complete orthonormal system {un} of eigenvectors with the corresponding eigenvalues {An}, i.e., Aun = \nun for all n.

326 5. Self-Adjoint Operators, the Priedrichs Extension, etc. We now use the functional calculus from Section 5.8. By definition, e'itAu := ]T e'itx" (un \ u)un, (124) n where u e D(e~itA) iff £n \e~itXn{un | u)\2 < oo. Since \eiOL\ = 1 for all real numbers a, we obtain D(e~ltA) = X. Moreover, \\e-itAu\\2 = J2 \e~itXn(un I u)\2 = J2 IK I u)\2 = \\u\\2 for all ueX. n n (125) As in the proof of Theorem 5.E, it follows from e-itxe-isx = e-i(t+s)\ for all t)5?AeR that e-itAe-isA = e-i{t+s)A for aU ^ s e R> That is, {e~ltA} represents a linear one-parameter group. Hence the operator S(t): X -» X is bijective, by Section 5.9. Moreover, relation (125) tells us that S(t) is unitary on X for each t. Set u(t) := e~itAu0 for all t e R. First let ii0 G X. According to (124), it follows from the majorant criterion (Proposition 3 in Section 5.8) that u = u(t) is continuous on R. Thus, {e~~ltA} is strongly continuous. Summarizing, we obtain that {e~ltA} is a one-parameter unitary group. Now let uq G D(A). Formal differentiation of (124) with u = uq yields U'$) = J2 e~iXnt(-iXn)(un I U0)un. (126) 71=1 Because of the majorant condition oo oo ]T \e-iXnt\n{Un I W0)|2 < XI \Xn(un I U°)!2 < °°' (127) n=l n=l formula (126) holds true for all t G R, and i/ is continuous on R. By (124) with u = u0 and (127), we get u(t) G D(A) for alU G R and 71=1 Hence Au(t) = ^2 e iXntK(un | u0)un for all t G R. 71=1 u'(t) = -i;4u(t) for all t G R. (128)

5.14 Applications to Quantum Mechanics 327 Finally, we want to show that the operator — %A is the generator of {e~itA}. Let C denote the generator of {e'itA}. By (124), (e~iAtu \v) = (u\ eitAv) for all u, v e X. Differentiating this with respect to t at t = 0, we obtain (Cu \v) = (u | -Cv) for all u,v e D(C). Thus, the operator C is skew-symmetric. By (128) with t — 0, C is an expansion of the skew-adjoint operator — iA, and hence C = —%A by Corollary 7 in Section 5.2. Case 2: In the general case where the operator A: D(A) C X -» X is merely self-adjoint, one has to use the general functional calculus. Then, we define /oo e-itXdExu for all u € X, -OO in the sense of Remark 4 in the upcoming Section 5.14, and the proof proceeds analogously to Case 1. The details can be found in Zeidler (1986), Vol. 2A, p. 186. □ Note that our applications of Theorem 5.G to the harmonic oscillator in quantum mechanics correspond to Case 1, for which we have given a full proof (cf. Section 5.14). 5.14 Applications to Quantum Mechanics We want to show that the theory of Hilbert spaces represents a proper tool for the mathematical description of quantum systems. 5.14.1 An Abstract Setting for Quantum Mechanics A quantum system, e.g., an atom or a molecule, is described by a complex Hilbert space X. (i) Physical states. The unit vectors ip of X are called states, that is, W> I VO = 1. The two unit vectors ip and <\> are called equivalent iff ip = Xcj) for some complex number A with |A| = 1. Intuitively, each physical state of the quantum system corresponds to a state. We assume that equivalent states represent the same physical state.8 8It was generally believed until 1952 that there exists a one-toone correspondence between "states" and "physical states." However, in quantum field theory

328 5. Self-Adjoint Operators, the Priedrichs Extension, etc. (ii) Physical quantities. The self-adjoint operators A: D(A) C X —> X on the Hilbert space X are called observables. Intuitively, each "physical quantity" of the quantum system corresponds to an observable. In particular, the energy of the quantum system corresponds to a self- adjoint operator H:D(H) C X —> X, which is called the Hamiltonian of the quantum system. (iii) Measurements. Suppose that we measure the observable A in the state ip. A fundamental feature of quantum physics is that, opposed to classical physics, the prediction of a measurement's outcome is only statistical. The numbers A:=(il>\ Ail)), ^ € D(A), and (AA)2:=\\Ail>-A<l>\\2, iieD{A) correspond to the mean value A and the dispersion (A.4)2 of the observable A in the state ^, respectively. Since the operator A is symmetric, the mean value A is real. Moreover, AA = \\Ail> - Ail>\\ >0.9 (iv) Dynamics. The equation i/>(t) = e-iJPil>o, t€R, (129) describes the time-evolution of the quantum system, i.e., if ^o € X corresponds to the state of the system at time t = 0, then i/;(t) corresponds to the state of the system at time t. Here, {e~V"} denotes the one-parametric unitary group generated by the skew-adjoint operator — ^ (cf. Theorem 5.G in Section 5.13). Recall that if ^o G D(H), then itwp'{t) = Hi/>(t) for all t e R. there exist so-called supers election rules, e.g., for charge and baryon number. These superselection rules say that there exist "states" (resp., "observables") that do not correspond to "physical states" (resp., "physical quantities"). For example, suppose that the two states ipi and ip2 correspond to a charged particle, with the charge e\ and e2, respectively, where e\ ^ e2> Then, the state aiipi 4- ot2ip2 with a\ ^ 0 and ct2 ^ 0 does not correspond to a "physical state." 9To motivate the definition of A*4, assume that ip G D(A2), i.e., ip G D(A) and A*l) € D(A). Then (A.4)2 = ((A - AI)iP I (A - AI)iP) = (</> I (A - AI)2i/>)9 i.e., (A*4)2 is the mean value of the observable (A — AI)2. This coincides with the definition of dispersion in probability theory.

5.14 Applications to Quantum Mechanics 329 This is the Schrodinger equation, where h := ^ and h denotes the Planck quantum action. If we measure length, time, and mass in meter, second, and kilogram, respectively, then h = 6.625 -HT34kg—. s Since the operator U(t) :— e-1*- is unitary for each t G 1, we get m) i w)) = (</>o i ^o) = i, i.e., if ipo is a state, then so is ip(t) for each time t. Therefore, we obtain the following crucial fact: Time evolution of quantum systems preserves states. 5.14-2 Discussion of the Abstract Setting Example 1 (Physical interpretation of eigenvalues). Suppose that a is an eigenvalue of the observable A with the corresponding eigenstate ip, i.e., Aip = aip, (ip I ip) = 1. Then A = (ip | Alp) = a and AA = \\AiP - Aip\\ =0. In this case we say that the observable A has the "sharp" value a in the eigenstate ip. Example 2 (Physical interpretation of Fourier coefficients and probability). Suppose that {ipn} is an orthonormal system of eigenvectors of the observable A, i.e., Aipn = anipn for all n. Set Y := closure of span {ipn}- Then, {ipn} represents a complete orthonormal system in Y. Thus, for each state ip G V, we get the Fourier expansion ^ = J^Wn I Wn- (130) n Since (ip | ip) = 1, n Therefore, the following physical interpretation of the Fourier coefficients makes sense. Suppose that the system is in the state ip €Y. Then \(ipn | ip)\2 = probability for the realization of the eigenstate ipn.

330 5. Self-Adjoint Operators, the Priedrichs Extension, etc. Assume now that ip e Y and Aip e Y. Then ^l=(V;|^) = ^an|(^n|^)|2, (131) n and (A^)2 = pv - ^vii2 = £k - A?\w« i ^)i2- (132) n Proof. Since Ai/> G V, M> = ^2(^n | Al/J^n = ^T(All)n | ^)^n = ^M^n \ 1p)^n. n n n Hence (l/) | i4^) = ^an(^n | ^)(^ | ^n). n This is (131). Moreover, (A^)2 = {{A - AI)il) | {A - AI)*p) = ( Xl(an " ^)(^ I W* I X](a™ " ^X^™ I W™ I ' \ n ra / which implies (132). □ More generally, let the quantum system be in the state ip G X. Let <\> be another state. Then, we define |(-0 | 0)|2 = probability for realizing the state <\>. This definition makes sense. In fact, by the Schwarz inequality, we have l(^l^)l2<W2IHI2 = i. If U: X —> X is a unitary operator, then {Uip \ U<p) = (ip | 0), and hence |(I70 | Uip)\2 = |(</> | <t>)\2 for all ^Gl Therefore, we obtain the following: Unitary operators preserve probability. In particular, since the time-evolution operator e-1*- is unitary, we obtain that Time evolution of quantum systems preserves probability. Proposition 3 (The uncertainty inequality). Let A and B be two observ- ables, and let ip be a state in the Hilbert space X such that il> G D(A) n D(B), Aip G D{B), and Bip G D(A).

5.14 Applications to Quantum Mechanics 331 Then AAAB > C, (133) where A A and AB correspond to the state ip, and C = 2-l((BA-AB)ip\ip). Roughly speaking, relation (133) tells us that // the two observables A and B do not commute, then it is impossible to measure precisely the corresponding two physical quantities at the same time. In Section 5.14.5 we will show that (133) implies the classical Heisenberg uncertainty principle: It is impossible to measure precisely position and momentum of a quantum particle at the same time. The proof of the fundamental inequality (133) will be based on the Schwarz inequality. Proof. For all A,BeR, {{BA- AB)ip\ip) = ({B -BI)(A- AI)ip\ip) - {{A- AI)(B -BI)ip\ip) = {{B - BI){A - AI)ip | V) - (ip I (B - BI){A - AI)ip) = 2i lm((B - BI)(A - A)ip \ ip). With A:= (i/>\ Aip) and B =: (ip \ Bip), the Schwarz inequality yields AAAB = \\(A - M)ip\\ ||(B - BI)ip\\ > \({A - AI)ip \ (B - BI)*P)\ > \lm((B - BI)(A - AI)ip | ip)\ > 2'1\((BA - AB)ip \ ip)\. D 5.14-3 A Look at the General Functional Calculus Remark 4 (General functional calculus). In the general spectral theory for self-adjoint operators, one shows that each self-adjoint operator (observable) allows a representation of the following form: /oo XdEx.n -oo Explicitly, this means that /oo Xd(v | Exu) for all u € D(A), veX, (134) -OO

332 5. Self-Adjoint Operators, the Priedrichs Extension, etc. and oo. /oo \X\2d(u | Exu) < -OO This way it is possible to define functions of the operator A through /oo F(\)dEx,» -oo for the given function F: K. —> C. Explicitly, this means that /oo F(X)d(v | £Au) for all u 6 I>(F(i4)), v G X, -OO where /oo |F(A)|2d(u | Exu) < oo. -OO In addition, /oo |F(A)|2d(u | Exu) for all u € D(F(A)). -OO Moreover, we get D(F(A)*) = D(F(A)), and /oo F(X)dEx," -OO i.e., /oo F(X) d{v | SAu) for all u G D(F(A)*), v € X. -OO The meaning of the Stieltjes integrals J... d(v | E'a'^) will be discussed ahead. This generalized functional calculus dates back to von Neumann (1932) who generalized the spectral theory of Hilbert (1912) for bounded symmetric operators to general self-adjoint operators. In terms of quantum theory, the formula /oo d(tp | Exip) -oo allows the following interpretation: Let J be an interval. Suppose we measure the observable A in the state ip. Let a be the measured value. Then / d(ip | E\i/>) = probability for a e J.

5.14 Applications to Quantum Mechanics 333 Using probability theory, the corresponding mean value A and the dispersion (A.4)2 are given through /oo Ad(V | Ex1>\ tl> e D(A), -oo /oo (\-A)*d{1>\Exil>), iJ€D(A2). -OO This implies A=(il>\ A*l>) and (A^)2 = (</> | (A-AI)2il>) = \\(A-AI)il>\\2, which coincides with the definition given earlier. This general spectral theory can be found in Riesz and Nagy (1955). Applications to quantum theory are contained in Reed and Simon (1972), Triebel (1972), and Prugovecki (1981). Remark 5 (Spectral family and Stieltjes integral). More precisely, the general functional calculus is to be understood as follows. For each self- adjoint operator A: D(A) CI-^I, there exists a unique spectral family {E\} having the following properties: (i) For each X £R, the operator Ey.X —> X is linear, continuous, and self-adjoint with E\ = Ex, i.e., Ex is an orthogonal projection. (ii) For each u £ X, the function A i-> (u | E\u) is nondecreasing on R. (iii) For each u £ X, lim E\u = 0 and lim E\u = u. A—> —oo A—>+oo (iv) For each u £ X and each /jl £R, lim E\u = Euu. A->/z-0 (v) The operator A allows a representation of the form (134). Since El = Ex and E$ = Ex, (u | Exu) = \\Exu\\2 for all u £ X, \£ R. Let u,v £ X and Ael. Then, for all u,v £ X, 4(v | Exu) = \\Ex(v + u)\\2 - \\Ex(v - u)\\2 + a\\Ex(v + au)\\2 - a\\Ex(v - au)\\2, (135)

334 5. Self-Adjoint Operators, the Priedrichs Extension, etc. where a — 0 or a = i if X is a real or complex Hilbert space, respectively. Since the function Ah ||£\w||2 is nonincreasing for each w G X, it follows from (135) that the function A h-» (v | E\u) is of bounded variation. That is, the integrals / ... d(v \ E\u) from Remark 4 are to be understood as Stieltjes integrals (see the appendix). Standard Example 6. Let A: D(A) C X —> X be a self-adjoint operator on the separable Hilbert space X, which possesses a complete orthonor- mal system {u\, u2,...} of eigenvectors with the corresponding eigenvalues {Ai,A2,...}, i.e., Aun = Xnun for all n. Then, where E\u = ^2 eA(An)(^n I u)un for all u G X, X e R, , v fO ifA</x if /x < A. Let the arbitrary function F:R —> C be given. Then, for all u G D(F(.A)) and v G X, we get /oo F(A)d(« | Saw) = J2F(Xn)(un | «)(« | «n). Here, u € D(F(A)) iff |F(A)|2d(u | Saw) - £ |F(An)|2|K | u)|2 < oo. / J — ( Thus, the functional calculus from Section 5.8 represents a special case of the general functional calculus from Remark 4. Example 7 (The multiplication operator). Let X := L2 (R). Define (j4iO(sc) := setose) for all x G R, ' where i£ G D(-A) iff ^ G X and /^ |xii(x)|2dx < oo. By Example 10 in Section 5.2, the operator A: D(A) C X -> J is self-adjoint. (i) The spectral family {E\} of A is given through (Exu)(x):={™ (x) if x < A if x > A, for all ii G X and each A G

5.14 Applications to Quantum Mechanics 335 (ii) For all ip G X and —oo < a < b < oo, rb rb f dty>|£^) = [ mx)\2dx. (136) J a J a Proof. Ad (i). For all u, v G X and A G R, /•A (v | £Au) = / v{x) u{x)dx = (JE7av | u). (137) «/ — oo In addition, /A /»oo |u(z)|2cte< / \u(x)\2dx = (u | u). -oo «/—oo Thus, the operator £"a:X —> X is linear, continuous, and self-adjoint. Obviously, El = Ex for all A G R. By (137), the function A i—> (?x | E\u) is nondecreasing on R. Let —oo < A < /x < oo. Then, for each u G X, /oo /./x l(£Au)(z)-(£Mu)(z)l2dz = / |ix(x)| dx —> 0 as A —> /x — 0, -oo «/A -oo «/A and hence lim Exu = Euu, \->fJL-0 * Similarly, we get lim En,u — 0 and lim Exu — u = 0 for all u G X. At—>-oo A—>+oo Let —oo<a<6<oo, and let F:R —► C be a measurable function. It follows from (137) and from (5) in the appendix that pb pb / F(X)d(v | Exu) = / F(A)v(A)u(A)dA, (138) J a J a provided u, v G X and the integral /^ F(X)v(X)u(X)dX exists. For all u G D(A) and «gX, /oo /»oo u(A)Au(A)dA = / Xd(v \ Exu). -oo J — oo In this connection, observe that u G D(-A) iff J_oq X2\u(x)\2dx < oo. Thus, the integral J v(X)Xu(X)dX exists, by the Schwarz inequality. Ad (ii). Use (138) with F = l. □

336 5. Self-Adjoint Operators, the Friedrichs Extension, etc. 5,14-4 Quantization of Classical Mechanics and the Schrodinger Equation In classical mechanics, the motion x = x{t) of a particle of mass m in M3 is governed by the following classical Newtonian equation: mx"(t) = K(x(*)). (139) Here, x denotes the vector of position. Let us suppose that the force field K = K(x) possesses a potential U, i.e., K(x) = -grad U(x). Then, the total energy E of the particle is given though E = ^ + U(x), (140) where p(t) := mx'(t) denotes the momentum vector at time t. Recall that |^ is called the kinetic energy and U is called the potential energy of the particle. If x = x(t) is a solution to (139), then E = const along the motion x = x(t) (conservation of energy), provided the potential U is C1 in a neighborhood of the trajectory. In fact, E'{t) = mx'{t)x"{t) + x'{t) grad U(x{t)) = 0. In quantum mechanics, the motion of a particle of mass m is described by the following Schrodinger equation: h2 ihil>t = -—A'<l> + Uil>, (141) 2m which Schrodinger formulated in 1926. We are looking for solutions ip = i/)(x,t) of (141) such that / Jr \i/>(x, t)\2dx = 1 for all times t £ R. (142) R3 The function ip describes the physical state of the particle. More precisely, ip allows the following interpretation: (i) Probability. Let G be a nonempty open set in M3. Then, L \ij)(x,t)^dx = probability of finding the particle in G the set G at time t.

5.14 Applications to Quantum Mechanics 337 (ii) Stationary particle states of fixed energy E. Substituting the ansatz ip(x,t) = 0(x)e-1^" into the Schrodinger equation (141), we get the stationary Schro- dinger equation £0=-A-A0 + t/0, (143) where the eigenvalue E corresponds to the energy of the particle in the state (j). The normalization condition (142) is equivalent to / \(j){x)\2dx = l. Formally, the Schrodinger equation (141) is obtained from the classic energy relation (140) by using the following simple substitutions: E => ih— and p => —ih grad. ot That is, when passing from classical mechanics to quantum mechanics, classical physical quantities (e.g., energy and momentum) are replaced with differential operators.10 In this connection, observe that in a Cartesian coordinate system we get grad f = e1d1f + e2d2f + e3<93/, where the basis vectors {ei,e2,es} form an orthonormal system. Hence -p2 = ft2(grad)2 = h2(dl + d\ + d\) = h2A. Remark 8 (Interpretation in terms of functional analysis). Let us introduce the Hilbert space X:=L%(R3). Then, the normalization condition (142) reads as follows: (i/)(t) | i/>(t)) = 1 for all t e R, i.e., il)(t) is a unit vector in X for each t. The differential operator H: D(H) CI->I given by D(H) := C$°(R3) and h2 H<p:= ~—A(p + U(x)(p A more detailed motivation can be found in Zeidler (1986), Vol. 4, p. 112.

338 5. Self-Adjoint Operators, the Priedrichs Extension, etc. is called the formal Hamiltonian of the particle. Integration by parts yields / (A^dx = f ^(A^)dx for all <£,^ € £>(«), and hence (0 \Hil>)= [ 4>H^dx = f jHfijifjdx = (Ht | V) for all 0,</> € Z?(W), provided the real function U = U(x) is sufficiently regular. Thus, the formal Hamiltonian H is a linear symmetric operator on the Hilbert space X. One of the main tasks of a rigorous mathematical approach to quantum mechanics consists in extending the formal Hamiltonian H to a self-adjoint operator H: D(H) CI->I, which is called the Hamiltonian of the particle. Then, the spectrum of H corresponds to the possible energy values of the particle. An important special case will be studied in the next section. In Problem 5.10 we will show that for the electron of the hydrogen atom the Hamiltonian H can be obtained as the Friedrichs extension of H. 5.14-5 Applications to the Harmonic Oscillator in Quantum Mechanics We want to explain how the abstract setting of quantum physics from Section 5.14.1 can be realized in the special case of a harmonic oscillator. In classical mechanics, a harmonic oscillator corresponds to a point of mass m > 0, where the motion x = x(t) in K. is governed by the ordinary differential equation mx"(t) = -mw2x(t) (144) for fixed u> > 0 (cf. Example 7 in Section 5.9). The total energy is given through 9 9 9 pz muj^x* where p(t) := mx'(t) denotes the momentum of the particle at time t. That is, . . p . . . momentum velocity of the particle = . mass In quantum mechanics, the motion of the harmonic oscillator is described by the Schrodinger equation h2 , muj2x2 , . A. ihtpt = -— Vz* + —o— ^' (145) 2ra 2 This is formally obtained from (144) by means of the substitutions p=>-ifi— and E^ih — . (146) CfX (jT

5.14 Applications to Quantum Mechanics 339 We are looking for solutions ip = ^{x,i) of (145) with /oo \ip(x,t)\2dx = 1 for all times t e R. -oo Using the ansatz i/>(x,t) = (j)(x)e-"LTr, from (145) we obtain the stationary Schrodinger equation E0 = - — 0" 4- —— x20 on R. (147) 2m 2 Let us introduce the Hilbert space X:=L%(R) with the inner product /oo _^ (/>(x)i/)(x)dx. -oo Each unit vector 0 G X is called a state of the particle (harmonic oscillator), i.e., /oo \4>{x)\2dx = 1. -oo Let —oo<a<6<oo. By definition, / \(j)(x)\2dx := probability of finding the particle (148) ^a in the interval [a, 6]. This will be motivated in Remark 16. Definition 9. The formal Hamiltonian H: D(H) C I -> I of the harmonic oscillator is given through where D(H) := 5. The space <S has been introduced in Section 3.7. Recall from Section 3.4 that the Hermitean functions un are defined through un(x) := an(-l)ne^ -^-, n = 0,1,2,... , (149) where __ 1 2 2 7T4(n!)2

340 5. Self-Adjoint Operators, the Priedrichs Extension, etc. Proposition 10. (i) The operator H is symmetric. (ii) For alln = 0,1,..., UK = En(t>n, (150) where <j>n{x) := un (j^j x~* wuh x0 := (~j)% and En = hw(n+~Y n = 0,1,2,... . (150*) (iii) T/ie eigenfunctions {(j)n} form a complete orthonormal system in X. By (iii), all the eigenvalues of H are given through (150*). In terms of physics, the numbers £"o, Ei,... are the only possible energy levels of the harmonic oscillator in quantum mechanics. This tells us that The energy of the simplest oscillating system is quantized. Planck made this fundamental discovery in 1900. He formulated such a quantum hypothesis in order to get the right radiation law. This marked the beginning of quantum physics. Relation (150*) is closely related to the following fundamental physical fact. In modern physics, one assumes that light consists of particles called photons. The energy AE of such a photon is given through 2ttc AE = En+\ - En = Hu), where u = ——. (151) A Here, c = velocity of light in vacuum and A = wavelength of the specific light. A derivation of Planck's radiation law from (151) and its applications to the expansion of our universe and the vaporization of black holes can be found in Zeidler (1986), Vol. 4. Proof of Proposition 10. Ad (i). For all 0, ip € <S, integration by parts yields /oo \N /»oo 4>"^dx= lim 4>'{x)il){x)\ - / fii/i'dx -oo iV->+oo \_N J_OQ 4>'ip'dx = / H"dx. -oo J — oo Hence (0 | Hi\)) = {H<\> | V) for all 0, </> € D(H). Ad (ii). By (149), un(x) = e xz x polynomial (x). Hence we obtain (j)n G S. A fairly simple computation yields (150).

5.14 Applications to Quantum Mechanics 341 Ad (iii). This has been proved in Section 3.4. □ Definition 11. The operator H: D(H) C X -» X defined through oo H4>:=Y,En{(t>n\(t>)(t>n 71=0 is called the Hamiltonian of the harmonic oscillator. Here, <\> G D(H) iff oo J2\En(<t>n |0)0n)|2<OO. 71=0 Proposition 12. (i) The Hamiltonian H: D(H) C X -* X is self-adjoint (ii) The operator H is an extension of the formal Hamiltonian H. Proof. Ad (i). This follows from Proposition 2 in Section 5.8. Ad (ii). Let (j) G D(H), i.e., (j) G S. By the definition of the space S in Section 3.7, H(j) G S. Since {(j)n} forms a complete orthonormal system in x, oo oo H<P = Y^(<Pn | Hcj>)<t>n = J2 En^n I M»' (152) 71=0 71=0 by the symmetry of H along with (150). Hence the series (152) is convergent, i.e., 0 e D(H). □ Since H C H, we get #0n = En(j)n, n = 0,1,2,... . According to Example 1, we say that the particle has the sharp energy En in the state <j>n. Suppose that the particle is in the state (j). By Example 2, |(07i I 0)|2 — probability of having the sharp energy En. Remark 13 (Dynamics of the harmonic oscillator). We are given ^o € X with (^0 | ^o) = 1. Suppose that the harmonic oscillator is in the state ?/>o at time t = 0. According to Section 5.14.1, the state ifj(t) of the harmonic oscillator at time t is given through il>(t) = e'^i/jo for all t e R.

342 5. Self-Adjoint Operators, the Priedrichs Extension, etc. Explicitly, oo i/)(t) = ^2 e_li^L (^n | ^o)^ for all t eR. n-0 This series converges in the Hilbert space X = L\ (R). In addition, if ^0 £ <S, then it follows from Theorem 5.G in Section 5.13 that ifhl>'(t) = Hi/)(t) for all t € R, ( . The abstract Schrodinger equation generalizes the classical Schrodinger equation (145). Next we want to study both the momentum operator A and the position operator B. Recall that X := l£(R). Definition 14. The operator .A: D(A) C X -+ X with (i40)(x) := -ih -r(t>(x) for all x e R ax is called the momentum operator. Here, Z?(.A) := {0 e X: (j)f e X}, where the derivative is to be understood in the generalized sense. The operator B: D(B) C X -+X with {B(j))(x) := z0(z) for all x G R is called the position operator. Here, D(B) := {(j) e X: B(j> e X}. According to Standard Examples 8 and 10 in Section 5.2, the operators A and B are self-adjoint. The definition of the momentum operator A and the position operator B can be motivated by (146) and (154), (155), respectively. A more detailed motivation can be found in Zeidler (1986), Vol. 4, p. 112ff. Remark 15 (Heisenberg's uncertainty principle). We are given the state (j) G 5 with (0|0) = 1. By Section 5.14.1, the mean position X and the corresponding dispersion (AX)2 of the particle in the state (j) are given through /oo x\<t>(x)\2dx (154) -oo and /oo {x-X)2\4>{x)\2dx. (155) -OO

5.15 Generalized Eigenfunctions 343 Furthermore, the mean momentum V and the corresponding dispersion (AV)2 in the state (j) are equal to V = {</>\ A<f)) and {AV)2 = \\A<f) - V(j)\\2. Obviously, AB<\> - BA<\> = ih{xct)f - {x<t>)') = -ihcj). Thus, it follows from Proposition 3 that APA* > ~. (156) Heisenberg formulated this famous uncertainty principle in 1927. Relation (156) tells us that it is impossible to measure exactly position and momentum (i.e., the velocity) of the particle at the same time. More precisely, we get the following: (i) If we localize sharply the particle (i.e., AX is small), then the velocity of the particle is highly uncertain (i.e., AV is large). (ii) Conversely, if we determine sharply the velocity of the particle (i.e., AV is small), then the position of the particle is highly uncertain (i.e., AX is small). Remark 16 (Justification of (148)). Let {E\} be the spectral family of the position operator B. We are given a state (j) of the particle, i.e., (j) G X and (cj) | (j>) = 1. By Remark 4, / b d((j) | E\4>) = probability of measuring the position x of the particle in the interval [a, 6], and Example 7 tells us that f d(<l> | Ex4>) = [ \<l>(x)\2dx. J a J a This yields (148). 5.15 Generalized Eigenfunctions In order to explain the basic idea in terms of physics, let us begin with the following simple relation: -ih — e^= pe1^ for all x G R (157)

344 5. Self-Adjoint Operators, the Priedrichs Extension, etc. I Ye |2 y - e y y + e (a) particle stream of (b) particle localized velocity v in a neighborhood of the point y FIGURE 5.10. and each fixed pel. That is, the function <j>p(x) := e1^ is an eigenfunc- tion of the differential operator —ih-^ that corresponds to the momentum operator A from Section 5.14.4. The point is that The function <\>p does not live in the Hilbert space X = L% (R). In fact, / J — c \4>p(x)\2dx = oo. Therefore, (j)p does not correspond to the state of a single particle. However, physicists use the following interpretation. The function <pp corresponds to a particle stream in R from left to right with the velocity P v= —, m where m > 0 denotes the particle mass (Figure 5.10(a)). In addition, the density p of the particle stream is given through p{x) := \(j)p{x)\2 for all x G R. That is, / J a b p(x)dx — number of particles in the interval [a, b]. Definition 1. Set X := l£(R). Let A:D(A) C X -+ X be a symmetric operator such that S C Z?(.A). Then the tempered distribution T E Sf with T 7^ 0 is called a generalized eigenfunction of the operator A with the eigenvalue A G R iff T(^0) = AT(0) for all 0 G S. The system {Ta}aG>/4 of generalized eigenfunctions of A is called complete iff Ta(0) = 0 for all a e A and fixed <\> e S

5.15 Generalized Eigenfunctions 345 implies 0 = 0. Lemma 2. We set /oo i/>(x) </>(x)dx for all 0 G 5, (158) -oo i.e., T(0) = ('0|0) for all 0 G <S. T/ien, £/ie following are met: (i) For each if; e X, T e S'. (ii) T/ie corresponding map if; \-> T is linear and bijective from the space X onto Sf. Proof. Ad (i). Let 0n —► 0 as n —> oo. By the Schwarz inequality, /oo /»oo \i>(x)\2dx \<t>n{x)-<t>{x)\ OO J —OO 2(l + x2)2dx {1 + x2)2 n2 < const sup(l + x2)|0n(x) — 0(x) LsceR 0 as n —* oo. Ad (ii). Let ^Gl,j = l,2. Since <S is dense in X, it follows from 7i(0) - T2(0) = (^i - ^2 | 0) = 0 for all 0 G 5 that ij)\ = '02- □ Proposition 3. Each eigenfunction if) G D(A) of the operator A from Definition 1 is also a generalized eigenfunction in the sense of (158). Proof. Let Ai\) = A-0 with -0 ^ 0 and A G M. For each 0 G <S, it follows from the symmetry of A that T(^0) = {$ | .A0) = (Aip | 0) = A(</> | 0) = AT(0). D In the following let us consider an interpretation of equation (157) in terms of generalized eigenfunctions. Standard Example 4 (Momentum operator). Let p G R. We set /oo </>p(x) </>(x)dx for all 0 G <S, (159) -oo where <j>p(x) := e^. By Standard Example 4 in Section 3.8, Tp G Sf.

346 5. Self-Adjoint Operators, the Priedrichs Extension, etc. Let A: D(A) C X —> X be the momentum operator from Definition 14 in Section 5.14. Then, {<t>p}Pem forms a complete system of generalized eigenfunctions for A, in the sense of (159). Proof. Recall that Aty ~ —ih^'. For each <\> G <S, integration by parts yields /oo /»oo ih(j)fp(x) (j)(x)dx = / p(j)p(x) (j)(x)dx -oo «/ —oo In order to prove the completeness of {Tp}, suppose that Tp(0) = 0 for all p G R and fixed 0 G <S. This means /•oo e~ ^ 0(z)cfo = 0 for all peR. I J — c Using the Fourier transformation F:S —> <S from Section 3.7, this implies F0 = 0, and hence (j) = 0. □ Standard Example 5 (The position operator). Let B:D(B) C X -* X be the position operator from Definition 14 in Section 5.14. Then, {Sy}yeR forms a complete system of generalized eigenfunctions for B. More precisely, 6y(B(/)) = y«„(<£) for all </> G 5, (160) Proof. Recall that (Bc/>)(x) = xcj>{x) for all x G R. Hence 6y(B(t>) = (Bct>)(y) = #(</) = y«y(0) for all 0 G 5. In order to prove the completeness of {6y}, let Sy((j)) = 0 for all y G R. Then, 0(?/) = 0 for all y G R, and hence 0 = 0. Because of (160), physicists regard the "Dirac delta function 6y" as a "state" of the particle in which the particle is localized at the point y G R. Such a "state" is approximated by a state i/;e G X, where ^ G Co°(R) and ^e(x) = 0 outside [y — e,y + e] for small e > 0, (161) along with (^ | ^e) = J^ \i/>e(x)\2dx = 1. By (161), the probability of finding the particle outside the interval [y — e, y + e] is equal to zero (Figure 5.10(b)). For the mean position Xe of the particle in the state i\)e, we get /oo x\^£(x)\2dx -* y as e —> +0. -oo

5.16 Trace Class Operators 347 5.16 Trace Class Operators Definition 1. Let X be a separable Hilbert space X over K. (i) The linear operator A: X —> X is said to be of the trace class iff the series tr A := Y^(yn I Avn) n converges for each complete orthonormal system {vn} of X and the value of the series is independent of {vn}. The number tr A is called . the trace of A. (ii) The linear continuous operator A: X —> X is called a Hilbert-Schmidt operator iff .AM is of trace class. Standard Example 2. Let A:X -+ X be & linear continuous symmetric operator on the separable Hilbert space X over K. Suppose that A possesses a complete orthonormal system {un} of eigenvectors with the corresponding eigenvalues {An}, i.e., Aun = \nun for all n. Then, the following hold true: (i) If An > 0 for all n and J2n ^n < °°> then the operator A is of trace class and tr A = ]TAn. n (ii) If J2n ^n < °°> tnen A is a Hilbert-Schmidt operator. Proof. Ad (i). Since the orthonormal system {un} is complete, the series u = ^2(un\ u)un n converges for each u e X, and hence 2_] IC^n I ii)|2 < oo for all u € X. n Let a > 0. Since the sequence (An) is bounded,11 it follows that J2 lAnK | u)\2 < oo for all ueX. nBy Proposition 2 in Section 1.25, |An| < ||i4|| for all n.

348 5. Self-Adjoint Operators, the Priedrichs Extension, etc. By the functional calculus from Section 5.8, we get D(Aa) = X and Aau = V^ ^n(un | u)un for all u e X. n In particular, the operator A* :X -» X is self-adjoint and A^A* = A. Let {vn} be an arbitrary complete orthonormal system in X. Then, Y An = Y \\A^\\2 = YY \(v™ i A^u")\2 n n n m n 77i Since this is a convergent double series with nonnegative terms, the summation can be interchanged. Hence Y Xn = Y Y \(A*y™ I un)\2 = Y M*vm||2 = ^2(Ahm | Ahm) n m n m m = Y(v™ iAvm)- m Ad (ii). Obviously, A2un = A(Aun) — \^un for all n. Since {un} is complete, all the eigenvalues of A2 are given through {A^}. Because of A*A = A2, the assertion (ii) is a special case of (i). □ 5.17 Applications to Quantum Statistics The true logic in this world lies in probability theory. James Clerk Maxwell (1831-1879) Don't trust any statistics that you didn't falsify yourself. Folklore 5.17.1 The Abstract Setting of Quantum Statistics Let T be a quantum system (e.g., a gas). Assume that the "physical states" of T correspond to states of the separable Hilbert space X. We want to describe the physical behavior of T in terms of statistics. (i) Statistical states. By a statistical state \P of the system T we understand the tuple * = Wi,pi;^2,P2;...), (162) where {^m} forms a complete orthonormal system in X, and pi, p2? • • • are real numbers with 0 < pm < 1 for all m and YPm = L 771

5.17 Applications to Quantum Statistics 349 Intuitively, we say that pm is the probability of finding the system T in the state ipm. In terms of statistics, roughly speaking, this means the following. Let us consider C copies of the system T, where the number C is very large. Then, pmC copies of the system T are in the state V™ • The statistical state # is called a pure state iff pmo = 1 for some fixed mo and pm = 0 for all m ^ mo. Otherwise, \£ is called a mixed state. (ii) Measurements. Let A be an observable of the system T, i.e., the operator A:D(A) C X -» X is self-adjoint. Suppose that we measure the "physical quantity" corresponding to A in the statistical state \£ from (162). Naturally enough, we assume that the outcome of this measurement is statistical. More precisely, we assume that the mean value A and the dispersion (A^l)2 of this measurement are given through12 771 and (A„4)2 := J>m||M» - Mm\?- 771 (iii) Entropy. By definition, the entropy S of the statistical state \£ from (162) is given through S = -k^pnlnpn. n Here, k is the so-called Boltzmann constant, where k = 1.380 Joule/Kelvin. (iv) Dynamics. Let H be the Hamiltonian of T, i.e., H: D(H) C X —> X is a self-adjoint operator which corresponds to the energy of the system T. Suppose that the system T is in the given statistical state #o = (^io,pi;^2o,P2;...) at time t = 0. Then, the system T is in the statistical state *(t) = (^i(t),pi;^2W,P2;...) at time tGl, where pm = const and ^m(t) = e~ ~^~ ifjmo for all times tGl and all m. That is, the time evolution of the state ^mo is identical to the time evolution of quantum states in Section 5.14.1. 12We assume tacitly that ^m £ D(A) for all m. Furthermore, note that (AA)2 is the mean value of (^4 — AI)2 provided ipm € D(A2) for all m. In fact, \\All>m ~ All>m\\2 = (Alpm - A^m \ A^m ~ Alpm) = (lpm | (A - Alfl/jm).

350 5. Self-Adjoint Operators, the Priedrichs Extension, etc. Let tGl. We have to show that &(t) represents a statistical state. In fact, since the operator e~1^~ is unitary and the given orthonormal system {V'rao} is complete in X, the set {^m{t)} also forms a complete orthonormal system in X. 5.17.2 Discussion of the Abstract Setting Definition 1. The operator p:X —> X is called a statistical operator iff the following are met: (a) p is linear, continuous, and self-adjoint. (b) p possesses a complete orthonormal system {^m} of eigenvectors with the corresponding eigenvalues {pm}, i.e., P^m = Pm^m for all m. (C) YsmPrn = 1 and 0 < pm < 1 for all m. Proposition 2. There exists a one-to-one correspondence between the statistical states *& of the system T from (162) and the statistical operators p, which is given through 9^ = ^Pmtym I ^rn for all ip e X. (163) 771 Proof. Let ^ be a statistical state. Since IHII2 = £ \PmWm | VOI2 < E IW™ I ^l2 = H^ll2 fOT a11 ^ € X' 771 771 the operator p defined through (163) is a statistical operator, by Proposition 2 in Section 5.8. Conversely, each statistical operator is of the form (163) and it determines uniquely a statistical state of the form (162). □ If p is a statistical operator, then p is of trace class and tr p = Y^Pm = 1- 771 In addition, by Section 5.8, (plnp)^ = ^Vm lnpm(^m | V#m for all ipeX. 771

5.17 Applications to Quantum Statistics 351 Hence the entropy of the statistical state \£ corresponding to p is given through S = —k tr(plnp). Let the operator A be given as in (ii) above. If pA is of trace class, then the mean value A is equal to A = tr(pA). In fact, tr(pA) = Y^i^rn | pAl/jm) = ^{p^m \ A<l/jm) = ^pm(^m | Al/)m). m m m Proposition 3. // the statistical state ty0 corresponds to the statistical operator po, then the time evolution t h> ^(i) corresponds to t \-+ p(t) with p{t) = U^poUit)-1 for all t e R, (164) where U(t) := e-1^"'. Equation (164) represents the basic equation of quantum statistics. A formal differentiation of (164) yields ihp'{t) - Hp(t) - p(t)H for all t e R. Proof. It follows from (163) that for each ^ G X m Since tpm{t) = U(t)^mQ and hence (tpm(t) \ ij)) = (^m0 | U(t)*i/>), we get p{t)ip = U(t)poU(t)*il> for all ip e X. Noting that U(t) is unitary and hence U(t)* = U^)"1, we obtain (164). □ 5.17.3 The Standard Model in Statistical Physics The huge field of statistical physics can be understood best by studying the following standard situation. Let X be an M-dimensional complex Hilbert space with the orthonormal basis {^i,...,^m}- We define the linear operators if, TV, p\ X -» X through H^m = Emil)m, N^m = Nmil)m, and p^m = Pm^m,

352 5. Self-Adjoint Operators, the Friedrichs Extension, etc. for m = 1,..., M, where Pm'^-^M ~T—ZT77ZZ- (165) ^flNm-E^/kT y-M efrNm-Em)/kT~ z_-/77i=l ^ Here, the positive numbers £"m and the nonnegative integers Nm are given. A motivation for the choice of pm will be given in Remark 8. Remark 4 (Physical interpretation). Let T be a physical system (e.g., a gas). Then, ipm corresponds to a state of T, where Em = energy of T in the state ipm, and Nm = particle number of T in the state ij)m. In addition, Pm = probability of finding T in the state ipm. Furthermore, the positive real parameter T and the real parameter /x possess the following physical meanings: T = absolute temperature of T; \i = chemical potential of T. By Section 5.17.1, M £ — mean value of energy of F = V^ Pm(n, T)Em, 771=1 M M = mean value of the particle number of T = Y_] Pm(/^ T)Nm. 771=1 This relates T and /x to £ and A/". In this connection, note that M M £ = tr(pH) = ]T (l/jm | pHl/jm) = 5Z Prn#m 771=1 771=1 and M A/" = tr(pN) = ]T PmNm> 771=1 Since \\Hi/>m - £tym|| = \Em - £\ and \\N^m - AA/>m|| = \Nm - A/"m|, we obtain that M (A£)2 = dispersion of the energy of r= yjpm(/x,T)|£'m — £|2; 771=1 M (AAf)2 = dispersion of the particle number of r= /] Pm(^T)\Nm — A/]2 771=1

5.17 Applications to Quantum Statistics 353 Finally, M S = entropy of F = -k ^ pm(/x, T) lnpm(/x, T). 771=1 iti( Dn 5. The function ZfaT): M = L e kT ra=l is called the partition function of T, and n(^,r):=-jfcrinz(^,r) is called the statistical potential of T. Obviously, Z(/x,T) = tre"ncT-. Proposition 6. j4ZZ important thermodynamical quantities of the system T can be computed from the function £1. We have dn Kr an n Kr ~ 8 fn Proof. Use simple computations. For example, M UT = -k]nZ-l^ = -klnZ+~f2^Nm-Em)eB^ m=l M = kY^Pm lnpm = -5. □ m=l Definition 7. Suppose that for each m the energy Em and the particle number Nm depend on the volume V of the system T. Then, the pressure V of T is defined through13 Remark 8 (Motivation of the fundamental formula (165) for the probability Pm)> Let us use the principle of maximal entropy for fixed mean energy 13 A motivation of this definition in terms of phenomenological thermodynamics can be found in Zeidler (1986), Vol. 4, pp. 387 and 400.

354 5. Self-Adjoint Operators, the Priedrichs Extension, etc. and fixed mean particle number. That is, let us consider the following maximum problem: M entropy S(p) := —k 2^. Pm lnpm = niax!, (166) 771=1 along with the side conditions M £(p) := ^2 p™e™ = const' 771=1 M N(p) :=z Yl P™N™ = const, (167a) 771= 1 M 771=1 and 0<pm<l, m = l,...,M. (167b) We are given the real numbers Em, Nm, m = 1,..., M, such that (Ei E2 - - Em \ JVi iV2 • • • NM = 3. (168) 1 1 ... 1 / We are also given the mean energy £ and the mean particle number J\f of the system Y. Suppose that p = (pi,... ,pm) is a solution of (166), (167) with 0 < pm < 1 for all m. By the Lagrange multiplier rule,14 there exist real parameters a, /?, and 7 such that ^=0, m = l,...,M, (169) OPm where £ := S(p) + a£(p) + (3Af(p) + jW(p). It follows from (169) that -k\npm - k + a^m + /?Nm +7 = 0- 14A rigorous justification of the general Lagrange multiplier rule can be found in Section 4.14 of AMS Vol. 109. Observe that condition (168) implies that p is a regular point of the side conditions (167a), i.e., T&nk(S'(p),Af'(p),W(p)) = maximal = 3. Cf. also Zeidler (1986), Vol. 3, p. 293.

5.17 Applications to Quantum Statistics 355 Hence aEm+(3Nm pm = const e k Using J2mPm = 1> we get (165) with T = - and /x = -/?T. a Therefore, we obtain the surprising fact that from a purely mathematical point of view temperature T and chemical potential \i are nothing more than Lagrange multipliers. 5.17.4 Bose-Einstein Statistics and Fermi-Dirac Statistics Suppose that the system T (e.g., a gas) consists of particles that may assume one of the energy values £1,... ,£m- By definition, a state of T is characterized through £i,£2,...,£j, ni,n2,...,nj. (170) This means that n^ particles of T have the energy £jr, 7 = 1,..., J. For each such state ^, the particle number N and the energy E are given through N = V^ rij and E = V^ n^-. j=i i=i Thus, the partition function Z and the statistical potential fi of T are equal to r j=i n3 and J n n(li,T) = -kT\nZ(fi,T) = Yl-kTlnJ2(e^) ° ' (171) Standard Example 9 (Bose-Einstein statistics). Suppose that Eac/i occupation number rij may assume the values 0,1,..., n. Using the geometric series, it follows from (171) that I _ e(n+l)(»-e3)/kT -/e'i'ln j=l n3=0 " " j=l ' » r HZU.^ A i_e(n+l)(p-e3)/k

356 5. Self-Adjoint Operators, the Priedrichs Extension, etc. To simplify computations, suppose that the maximal occupation number n is very large and /x — ej < 0 for all j. Letting n -» oo, we get By Proposition 6, the mean particle number M and the mean energy £ of the system T are given through Af — —ft^ and £ = /xA/" — T2 (p)T- Hence j j ^=E^- f = E^*i' (172) where M := t wt^r. (173) By (172), A/j is the mean occupation number of the energy level Sj. At —£j In the special case where e *t is very small (e.g., the energies £i,..., ej are very large for fixed \i and T), we approximately obtain tf. = €ri*L% (173*) This corresponds to the classic Maxwell-Boltzmann statistics. Standard Example 10 (Fermi-Dirac statistics). In contrast to Standard Example 9, we now assume that Each occupation number rij may only assume the values 0, 1. This corresponds to the Pauli principle. By (171), we get j i=i Hence the mean particle number M and the mean energy £ of the system T are given through J\f = —fiM and £ = fjj\f - T2 (p)T, i.e., j j i=i i=i where M = ' l + e(M-^)/feT* If e kT is very small, then again the mean particle number A/} corresponds approximately to the Maxwell-Boltzmann statistics from (173*).

5.18. C*-Algebras and the Algebraic Approach to Quantum Statistics 357 In physics, the Bose-Einstein statistics can be applied to particles with integer spin (e.g., photons), whereas the Fermi-Dirac statistics can be applied to particles with half-numberly spin (e.g., electrons, protons, neutrons, etc.). Interesting applications to cosmology (the Planck radiation law, the Big Bang and the expansion of our universe, white dwarfs, etc.) can be found in Zeidler (1986), Vol. 4. 5.18 C*-Algebras and the Algebraic Approach to Quantum Statistics Banach algebras have been defined in Section 1.23. Recall that we always assume that such a Banach algebra contains a unit element E, which we also denote by /. In the following, let us introduce special Banach algebras, where we have the following implications: von Neumann algebra => C*-algebra ■=> Banach algebra. Definition 1. By a C*-algebra 21, we understand a Banach algebra over C such that there exists a map A i—> A* from 21 to 21 having the following properties for all A, B G 21, and all a, /? G C: (i) (A*)*=A; (ii) (a A + /3B)* = a A* + 0B*; (iii) (AB)* = B*A*; (iv) ||AM|| = P||2. In addition, 21 is called commutative iff AB = BA for all A, B G 21. An element A of 21 is called self-adjoint (resp., unitary or normal) iff A = A* (resp., AA* = A"A = I or AA* = AM). Standard Example 2. Let X be a complex Hilbert space. Then, the Banach space L(X, X) of all linear continuous operators A: X —> X is a C*-algebra if A* denotes the adjoint operator. Proof. Properties (i)-(iii) follow easily from the definition of the adjoint operator. Let us prove (iv). Assume that X ^ {0}. Since (AM)* = AM** = A* A,

358 5. Self-Adjoint Operators, the Friedrichs Extension, etc. the operator A* A is self-adjoint. By Proposition 3 in Section 4.1, PM||= sup \(A*Au\u)\= sup \(Au \ Au)\ IMNi IMI=i = sup ii^h2 = \\a\\2. n IMI=i Standard Example 3. Consider the Banach space C[a, b]c of all continuous functions /: [a, b] —> C equipped with the norm ||/|| := maxa<x<t |/(#)|. Letting f*(x) ~f(x) for all x G [a, 6], C[a, 6]c becomes a commutative C*-algebra. Definition 4. Let 21 and 05 be two C*-algebras. By a *-homomorphism, we understand a linear map (j>: 21 —> 05 such that, for all A, S G 21, ^(.AB) = 0(^)0(5) and 4>{A*) = <j>(A)*. If, in addition, <\> is bijective, then </> is called a *-isomorphism between 21 and <B. Each *-isomorphism (j>: 21 —> 21 is called a *-automorphism of 21. The following notion is crucial for quantum statistics. Definition 5. By a state a; of a C*-algebra 21, we understand a linear continuous functional a;: 21 —> C such that u;(J) = 1 and u(A*A) > 0 for all A G 21. A state u; is called mixed iff there exist two different states u)\ and o;2 of 21 such that a; = Xuji + (1 — X)uj2 for some A G ]0,1[. Otherwise, the state w is called pure. Standard Example 6. Let 21 := L(X,X), where X is a complex Hilbert space, and X ^ {0}. For fixed u e X with ||^|| = 1, we set u(A) := (u | Au) for all A G L(X, X). Then, a; is a state of 21. Proof. Obviously, uj(I) = 1 and w(i4M) = (i4u | Au) > 0 for all A G L(X, X). D

5.18. C*-Algebras and the Algebraic Approach to Quantum Statistics 359 Let us now define von Neumann algebras based on the algebraic relation 05 = 05". (174) Definition 7. Let X be a complex Hilbert space. A subset 05 of L(X, X) is called a C*-subalgebra of L(X, X) iff 05 is a linear subspace of L(X, X) and A,B eB implies AB e*B and A* e 05. The set 05' := {A e L(X,X): AB = BA for all B G »} is called the commutant of 05. We also set 05" := (05')'- By a von Neumann algebra, we understand a C*-subalgebra of L(X, X) such that (174) holds. Operator algebras represent an impoi ant tool of modern mathematical physics. John von Neumann introduced von Neumann algebras in the 1930s. The theory of C*-algebras was developed by Gelfand and Naimark in the 1940s. In particular, the famous Gelfand-Naimark theorem says that each C*-algebra is *-isomorphic to a C*-subalgebra of L(X, X) for some complex Hilbert space X (cf. Problem 1.20 of AMS Vol. 109). 5.18.1 The Algebraic Setting for Quantum Statistics A physical system V (e.g., a gas) is described by a C*-algebra 21. (i) The self-adjoint elements A of 21 are called observables. They correspond to physical quantities like energy, particle number, and so forth. (ii) The states lj of 21 correspond to physical states of I\ (iii) We define uj(A) :— expectation value A of the observable A in the state uu; lj((A - ou(A)I)2) := dispersion (AA)2 of the observable A in the state uj. (iv) Dynamics. We postulate that there exists a one-parameter family {4>t}teM. of *-automorphisms <pt:21 —> 21 such that, for all t,seR, (j)t+s = <j>t<j>8 and 0o = identity. This allows the following physical interpretation. If the system Y is in the state u at time t = 0, then it is in the state ut := u o cj)t

360 5. Self-Adjoint Operators, the Friedrichs Extension, etc. at time t. We have to show that ut is a state. In fact, for each A G 21, we get <j)t(A*A) = B*B, where B := </>t(A), and hence vt(A*A) = uj(B*B) > 0. Definition 8. Let (3 G R. A state u is called a (3-KMS-state iff / f{t)uo{A(j>t(B))dt = [ f(t + iPh)uo{(j>t{B)A)dt (175) for all A, i? G 21 and for all continuous functions /: R —> R whose Fourier transform belongs to Cq°(R). In particular, condition (175) is satisfied if uj{A(j>t{B)) = u;((j>t.m(B)A) for all t G R, (175*) and these functions are continuous on R with respect to time t. It was discovered around 1960 by the physicists Kubo, Martin, and Schwinger that such states may describe thermodynamic equilibrium states. From the mathematical point of view, we postulate that the KMS-states correspond to thermodynamical equilibrium states of the system T. The corresponding temperature T is given by fc/T where k denotes the Boltzmann constant. A motivation will be given ahead, where we also introduce a more general concept of KMS-states that includes systems with variable particle number (i.e., there is a nonvanishing chemical potential fi). By (175) and (175*), we obtain the surprising fact that The temperature T of thermodynamic equilibrium states is related to imaginary time i(3h, where (3 = -^. 5.18.2 Applications to the Standard Model of Statistical Physics Let us reconsider the situation from Section 5.17.3. We are given a finite- dimensional Hilbert space X with the orthonormal basis {V>i,...,V>m}. Let Pm-^i 177Z ~> ra = l,...,M, e/3(/iJVm-£m) where (3 := ^. The operators H, N, p:X —> X are defined through #Vm = Emi>m, Nl/j^ = Nm^m, and p^m = Pml/Jm

5.18. C*-Algebras and the Algebraic Approach to Quantum Statistics 361 for allra = 1,...,M. Constant particle number. Let us first consider the case where \i = 0, i.e., the particle numbers Nm are constant for each m. (i) Algebra. Set 21 := L(X,X). (ii) States. If we define u(A) := tr(pA) for all A e 21, (176) then a; is a state. In fact, it follows from M UJ(A) = ]P PmiAljJm | VVi) 771=1 that M V(A*A) = ]T Pm(A*Atl>m | Vm) m=l M = ^ Pm(A>ilJm \ A^m) > 0 for all A e 21 771=1 and u;(7) = 1. The state lj from (176) is mixed provided we have pj > 0 and pk > 0 for two different indices j and fc. (iii) Dynamics. Define 4>t{A) := e^ Ae"^ for all A G 21 and all t e R. (177) Then, 0t: 21 —> 21 is a *-automorphism for each time tel. In fact, it follows from (e nr) = e^ h and (AB)* = B*A* that &(AB) = MA)MB) and &(A)* = &(A*) for all A,B G 21, and all * G R. Furthermore, </>t(A) = B implies A = <i>-t{B), by (177). Hence ft1 = <j>-t. To motivate (177), observe that M uj((j>t{A)) = ]P Pm{Am^m(t) \ ^m{t)), where i/>m(t) := e-^i/jrn. 771=1

362 5. Self-Adjoint Operators, the Friedrichs Extension, etc. This coincides with the time evolution from Section 5.17.1. From the physical point of view, the state u from (176) corresponds to a physical state, where the total energy Em is realized with the probability Pm. Intuitively, we expect that such a state corresponds to a thermodynamic equilibrium. The following proposition justifies this in terms of the KMS- condition. Proposition 9. Each state uj from (176) is a (3-KMS-state with respect to <t>t, i-e., uj(A<j>t{B)) = uo{(j>t-i(3h(B)A) for all A, B e 21 and all t e R. (178) The temperature of this state is given by T = A. Proof. For simplifying notation, let us use such units of time that ft = 1. Observing that tr(C) := Ylm^i^m \ Cijjm) for all C G 21, we get tr(OD) = tr(DC) for all C, D e 21 (179) (cf. Problem 5.12). This implies (178). In fact, we have u(C) = tr(pC) = Z'hrie-^C), where Z := tr e~PH. By (179), Zu(<Ih-0(B)A) = tr (e-Wei(t-imBe-i(t-iP)HA^ = ti(eitHBe-itHe-PHA) = tr (e-PHA eitHB e~itH) = Zuo{A(j>t{B)). U Variable particle number. Let us now consider the more general case where j^^O. Here, the state u from (176) corresponds to a physical state where the particle number Nm and the total energy Em are realized with probability pm. We expect again that this represents a thermodynamic equilibrium state. Introducing ^(A) := e~*Ae ^ for all A e 21, where H{fi) := H — fj,N, we obtain the following generalization of Proposition 9. Proposition 10. The state uj from (176) is a /3-KMS-state with respect to uj(A^(B)) = uj Ut-i(3h(B)A) f°r al1 A,B e$l andteR. This equilibrium state corresponds to the temperature T = -K and to the chemical potential \x. Proof. Replace H with H{fi) and use the same argument as in the proof of Proposition 9. □

5.19. The Fock Space in Quantum Field Theory 363 5.19 The Fock Space in Quantum Field Theory and the Pauli Principle In quantum mechanics the number of particles is fixed. Quantum field theory describes the interaction between elementary particles where the number of particles is not fixed. The Fock space allows us to describe such a situation. Experience shows that there exist two completely different kinds of elementary particles in nature, namely, (i) bosons (i.e., particles with integer spin like photons) and (ii) fermions (i.e., particles with half-numberly spin like electrons or quarks). The Pauli principle postulates the following: It is impossible for two identical fermions to be in the same state. Furthermore, we have the following principle of indistinguishability for both bosons and fermions: It is impossible to distinguish between n identical particles. This means the following. For example, consider two electrons and let A and B be two one-electron states. In classical physics, we can distinguish between the following two-electron states: AXA2, AXB2, A2BU BXB2. Here, A\B2 means that electron 1 is in state A and electron 2 is in state B. In quantum physics, only the following states exist: AA, AB, BB, where AA means that the two electrons are both in state A, and so on. In quantum statistics, the number of different states is of fundamental importance. A different counting of states yields completely different physical results. In fact, the two preceding principles were discovered by physicists via quantum statistics. The completely elementary proofs of the following propositions are left to the reader as exercises. 5.19.1 The Fock Space for Bosons We start with the Hilbert space Xn := Z|(M3n), n = 1,2,..., with the inner product (/ I 9)n := / f(x)g(x)dx, 7R3n

364 5. Self-Adjoint Operators, the Priedrichs Extension, etc. _ i and we set X0 := C with (/ | g)0 - fg. As usual, let ||/||n := (/ | /)£. Definition 1. The bosonic Fock space X consists of all the sequences (/0n)n=O,l,.. SUCn tnat oo ]£lhMn<00. n=0 Here, we assume that each function ^n = ^n(^i»• • • > xn) is symmetric with respect to all arguments x±,...,xn We also set Vn := C§°(R3n), n > 1, and £>(X) := {(V>n) eI:iG Vn for all n > 1}. Proposition 2. T/ie bosonic Fock space X is a complex Hilbert space with respect to the inner product oo C0 I 0) := X^n I 0n)n- n=0 Definition 3. We set D(N) := {V> € X: £JJL0n2II^M2 < oo} and define the particle number operator N: D(N) CI-^I through N(il>n) = (nV>n). Example 4. Let ^n £ -^n with ||^/;n|| = 1 for fixed n — 0,1, Set ^:=(0,...,0,^n,0,...), where ij)n stands at the nth place. Then, ||^/;|| = 1 and Ni/j — ni/j. We say that the state tj) corresponds to n identical bosons (e.g., n photons). The state fi:= (1,0,0,...) is called the vacuum (or the ground state). Obviously, NQ — 0. Definition 5 (Creation operators 6+ and annihilation operators b). We are given / G V\ and i\) G T>{X). Let us define the operators b±{f):V{X)QX^V{X) in the following way. For all xi,..., xn Gi3 and all n — 1,2,..., let n (M/MnOzi,... ,xn) := n"i ^/(arjj^n-i^i, • • •,^-1,^+1,... ,xn)

5.19. The Fock Space in Quantum Field Theory 365 and (b-(f)ilj)n(xi,...,xn) := (ra + l)i / f(x)^njri(x,xi,... ,xn)dx. Furthermore, for n — 0, we set (&±(/)V0o = 0. Example 6. If /, g e Pi, then M/)" = (0,/,0,...) and 6+(^)6+(/)fi = (0,0,/i,0,...), where /i(xi,x2) := 2-s(#(xi)/(x2) + f(x1)g(x2)). The following result is crucial. Proposition 7 (The commutation relations). For all /, g G V\ and rf,<p e V(X), we get M/)Mff)V> - b+(g)b+(f)rl> = 0, (180) b-(f)b-(g)rl> - b-(g)b-(f)rl> = 0, (181) M/)MffW ' b+(g)b-V)1> = (/ I ff)i^ (182) Furthermore, (M/)lM 0 = (iH M/M- Example 8. Let f,g € Vx with ||/||i = ||#||i = 1. Since M/)fi = °» ^ follows from (182) that M/)M/)n = M/)M/)« + (/1 /)ifi = n and M/)M/)M/)n = i+(/)ft_(/)ft+(/)n + M/)n = m/)M/)M/)« + 2ft+(/)n = 26+(/)n. Moreover, from (180) we get M/)M$)fl = b+(g)b+(f)n. (183) Physical Interpretation 9. Let /,/i,...,/m E Pi with ||/||i = 1 and ||/j ||i = 1 for all j. Then, we regard /j as the state of one particle. Suppose that V> ^ 0, where ^:=am6+(/i)6+(/2)---6+(/m)fi. Choose am G C in such a way that ||V>|| = 1. Then, we regard i\) as a state of m identical particles (bosons) in the states /i,..., /m. It follows as in

366 5. Self-Adjoint Operators, the Priedrichs Extension, etc. (183) that ij) remains unchanged under a permutation of /i,..., fm. This reflects the principle of indistinguishability of identical particles. Observe that Ni/j = mi/j. We say that the operator &+(/) creates one particle in the state / from the vacuum. Furthermore, by Example 8, M/)M/)fi = a Therefore, we say that &-(/) annihilates one particle in the state /. 5.19.2 The Fock Space for Fermions In constrast to the bosonic Fock space, the functions i/jn are now antisymmetric. As we will show, this forces the Pauli principle. Definition 10. The fermionic Fock space Y consists of all the sequences (/0n)n=O,l,... SUCn tnat oo 5ZlhMn <00. n=0 Here, we assume that each function ij)n = rfn(xi,..., xn) is antisymmetric with respect to all arguments xi,..., xn e M3. Then, Y is a complex Hilbert space with respect to the inner product oo n=0 Let V(Y) := {(V>n) e Y: ijjn e Vn for all n > 1}. The particle number operator N: D(N) C Y —> Y is denned through N(i/jn) = (ni/jn) and D(N) := {^ € F: E^o^ll^ll2 < oo}. Definition 11 (Creation operators a+ and annihilation operators a_). We are given f £V\ and ^ 6 P(y). Let us define the operators a±(f):V(Y)QY^V(Y) in the following way. For all x\,..., xn G M3 and all n = 1,2,..., let n (a+(/)V0n(*i,..., *n) := n-i ^(-l)^1/^) Xlpn—l(^lj • • • )Xj — i, Xj+i) . . . , XnJ

5.19. The Fock Space in Quantum Field Theory 367 and (a-(f)il>)n(xi,...,xn) := (ra + 1)* / f(x)tl>n+i(x,xi,... ,xn)dx. Furthermore, for n = 0, we set (a±(/)V>)o = 0. Example 12. If / e £>i, then a-(/)(0,/,0,0,...)-(/|/)iO. Proposition 13 (The anticommutation relations). For all f,g € T>\ and ip,4>€V(Y), we get a+(f)a+(g)iP + a+(g)a+(f)i> = 0, (184) a_ (/)a_ {g)i> + a_ (<?)a_ (/)V = 0, (185) a-(f)a+(g)il> + a+(ff)a_(/)V> = (/ | g)nl>. (186) Fwrt/ierTnore, (a+(/)^|0) = (^|a_(/)0). Example 14. Let f,g eT>x with ||/||i = ||#||i = 1. Since a_(/)ft = 0, it follows from (186) that o_(/)o+(/)n = -o+(/)a_(/)n + (/1 /)ifi = n and a_(/)a+(/)a+(/)fi = -o+(/)o_(/)o+(/)fi + a+(f)Q = a+(f)a+(f)a-(f)tt + 2a+(f)fl = 2a+(f)fl. From (184) we get a+(/)a+(/)fl = -o+(/)a+(/)n = 0. (187) Physical Interpretation 15. Let /i,..., fm G V\ with ||/j||i = 1 for all j. Then, we regard fj as the state of one particle. Suppose that V ¥" 0> where V> := ama+(/i)a+(/2) • • • a+(/m)fi. Choose am E C in such a way that ||V>|| = 1. Then, we regard ^asa state of m identical particles (fermions) in the states /i,..., /m. It follows as in (187) that i\) passes to i\) (resp., — V>) if we perform an even (resp., odd)

368 5. Self-Adjoint Operators, the Friedrichs Extension, etc. permutation of /i,..., fm. This reflects the principle of indistinguishability of identical particles. If f\ — $2 = * • * = /m? then it follows as in (187) that a+(/)a+(/)-.-a+(/)fi = 0. This is the Pauli principle. Remark 16. Physicists use a heuristic machinery (e.g., the path integral) in order to compute physical effects in particle accelerators with a high accuracy. Unfortunately, there is no rigorous mathematical justification of the arguments of physicists. It is one of the most important challenges of mathematics to construct a rigorous quantum field theory which describes realistic physical situations. Rigorous application of Fock spaces to quantum statistics can be found in Bratteli and Robinson (1979), Vol. 2. Mathematical models of quantum field theory are studied rigorously in Glimm and Jaffe (1981), and in Grosse (1995). 5.20 A Look at Scattering Theory Scattering theory studies the motion of particles that move like free particles as time t goes to ±oo. By a free particle, we mean a particle that is free of forces. In classical mechanics, free motion corresponds to a uniform motion on straight lines. For example, in celestial mechanics the motion of a comet with a hyperbolic trajectory is free as t —> ±oo (cf. Figure 5.11). In particle accelerators, scattering experiments are performed in order to study properties of elementary particles. In the following, let us study scattering processes in quantum physics. We are given a complex Hilbert space X along with the self-adjoint operator H: D{H) CI->I such that H = Hq + H\, and the motion of a particle is given by ${t) := e-^i/j(0) for all t e R. We assume that HjiD(Hj) C X —> X, j = 0,1, are self-adjoint. In terms of physics, we regard the motion <0O(£) := e-^Vo(O) for all* G R as a free motion, whereas Hi corresponds to the action of forces. Definition 1. The motion V> = i/j(t) is called asymptotically free as t —> +oo (resp., t —> —oo) iff there exists a V'o(O) £ X such that lim ||V(i)-Vo(i)ll=0 t—►-{-oo

5.20 A Look at Scattering Theory 369 FIGURE 5.11. (resp., limt-.-oo \\^{t) - ^o(0ll = °)- The motion ijj = i/;(t) is called asymptotically free (or a scattering motion) iff there exists a ^(0) G I such that lim ||^(0 —-0o(OII =0 and lim ||0(O-0o(OII = 0. t—»+oo t—* — oo Let us also define the wave operators W±:D(W±) C X —> X in the following way. We set W±tl>±(0) := lim e^e-^^MO), (188) t—»-±oo where V±(0) G D(W±) iff the limit (188) exists. Proposition 2. Le£ ^i(O) 6 X. Then the motion i/>(t) = e"^0(O) /or fll/tGl (189) is asymptotically free as t —► +oo (resp., t —> —oo) ijf ^(0) G i2(VF+) (reap., ^(0)6 £(1^-)). More precisely, if i/)(Q) = W±i/j±, then lim ||V(*)-Vo(*)|| = 0, (190) t—*±oo w/iere Vw) := e ~K^^±- In particular, if D(W+) = X (resp., D(W_) = X), then for each free motion V>o = V>o W> there exists a motion ^ = -0(0 under the action of the "force £Ti" such that (190) holds as t —> +oo (resp., £ —> —oo). Proof. Since the operator e^~: X —> X is unitary for each tGl,

370 5. Self-Adjoint Operators, the Friedrichs Extension, etc. le-^V(O) - e-^Vo(0)|| - |V(0) - e^e-^^o(0)|l. (191) □ At the same time, relation (191) tells us the following. Corollary 3. Let i/j(0) G X. Then, the motion (189) is asymptotically free iff*/j(o)eR(W+)nR(W-). Example 4 (One-dimensional classical scattering motion). Let us first consider the classical motion x = x(t) of a particle of mass m > 0 on the real x-axis under the action of the force K(x) :=-U'(x), xeR in the direction of the positive x-axis. This motion is governed by the Newtonian equation mx"(t) = K(x(t)) for all teR, , . x(0)=x0, x'{0) = x1. (iy^ We assume that (A) The potential U: R —> R is C1 and vanishes outside some compact interval. Since K is bounded on R, the classical theory of ordinary differential equations tells us that, for given real numbers xo and xi, problem (192) has a solution for all times t. If x = x(t) is a solution of (192), then we get i- (2~1mx\t)2 + U(x(t)) = [mx"(t) - K(x(t))]x'(t) = 0, at and hence 2~1mx,(t)2 + U(x(t)) = const = E for all times t, (193) where E = energy. This means conservation of energy. Case 1: Bound motion. Let E < 0 and let U be as given in Figure 5.12. Then it follows from (193) that the motion is only possible in the region {xeR:E- U{x) > 0}. Case 2: Asymptotically free motion. Let E > 0 and let U be as given in Figure 5.12. By (193), x'(t) ^ 0 for all times t e R. Thus, if xx > 0, then x'(t) > (2f) * > 0 for all times t > 0, and hence x(t) > f — J t + x0 for all* > 0.

5.20 A Look at Scattering Theory 371 particle E >x ►—-*5^- *—> ^~- >x U U (a) bound motion (E = energy) (b) asymptotically free motion FIGURE 5.12. Replacing t with —t, this implies x(t) —> ±oo as t —> ±oo. Recall that the classical momentum p is given by p = mx'(t). If U = 0, then the force K vanishes and the energy E of the free motion x = x\t + xo is equal to * = t- 2m Since each real value p is possible, we obtain the following: The energy values E of classical free motion fill up the interval [0, oo[. Standard Example 5 (One-dimensional scattering motion in quantum mechanics). Parallel to the classical motion (192), let us now study the corresponding motion in quantum mechanics described by the one-dimensional Schrodinger equation ihil>'(t)=Htl>(t), (192*) where H = Hq + Hi with the free Hamiltonian, the momentum operator pi\) := —ihijj', and H\ijj := Uijj. Let X := I^(R) along with D(H) := {i/j e X: ijj', if," e X}. Assume (A). Then, the following hold true: (i) The Hamiltonian H: D(H) C X -> X is self-adjoint. (ii) Let E(H) denote the linear hull of the eigenvectors of H. Then D(W±) = X and R(W±) = E{H)L, where E^)1- denotes the orthogonal complement to E{H).

372 5. Self-Adjoint Operators, the Friedrichs Extension, etc. (iii) The spectrum <r(H) of H satisfies the relation [0,oo[ Qor(H) CM, where the essential spectrum of H is equal to [0, oo[. If A is an eigenvalue of iJ, then A < 0 and A has finite multiplicity. Each eigenvalue of H is isolated and the only possible limit points of the set of eigenvalues of H are A = 0 and A = — oo. (iv) If Jm U(x)dx < 0, then H has at least one eigenvalue. (v) If U = 0, then H = Ho has no eigenvalues and the spectrum cr(Ho) is equal to the essential spectrum [0, oo[. Proof. The sophisticated proofs can be found in Schechter (1981). □ By (iii) and (v), the essential spectrum of the free Hamiltonian Hq is stable under the perturbation Hi corresponding to the potential V. Generally, it is a fundamental property of the essential spectrum that it is stable under reasonable perturbations. In terms of physics, the eigenvalues of H are the energy levels of bound states of the particle. These energy levels do not belong to the essential spectrum. From (ii) we get the following. Let ^(0) G X. Then, it follows from R(W+) = R{WJ) = E(H)1- and Corollary 3 that V>(0) is the initial state of an asymptotically free (scattering) motion iff i/j(0) is orthogonal to all eigenvectors (bound states) of H. In order to motivate the spectral property (r(H0) = [0, oo[, observe that the function <pp(x) := e^ is an eigenfunction of Ho := — (^) ^ with 2 the eigenvalue ^, i.e., P2 Ho(t>P = 7T-(t>p for all p G R, 2m but 4>p does not live in the Hilbert space X = L^ (M). However, for each p G R, (j)p is a generalized eigenfunction of Ho in the sense of Section 5.15. In fact, if we set Tp((j)) := / (j)p(j)dx for all <\> G <S, Jr then Tp G S' for all p G R and integration by parts yields p2 Tp(H0<p) = ^—Tp(4>) for all <£ 6 S and all p € E.

5.21. The Language of Physicists 373 5.21 The Language of Physicists in Quantum Physics and the Justification of the Dirac Calculus If one does not sometimes think the illogical, one will never discover new ideas in science. Max Planck, 1945 "I think this is so," says Cicha, "in the fight for new insights, the breaking brigades are marching in the front row. The vanguard that does not look to left nor to right, but simply forges ahead—those are the physicists. And behind them there are following the various canteen men, all kinds of stretcher bearers, who clear the dead bodies away or, simply put, get things in order. Well, those are the mathematicians." From the criminal novel Dead Loves Poetry of the Czech physicist Jan Klima (born in 1938) In this section, we try to build a bridge between the language of physicists and mathematicians. Let X be a separable Hilbert space over K with the inner product (u | u), where u, v G X. Physicists write \u) instead of u. This forces \au) = a\u) for all a G K and uGl. They also use the symbol (v\ along with the formal multiplication (v\ • \u) = (v I u) := (u | v) for all u,v e X. This forces (av\ = a (v\ for all a G K and u G X. Furthermore, if A is an operator on X, then this formal multiplication yields (v\ • A\u) = (v\ A\u) = (v | Au). Finally, the symbol \u){v\ stands for the operator B: X —> X defined through Bw := (v | w)u for all w G X. In fact, formally we get \u)(v\ • \w) = \u) (v \w). As each perfect calculus, The Dirac calculus works on its own.

374 5. Self-Adjoint Operators, the Friedrichs Extension, etc. 5.21.1 The Discrete Dirac Calculus for Complete Orthonormal Systems Let {uj} and {vj} be two complete orthonormal systems in X. The Completeness and Orthogonality Relation 1. Physicists write \uj)(uj\=I (completeness relation) (194) 3 and (uj | Uk) = 6jk fc>r all j, k (orthogonality relation). (195) Formal Consequences 2. Using the formal rules introduced previously, we conveniently get the following relations from (194): \u) = yz \uj)(uj iu) f°r a^u ^ ^ (196) (v I u) = Y2(v | Uj)(uj | u) for all u,vGl. (197) 3 Along with V • |vj)(vj| = I and (194), we also obtain 2Z(V I uj)(u3 I vk)(vk \u) = (v \u) for all w, v G X (198) 3,k Justification of the Dirac Calculus 3. Since {uj} is complete, U = J2(U3 I U)U3 for a11 U e X- (196*) 3 This is (196). From (196*) we obtain (v\u) = ]P(t; I Uj)(uj | u) for all u,v e X. (197*) This is (197). Finally, since {vk} is complete, we have Uj = ^(Vfc | Uj)Vife, k and from we obtain (v | u) = J^(t; | Uj)K' I Vfc)(vfc I w)» (198*) noting that (aw | v) = a(w | v) for all aGK. This is (198). E

5.21. The Language of Physicists 375 5.21.2 The Continuous Dirac Calculus, the Fourier Transformation, and the Momentum Operator Physicists also use formally the Dirac calculus in the case where a continuum of indices appears. As an example, let us introduce the function <j)k(x) := (27r)~ieikx for all x e R and each index k e R. Physicists set \k) := cj)k for all k e R and Recall also the formal use of the Dirac ^-function, namely, Tf(k)6(k - k') = / f(k)S(k - k')dk = /(*'). k J^ This relation tells us the following: The Dirac function 8(k — k') can be regarded as a continuous version of the Kronecker symbol 8kk' • The Completeness Relation and the Orthogonality Relation 4. Physicists write y^ \k)(k\ — I (completeness relation) (199) k and (k' | k) = 6(k' - k) for all k\ k e R (orthogonality relation). (200) Formal Consequences 5. Prom (199) we get \u) = ]T |jfc>(jfc | u) for all u € X (201) k and (v\u) = Y^(v \k)(k\ u) for all u,v e X. (202) k Furthermore, it follows from ^2k, \k')(k'\ — I that (t;| = ]jT(t; | fc')(fc'| and \u) = ]T \k)(k I u)-

376 5. Self-Adjoint Operators, the Friedrichs Extension, etc. Hence, by the orthogonality relation (200), (v\u)= ]jT(t; I W \k)(k\u) = Y,(v I k')S(k' -k)(k\ u) k This coincides with (202). Thus, the orthogonality relation (200) guarantees that the computation of (v | u) can be based on the multiplication of (v\ by \u). Justification 6. Let X := l£(R). Then, (u\v) = (u\v)= / u{x)v(x)dx for all u,vGX jr Now let u,v eS. Then (k\u) = (</>k \u)= (j>k(x)u(x)dx = (27r)"~s / e~lkxu(x)dx. JR JR Consequently, the function k *-+ (k \ u) represents the Fourier transform of u. Hence the inverse Fourier transformation yields u(x) = (2tt)-* / eikx(k | u)dk for all xeR. JR This can be written as u(x) = / (/>k(x)(k | u)dk, JR which is (201). Furthermore, since the Fourier transformation represents a unitary operator on X, we get (u | v) = / (k | u) (k | v)dk. JR Because of (k \ u) = (k \ u) = (u \ k) = (u \ fc), this is (202). Example 7 (The momentum operator). As in Section 5.15, let us consider the momentum operator P — -' — dx' Here, we use such physical units that fi = 1. The equation Peipx = peipx for all x e R and each p € R

5.21. The Language of Physicists 377 can be written as P\p)=p\p) for all p e R. Because of the completeness relation (199) and the orthogonality relation (200), physicists say that {\p)} forms a "complete system" of eigenstates of the momentum operator P. Observe that \p) does not live in the Hilbert space X. But, recall from Section 5.15 that, in terms of mathematics, the system {\p)} forms a complete system of generalized eigenfunctions of P. 5.21.3 The Continuous Dirac Calculus and the Position Operator In order to deal with the position operator in a similar way, physicists introduce the formal state \x) and they set u(x) := (x | u) for all xGi. The Formal Completeness and Orthogonality Relation 8. Physicists write formally y^ \x){x\ = I (completeness relation) (203) X and (xf | x) = 8{x' — x) for all x',x eR (orthogonality relation). (204) Moreover, they set ^2X • • • := JR • - • dx. Formal Consequences 9. Prom (203) we get (v | u) = ^2{v\x)(x\u). (205) X This is identical to the rigorous expression (v | u) = (v I u) = / v(x) u(x)dx for all u, v € X, Jr where X := l£(R). From (v\ = Yj(t> | x'){x'\ and \u) = Vj \x)(x \ u) x' x

378 5. Self-Adjoint Operators, the Friedrichs Extension, etc. along with the orthogonality relation (204), we obtain (v | u) = y^(^ | x')(x' I x)(x I u) = 2Z(V I x')b(x' — x)(x,u) x,x' x,x' X This coincides with (205). Formally, u{x) = (x | u) = / u(y)6(y — x)dx. JR Therefore, physicists write \x) = 6X, where 6x(y) := 6(y — x) for all y G R. Finally, we want to show that the Dirac calculus is so powerful that it automatically leads to the right formula for the inverse Fourier transformation. In fact, from iu> = X>x*i«> k with \k) = (j>kt we obtain (x | u) = ]T(x | jfc>(jfc | u) = ]T<^(:r)(fc I u). (206) Furthermore, J2X \x)(x\ = I yields (k\u) = ]T(fc | x)(x | u) = ^"(^P) <& I u) = Y^<t>k{x)u{x). (207) CC X X Letting a(k) := (fc | w) and recalling that <f>k{x) := (2tt)~*elkx, relation (207) is identical to the Fourier transformation a(k) = / </>k(x) u(x)dx for all k G R, and (206) is identical to the inverse Fourier transformation u(x) = / (j>k(x)a(k)dk for all x G R. Jr Example 10 (The position operator). Let Q be the position operator defined by W)(V)'.= Wl>(v) for all y G R. Formally, y*>x{y) = y$(y -x) = x6(y -x)= x6x(y) for all y€R,

5.22 The Euclidean Strategy in Quantum Physics 379 since "6(y - x) = 0 if y ^ x.n Hence Q | x) = x | x) for all x G R. Because of the completeness relation (203) and the orthogonality relation (204), physicists say that {\x)} forms a "complete system" of eigenstates of the position operator Q. Observe that \x) does not live in the Hilbert space X. But recall from Section 5.15 that, in terms of mathematics, the system {\x)} forms a complete system of generalized eigenfunctions of Q. Remark 11 (An artificial barrier between mathematics and physics). In many math textbooks, the inner product on a complex Hilbert space X is denned in such a way that (au | v) = a{u | v) for all a G C and all u, v G X. Hence {u \ av) = a(u \ v). However, the Dirac calculus used in all physics textbooks forces the convention (u \av) = a(u\ v) for all a G C and all u, v G X, (208) which is used in the present book. In the future, mathematicians should pass to the convention (208) in order to avoid an artificial barrier between the language of physicists and mathematicians. The beauty and elegance of the Dirac calculus will become clear in the next section. The relation between the general Dirac calculus and rigorous mathematics (namely, the general spectral theorem due to von Neumann), is discussed in Zeidler (1986), Vol. 5, Chapter 89. 5.22 The Euclidean Strategy in Quantum Physics 5.22.1 Diffusion Let us start out from the diffusion equation ut = auxx for all x G M, t > 0, (209) u(x, 0) = uo(x) for all x G R (initial condition). This equation describes the mass conservation of a diffusion process on the real line, where u(x, t) := mass density at the point x at time t.

380 5. Self-Adjoint Operators, the Friedrichs Extension, etc. Hence rd i: u(x, i)dx = mass on the interval [c, d] at time t. The fixed positive number a is called the diffusion coefficient. A physical motivation of (209) can be found in Zeidler (1986), Vol. 4, Section 69.1. By a classical solution of (209), we understand a bounded continuous function u:Rx [0,oo[ —> R which is C1 on Rx]0,oo[ and satisfies (209). Proposition 1. Let the bounded continuous function uq:TSL —> R be given. Then, the initial-value problem (209) has a unique classical solution given by {( — "i2 (Anat)-^ f^ e" *«« u0(y)dy if x e R, * > 0, wo (a:) if are R, t = 0. In addition, u is C°° on R x ]0, oo[. The well-known classical proof of Proposition 1 can be found in John (1982), Chapter 7. Remark 2 (Probabilistic interpretation of the diffusion process via Brow- nian motion). Suppose that the diffusion process (209) corresponds to the motion of a large number of identical particles of mass m > 0. Define fd _i (*-i/)2 (P) / (Airta) 2 e 4ta dx = probability of finding c the particle in the interval [c, d\ at time t > 0 provided the particle is at the point y at time t = 0. Naturally enough, this corresponds to a Gauss distribution with the mean value y and the dispersion a1 — 2ta. We want to show that (P) implies Proposition 1. Let the mass density ^o be given at time t = 0. For small Ay > 0, the number N of particles in the interval [y, y + h] at time t = 0 is approximately equal to N= Mv)&v m These N particles spread over the real line during the time interval [0,£]. Let M of them be in the small interval [x,x + Ax] at time t. By (P), _ 1 (ar-y)2 M = (Anta) * e ^ Ax • N. This corresponds to the partial mass density ^^ at the point x at time t. The total mass in the interval [x,x + Ax] at time t is obtained by summing

5.22 The Euclidean Strategy in Quantum Physics 381 over all the contributions coming from all the possible intervals [y, y + Ay], Letting Ax —> 0 and Ay —> 0, we get the formula from Proposition 1 for the mass density u(x, t) at the point x at time t. 5.22.2 The Schrodinger Equation as a Diffusion Equation in Imaginary Time By (141), the Schrodinger equation for a free particle of mass m > 0 on the real line reads as follows: ipt = iai/jxx for all x € R, t> 0, (209*) V>(x, 0) = rfo(x) f°r all x € R (initial condition), ^ ' where a := ~^. Recall that \rf(x, t)\2dx = probability of finding the particle in the interval [c, d). Comparing (209*) with (209), we obtain the following fundamental result: If we pass from the real time t to the imaginary time it, then the diffusion equation passes to the Schrodinger equation. This leads us immediately to the so-called Euclidean strategy of physicists: In order to compute quantum processes in nature, consider first diffusion processes and pass then to imaginary time. The theory of diffusion processes on a microscopic level (the Brownian motion) was created in rigorous mathematical terms by Norbert Wiener in 1923. Roughly speaking, we have the following: Brownian motion in real time => the Wiener path integral =^ diffusion process in nature. Formally using the Euclidean strategy, we obtain the Feynman approach to quantum processes discovered in the 1940s: Brownian motion in imaginary time => the Feynman path integral => quantum process in nature. Unfortunately, the latter approach frequently works only on a formal level. However, from the physical point of view, the Feynman approach provides us with deep insight concerning the relation between classical physics and quantum physics. Let us first use the Euclidean strategy in order to solve the Schrodinger equation (209*) in terms of classical mathematics. The Feynman path integral will be studied in the next section. Replacing t with it and letting

382 5. Self-Adjoint Operators, the Priedrichs Extension, etc. a := ^, from Proposition 1 we obtain the following formal solution of (209*): / TU \5 f°° m(-,)» -OO for all x € R and £ > 0. Here we choose i* := e™. To justify this, let us write the Schrodinger equation (209*) in the usual operator form ihip'(t) = H0ip(t) for all t e R, (209**) ^(0) = V>o, with the Hilbert space X :— L^M.) and the free Hamiltonian Hq: D(Hq) C X ^ X denned through D(#0) := {V> € X: ?//,V" € X} and #0V> := ~ (5m) ^", ^ Problem 5.7, Hq is self-adjoint. Thus, the quantum motion corresponding to (209**) is given through ip(t) := e-^Vo for all t e R. Finally, let us introduce the Gauss function UaAx) := e-^x~a)2 for all x € R and define D := {ua^: a € R, /3 > 0}. It has been proved in Problem 3.6 that span D is dense in X. We shall show ahead that (Ai/jo)(x,t) can be continued to all times t G R if V>o € span D. By a classic solution of the Schrodinger equation (209*), we understand a C°°-function V = ^(a;, *) on R2 that satisfies (209*). Proposition 3. Let V>o £ span D be given. Then, the following hold true: (i) A^o is a classical solution of the Schrodinger equation (209*). (ii) Ai/jq coincides with the corresponding quantum dynamics, i.e., (Ai/jo) (t) = e~- ^ Vo for all t e R. Proof. Ad (i). Let ua,p € D. For simplifying notation, set m = h = 1. Computing the classical integral, we obtain 1 0(x-a)2 (Aua b)(x, t) = e w»* for all a; G R, t > 0. V1 + Aijj But the right-hand side also makes sense if t < 0. Therefore, we define 1 0(x-a)2 (Aua 0)(x, t) := „ e" *+4^ for all x € R, t< 0.

5.22 The Euclidean Strategy in Quantum Physics 383 More precisely, we have — f < arg(l + 4i/3) < ^ for alH G R, and we choose that branch of the square root where 7T -j < arg^l + 4ifit < - for all t G R. To simplify notation, set B(x, t) := (Aua,p)(x, t). For all x, t G R, 27*/? flfcc-a)2 n ; (l + 4z^)Vl + 4i/?r V ' J Computing the partial derivative Bxx the same way, we obtain Bt(x, t) = iBxx(x, t) for all x, t G R. Thus, B is a classic solution of the Schrodinger equation (209*). Ad (ii). Since the operator Ho is linear, it is sufficient to prove the statement for -00 £ D. Let i/jq :— ua^. Define i/j(t) :=B(-,t) for alUGR, i.e., ij)(t) represents the function x *-+ B(x, t) on R. Let us show the following: (a) i/j(t) G D(H0) for all t G R. (b) The time derivative i/)'(t) exists in the Hilbert space X for all iGl. Explicitly, i/j'(t) = Bt(;t) for all t G R. (c) The function t »-> V'CO is continuous from R to X. (d) i/j is a solution of the Schrodinger equation (209**). This follows easily from the explicit expression for B(x,t) by using ma- jorants for parameter integrals based on J — c e-P(x-oc)\kdx < ^ (M) for all a € R, 0 > 0, and k = 0,1, .... Proof of (a). By (M), f00 (drB(x, 7-ocV dx- t)\2 ' dx < oo, r = 0,1,... , for all t € R. Thus, the functions B(-,t), Bx(-,t), and Bxx(-,t) belong to X = Z|(R). Hence V £ £(#<))•

384 5. Self-Adjoint Operators, the Friedrichs Extension, etc. Proof of (b). For all* G R, ,. f°° fB(x,t + h)-B(x,t) n, A2, This follows by using a majorant of type (M) (cf. Parameter Integrals in the appendix). Proof of (c). Similarly, for all * G R, /oo (Bt(x, t + h)- Bt(x, t))2dx = 0. -oo Proof of (d). Observe that B is a classic solution of the Schrodinger equation (209*). In summary, i/j represents a C1 -solution of the Schrodinger equation (209**). Therefore, assertion (ii) follows from Theorem 5.G in Section 5.13. □ Remark 4 (The generalized quantum dynamics). Let the initial state V>o G X be given. Then the corresponding quantum motion reads as follows: ip(t) := U(t)ip0 for all * G R with the unitary operator U(t) := e"1^*. Hence ||/7(*)|| = 1 for all * G R. Since the set span D is dense in the Hilbert space X, there exists a sequence (V>on) in span D such that Von —► V'o in X as n —> oo. This implies V>(*) = lim (A0on)(-? *) in X as n —> oo, n—>oo for all * G R. In fact, l^(t) - (^0n)M)|| = \\U(t)(^0 ~ lfon)|| < II^OOII 11^0 ~ V^OnH = ||^0 ~ V^OnH ~> 0 as U -> OO. The preceding considerations show that the formal integral expression for (Ai/)q)(x, *) corresponds to the rigorous quantum dynamics provided we use analytic continuation and the approximation argument from Remark 4. It happens frequently that formal solutions lead to rigorous solutions after discovering the proper mathematical interpretation of the formal solution.

5.23 Applications to Feynman's Path Integral 385 5.23 Applications to Feynman's Path Integral Dick Feynman (winner of the Nobel Prize in physics in 1965) was a profoundly original scientist. He refused to take anybody's word for anything. This meant that he was forced to rediscover or reinvent for himself almost the whole of physics. It took him five years of concentrated work to reinvent quantum mechanics. At the end, he had a version of quantum mechanics that he could understand. The calculations I did for Hans Bethe, using the orthodox theory (via the Schrodinger equation), took me several months of work and several hundred sheets of paper. Dick could get the same answer, calculating on a blackboard, in half an hour. In orthodox physics it can be said: Suppose an electron is in this state at a certain time, then you calculate what it will do next by solving a certain differential equation (the Schrodinger equation from 1926). Instead of this, Dick said simply: "The electron does whatever it likes." A history of the electron is any possible path in space and time. The behavior of the electron is just the result of adding together all the histories according to some simple rules that Dick worked out. I had the enormous luck to be at Cornell in 1948 when the idea was newborn, and to be for a short time Dick's sounding board Dick distrusted my mathematics and I distrusted his intuition. Dick fought against my scepticism, arguing that Einstein had failed because he stopped thinking in concrete physical images and became a manipulator of equations. I had to admit that was true. The great discoveries of Einstein's earlier years were all based on direct physical intuition. Einstein's later unified theories failed because they were only sets of equations without physical meaning Nobody but Dick could use his theory. Without success I tried to understand him For two weeks I had not thought about physics, and then it came bursting into my consciousness like an explosion. Feynman's pictures and Schwinger's equations began sorting themselves out in my head with a clarity they had never had before. I had no pencil or paper, but everything was so clear I did not need to write it down. Feynman and Schwinger were just looking at the same set of ideas from two different sides. Putting their methods together, you would have a theory of quantum electrodynamics that combined the mathematical precision of Schwinger with the practical flexibility of Feynman. Freeman J. Dyson in Disturbing the Universe (Harper & Row, New York, 1979) In this section, we only use purely formal arguments in the spirit of the two great physicists Dirac and Feynman. We hope that our detailed pre-

386 5. Self-Adjoint Operators, the Friedrichs Extension, etc. sentation helps mathematicians to understand the thoughts of physicists. Our goal is to compute the Green function G = G(x, t; y, s). If the Green function G is known, then the process u = u(x, t) is known for each initial state u(x, s) = uq(x) at the initial time s, namely, /oo G(x, t\ y, s)u(y, s)dy for all x G R, t > s. (210) -oo This will be shown later. Therefore, the Green function plays a fundamental role in physics. It is decisive that we will compute the Green function without referring to the results from Section 5.22. To this end, we will use the Dirac calculus and the Euclidean strategy. Recall from Section 5.21 that \p) := (j)p and \x) := 6X for all x,p G R, (211) where (j)p(x) := (27r)-£eipa: for all xei 5.23.1 Diffusion and the Wiener Path Integral Let us first consider the following generalized diffusion equation: Ut(x,t) = auxx(x,t) — U(x)u(x,t) for all x G M, t > s, , , u(x, s) = Uq(x) for all x G M, ^ ' and fixed initial time s. Here, u(x, i) denotes the mass density at the point x at time £, and a > 0 is the so-called diffusion coefficient. Introducing the operator (Hv)(x) := -av"(x) + U(x)v(x) for all x G R, equation (212) can be written as an operator equation of the following form: u'(t) = -Hu(t) for all t > s, _„. u(s) = u0 ^ ' with the solution u(t) = S(t, s)uo for all t > s, where 5(^, s) := e~^~s^Huo- Here, u(t) stands for the function x i-> u(x,t) onl. Formal Definition 1. The function G defined through G(x, t\ y, s) := (x | S(t, s) \ y) for all x, y G M, t > s

5.23 Applications to Feynman's Path Integral 387 is called the Green function (or the propagator) of the diffusion equation (212). Formal Proposition 2. For given u$, equation (210) yields the solution u = u(x, t) of the initial-value problem (212) for the generalized diffusion equation. Formal Proof. By the Dirac calculus, J2y \y)(y\ — I- Hence u(x,t) = (x | u) = (x | S(t,s) | u0) = Y2(x I S(t,s) \y)(y\ u0) y /oo G(x,t\y,s)u0(y)dy. D -oo Formal Proposition 3. The Green function G satisfies the following fundamental relation: /oo G(x,t;z,T)G(z,r;y,s)dz (213) -OO for all x, y G R and all times t >r > s. Formal Proof. Observe that e-(ts)H = e-(t-r)He-(r-s)H if t > T > S. (214) Hence (x | e-(t-s^ | y) = ]T(z | e-^-T^Hz)(z | e^T^Hy). z This is (213). D The proof shows that (213) can be regarded as a localized version of the semigroup property (214). Formal Proposition 4. Let At > 0 be small. Then, for all x,y,t G R, /oo e-At(aP2+U(y))+i(x-y)pdp^ (215) -oo lip £o terras of order (At)2. Formal Proof. Since e~tH = I - At • H + £M^! , we get G(x, * + A*; y, t) = (x \ e~AtH \y) = (x\y)- At(x \ H \ y) + 0((At)2).

+ At^2{x\p')(p'\a1-^\p){p\y). p,p' 388 5. Self-Adjoint Operators, the Priedrichs Extension, etc. It follows from H = U-$ that A:=(x\y)- At(x \H\y) = (x\y)- At(x \U\y) ldx* Observing \y) = 6yi we obtain (x \ U \ y) = U(y)(x \ y) and V' ~h1 p) = (</>* IK) = -p2^p' I <t>p) = -p2(p' I p) = -*W - p). as well as (x \ y) = J2p(x \p)(p\y). Hence A = £> I p){p I y){\ - At(ap2 + U{y)) P = E<x I P)<P I y)e-At^2+u^ + 0((Atf). V Thus, up to terms of order (A£)2, /oo 0p(x)</»p(y)e-A*(^ +U^dp. -OO This is (215). □ Formal Proposition 5. // we set At := ^|, #n+i :— ^> ^0 :— V) and n+l r / \ a I / rp . rp ■ t \ _ I (216) s ■■= £ {* (^^zi) ^ -a^ - ^i-o} ha')> then /oo ei,s(27r)"n"1dxi • • • dxndpi • • • dpn+1 (217) -00 /or all x,|/Gl and £ > s. Formal Proof. Let tk := fcA£. Then, s := to < t\ < • • • < tn+i := t. By (213), /OO Gr^X; t\ Xn, tn)(jr{Xn, tn\ Xn—\, tn—\) -00 - - G(xi,ti;y,s)dxn - -dxi. Using (215) and letting A* -> 0, we obtain (217). D Formal Example 6. If U = 0, then G(ar, *; y, 5) = (47ra(£ - a))-*c"^^. (218)

5.23 Applications to Feynman's Path Integral 389 Formal Proof. By Example 2 in Section 3.7, the Fourier transformation yields the following rigorous formula: e-i0pe-<*p2dp= (47ra)-5e-fe -OO for all a > 0 and (3 € R. Furthermore, we will use the formal relation15 /oo (2n)-1eix^k'-kUx = ]T<jfc I *><* I **> = «(* - *')• By (216), n n+1 z<S = ipn+ixn+i - ipix0 + i ]T] Xj(pj - pj+i) - ]P ap^At, where (n + l)At = t — s. Hence /oo e^dxi • • • dxndpi • • • dpn+i -oo /oo ra+1 n gtfon+lX-piy) JJ e-aPj2Atrfft. JJ e™APr-Pr+l)dXrt ■°° j=l r=l Integrating first over eter, we get /oo w+1 n ci(p„+i*-piy) "Q e-ap23Atdpj JJ (5(pr+i _ ^ -°° j=l r=l Observing that J^ 6(p-q)f(p)dp = /(<?), integration over pn+u pn,... ,p2 yields /oo eip1(x-y)e-ap5(n+l)Atdpi -oo _ i _ (g~y) = (47ra(*-s)) 2e *«(*-•). The assertion follows now from (217). □ Observe the following: The Green function G from (218) coincides with the classical Green function from Proposition 1 in Section 5.22. Definition 7. We are given Ax > 0 and Ap > 0. By a discrete path, we understand a curve x = xn(r), p = pn(r), s<r<t, 15The rigorous version of this formula can be found in Standard Example 5 of Secton 3.7.

390 5. Self-Adjoint Operators, the Friedrichs Extension, etc. where xn,pn : [s,t] —> R are piecewise linear, continuous functions such that xn(tj) = integer • Ax, pn(tj) — integer • Ap for j = 1,..., n. Furthermore, we postulate that xn{-) connects the fixed initial point y with the fixed end point x, i.e., we set %n(s) '= V, xn(t) := x, and pn(s) := 0, pn(t) := integer • Ap. Let Vn denote the set of all these paths. According to (216), we define S{xn,pn) := ^j^^-xnj-!)^ _ ap2^ _ y.(Xnj_i)| {_iAt)_ (219) Here, xnj := arn(fy) and pnj- := pn(fy). By an admissible path, we understand a curve x = x(r), P = p(t), s<r<t, which connects the fixed initial point y and the fixed end point x, i.e., x(s) :— y and x(t) := x. Furthermore, let p(s) := 0. In addition, we demand that the integral S(x(-),p(-)) ■= J {^(r)p(r) - ap{r)2 - U{x{r))}{-idt) (220) exists. The set of all these admissible paths is denoted by V. Observe that (220) is the limit of (219) as A* -> 0. Formal Observation 8 (The path integral). Replace the integrals J ...dx and J .. .dp with sums ^ • • • Ax and ^ • • • Ap> respectively. Then, the integral from (217) has to be replaced by '» £*•<***(£) n+1 Here, the function <S has to be taken at all the possible node points, i.e., Xj := mAx, pr := fcAp, for all integers m, fc, and j = 1,..., n, r = 1,..., n+1. Now to the point. A simple combinatorial argument shows that j=5>*<*»-*">(as)*(^Y

5.23 Applications to Feynman's Path Integral 391 where we sum over all discrete paths (xn,pn). Therefore, from (217) we get the following fundamental formula: G(x,t;y,8) = "lim"2e^n'Pn)(Ax)n (^V , (221) where the symbol "lim" stands for the following formal limiting process: n -> oo, A* -> 0, Ax -> 0, Ap -> 0. Instead of (221), physicists write G(x,t;y,s) = [ eiS{x^^VxVp. (221*) Jv Here, the "integral" has to be taken o\ t all paths (#(•),p(-)) G P and VxVp denotes a "measure" on the path space V. Remark 9 (Rigorous approach). The integral from (221*) can be given a precise meaning by constructing a measure on V. This is the so-called Wiener measure introduced by Norbert Wiener in 1923. In this paper, Wiener created a mathematical theory for the Brownian motion. In terms of physics, the Brownian motion was first studied by Einstein in 1905. A rigorous justification of the so-called Feynman-Kac formula (221*) can be found in Reed and Simon (1972), Vol. 2, Section X.ll. See also Albeve- rio and Brezniak (1993) for a rigorous justification of the Feynman path integral on the level of quantum mechanics. 5.23.2 Quantum Mechanics and the Feynman Path Integral Parallel to the generalized diffusion equation (212), let us consider the Schrodinger equation —ihut(x,i) = auxx — U{x)u{x,t) for alHGi,t>s u(x, s) = uq(x) for all xGl, (222) for the fixed initial time s. Here, a := ^. According to (141), this equation describes the motion of a particle of mass m on the real line, under the action of the force —U'(x) at the point x G R in direction of the positive x-axis. Recall that Jc \u(x, t)\2dx = probability of finding the particle (22^ in the interval [c, d], provided j_Qo \u(x,t)\2dx = 1. The Hamiltonian (i.e., the energy operator) is given by p2

392 5. Self-Adjoint Operators, the Friedrichs Extension, etc. where P := ihJ^ denotes the momentum operator. Hence (Hv)(x) = -av"(x) + U(x)v(x), i.e., the Hamiltonian coincides with the operator H introduced in Section 5.23.1 for the diffusion equation. Using the Hamiltonian, the solution of (222) is given by %{t-s)H u(t) = e h uq for all teR. First choose such physical units that ft = 1. Then, all the formulas from Section 5.23.1 can be applied to (222) if we replace the real time t by the imaginary time it. In order to get formulas that display the role of Planck's quantum of action ft, forget about the convention ft = 1. Then, the rescaling t => ^ and p => | yields the following basic formulas: (i) The Green function G of the Schrodinger equation (222) is denned through G(x, t\ y, s) := (x \ S(t, s) \ y) for all x, y,t,se R, (224) where S(t, s) \—e~ » (ii) The solution of the initial-value problem (222) for the Schrodinger equation is given by /oo G(x, t\ y, s)u0(y)dy for all x,teR. (225) -oo (iii) By definition, the set V of admissible paths is given by the curves x = x{t), p = p(r), s<r<t, (226) which connect the fixed initial point y and the fixed end point x, i.e., x(s) := y and x(t) := x. Furthermore, we postulate that p(s) = 0 and that the integral $(*(.),»(•)) := J (p(t)x'(t) - H{t))dt (227) exists. (iv) In terms of classical mechanics, the integral (227) represents the action1^ of the admissible path (#(•),p(-)). Here, x{t) := position at time 16The physical meaning of action is discussed in great detail in Zeidler (1986), Vol. 4, Sections 58.19fF. Observe the following peculiarity. If x = x(t) describes the motion of a particle of mass m, then the momentum at time t is given by p(t) := mxf(t), where x'{t) equals the velocity of the particle at time t. However, we also consider such paths (224) in the (cc,p)-phase spaces, where p(-) is completely independent of x(-).

5.23 Applications to Feynman's Path Integral 393 £> pW •'— momentum at time £, and H(t) := ^- + I7(a;(*)) = energy at time t. (v) The Green function G can be expressed by the following Feynman path integral: G{x,t;y,s)= / e ** £>x£>p, (F) where we "integrate" over all admissible paths (#(•),£>(•)). The Feynman formula (F) is one of the most wonderful formulas of physics. In fact, (F) explains the relation between classical mechanics and quantum mechanics. Namely, according to (223) and (i), the Green function G describes the propagation of probability in quantum mechanics. This propagation is obtained in the following way. Let the particle perform all the admissible paths x = x(r), p = p(r) in the (x,p)-phase space. Then, by (F), the Green function is i , if i i tS(a:( ),p(Q) the mean value over all the numbers e h , where S denotes the classical action along the admissible path x = x(r), P — P(T) m tne (x,p)-phase space. This action is measured in units of ft, which corresponds to the quantization of action in quantum mechanics. This approach goes back to Feynman's Princeton dissertation in 1942. Feynman's universal method also works in quantum field theory. This can be found in Kaku (1993), Sterman (1993), and in Zeidler (1986), Vol. 5, Chapter 92. Formal Example 10 (Free motion). Let U = 0. Then, the corresponding Green function Go is given by G0(x,t;y,8) = ( ^ LJ e ™(t-.) . Formal Proof. This follows from Formal Example 5 by letting a := ^ and replacing t with ^ according to our general strategy. □

394 5. Self-Adjoint Operators, the Friedrichs Extension, etc. 5.24 The Importance of the Propagator in Quantum Physics In the following we want to show that The propagator «S'(-,-) contains all the information about the quantum system. Namely, this information includes (i) time evolution (the Dyson formula); (ii) bound states of fixed energy E\ and (iii) transition probabilities. We will apply this to (a) time-dependent scattering theory, the Feynman diagrams, and the Heisenberg S-matrix; and (b) time-independent scattering theory and the Lipman-Schwinger integral equation. Furthermore, observe that The knowledge of the propagator S'(-, •) is equivalent to the knowledge of the Green function G. In fact, if the propagator S(t, s) is known, then by definition the Green function G is given through G(x,t',y,s) := (x | S(t,s) \y). Conversely, let the Green function be given, and let {(j)a} be a complete orthonormal system. Then S(t,8)<l>:=^2Sa(t,8)(l>a, Ot where /oo (j>a(x)G(x,t;y,s)(l>(y)dy. -oo This follows from sa(t,s) = ((t>a I s(t,s) | <t>) = J2(<t><* I x)(x I sfo*) I y)(v I <£>• x The following considerations possess a "universal" character. They can be generalized directly to three-dimensional quantum mechanics. Moreover, the same approach also applies to quantum field theory. This can be found in Mandl and Shaw (1989), and in Zeidler (1986), Vol. 5, Chapters 89ff. As in the preceding section, we restrict ourselves to purely formal arguments.

5.24. The Importance of the Propagator in Quantum Physics 395 5.24-1 Time Evolution Let us consider the Schrodinger equation ihut = H(t)u for all x € R, t > s, (oo%\ u(x, s) = uq(x) for all x G R, where H(t):=H0 + U(-,t), and Hq := — (~) ^. Observe that the potential U = U{x, i) may depend on time t. Let u = u(t) be the solution of (228). Then, the propagator S(t, s) is denned through u(t) = S(t,s)u0. (229) Formal Theorem 1. The propagator is given through the fundamental Dyson formula: 5(4, s) = / + J2 T^y^j / TH{t{)H{t2) • • • ff(*n)d*i • • • d*n (230) n=l ^ '- J s for all t,sGl wz£/fc t > s. Here, T denotes the chronological operator, i.e., More generally, TH(h)H(t2) • • • ff(*n) := H(tv)H(t2') • • • J5T(*n0, where ^/,..., tn/ denotes a permutation of £i,..., tn with ty > t2' > • • • > tn/. Instead of (230), physicists formally write S(t, s) = T exp (\- f H(r)drj . (230*) Formal Proof. From (228) we get the integral equation u(t) = u0 + (ih)-1 / H(r)u(r)dr. Using the iteration method un+i(t) = u0 + (ifi)"1 / H{r)un{r)dT, n = 0,1,... ,

396 5. Self-Adjoint Operators, the Friedrichs Extension, etc. we obtain u(t) = u0 + J2(ih)~n / #(*0 • • • H(tn)u0, where / := f* dh J*1 d*2 • • • /^"_1 d*n. This yields (230). In fact, for example, we get J:= f dh f ' dt2H(h)H(t2) = l f H(t1)H(t2)e(t1-t2)dt1dt2, J s J s Js J s where 6(x) := 1 if x > 0 and 6(x) := 0 if x < 0. Using a permutation of indices, we get rt rt -, tidt2 j = f f ^{Hit^H^e^-^ + Hi^H^e^-h^dh = j j ^TH^H^dhdh. D Formal Proposition 2. The evolution operator S(t, s) is unitary for all times t and s. Formal Proof. By (229), jt{u{t)\u{t)) = {u\t)\u{t)) + {u{t)\u\t)) = {(ih)-lHu(t) | u(t)) + (u(t) | (ih)-lHu(t)) = -(ifi)-1^) | u(t)) + (tfi)-1^) | Hu(t)) = 0. Hence (u(t) \ u(t)) = (u0 \ u0) for all t£R. □ 5.i^.£ Sound States Suppose that the potential U is independent of time t. Then, it follows from (230) that 00 1 S(t, s)=I + Y jT^—.it - s)nHn v J ^ (ih)nn\v ; n=l v 7 i(t-s)H Recall that by definition the state (j) is called a bound state of fixed energy iff H(j) = E(j), i.e., ihSt(0,0)(j) = E(j). (231) In terms of the Green function G(x, t; y, s), this means that ^£^<* I 5(t,0) | y)(y | ^)|t=0 = £<* | <t>), y

5.24. The Importance of the Propagator in Quantum Physics 397 i.e., equation (231) is equivalent to /oo Gt(x, 0;y,0)(/>(y)dy = E</>(x) for all x e R. -OO 5.24-3 Transition Amplitude and Transition Probabilities Formal Definition 3. Let (j) and i\) be two states. Suppose that the quantum system is in the state (j) at time s. Then |(V> | S(t, s) | (j))\2 = probability of finding the system in foQo^i the state ij) at time t. ^ ' Moreover, the complex number (i/j \ S(t, s) \ (j)) is called the transition amplitude from the state (j) at time s to the state i\) at time t. Definition (232) makes sense since S(t, s) is unitary. Hence \\S(t, s)</>\\ = \\<p\\ = 1, i.e., S(t, s)(j) is a state. Formal Proposition 4. We have /oo /»oo / ip(x)G(x, t\ y, s)(j)(y)dxdy. -OO J —OO Formal Proof. By the Dirac calculus, (V | S(t,s) | <j>) = £> | x)(x | S(t,s) | y)(y | </>). D x,y Formal Proposition 5 (The fundamental Feynman relation for transition amplitudes). Let {(j>a} be a "complete orthonormal system" in the sense of the Dirac calculus, i.e., J2a l^a) (<l>a\ = I- Then (V | S(t,s) | </>) = £<V | S(t,a) | <f>a)(<f>a | S(<r,s) \ </>) Ot if s <a <t. Formal Proof. By (228) and (229), S(t,s)(j> = S(t,<j)[S(<j,s)(j)}. Hence S(t, s) = S(t, ct)S((t, s). Now use the Dirac calculus. □ Formal Example 6 (Complete orthonormal systems). First let {4>a} be a complete orthonormal system, i.e., {<j>ot I <f>p) = 8aj3-

398 5. Self-Adjoint Operators, the Friedrichs Extension, etc. Set WQ)/3 := transition probability from the state <\>$ at time s to the state <j>a at time t. Suppose that ((/)(3 | S(t,s) | </>a) =y}T/Q>j8j33a. 3 By (232), W^ = {ifli|2 ITA for all,. <*»> Now let {(/>a} be a "complete orthonormal system" in the sense of the Dirac calculus, i.e., {(t>a | <j>p) =6(a-0), and suppose that {<i)p i s(t, s) i </>a) = Y^aJ6(f3j - «)• 3 Then, (j)a is a "generalized state" and physicists assume that (233) remains valid. This is motivated by the philosophy that 6(a — /?) represents a continuous version of the Kronecker symbol 6ap. This argument will be used later in scattering theory (cf. Remark 10). 5.24-4 Applications to Time-Dependent Scattering Theory and the Feynman Diagrams Consider first a classical particle of mass m on the real line which moves with the velocity v from left to right (Figure 5.13(a)). Such a particle has the momentum p = mv and the energy In terms of quantum mechanics such a particle is described by the function i/jp(x,t) :— ae K~~(j)p(x), where </>p(x) := (27rft)"~2e'* . More precisely, ijjp represents a particle stream of velocity v = 2- and particle density p(x,t) — \ij)p(x,t)\2, i.e., fb / p(x, t)dx = number of particles in the interval J a [a, b) at time t.

5.24. The Importance of the Propagator in Quantum Physics 399 U • ► —^ ^ • ► p q (a) scattering process Uih) <t\<t P (b) <q\SX{t,s)\p> mti) < u(t2) -^* S < t\ < t2 < t p Pi q (c) <q\ S2(t, s)\p> FIGURE 5.13. In the following, we set a := 1 and fi := 1. We also write Recall that {(j)p} forms a complete orthonormal system in the sense of the Dirac calculus, i.e., /oo dP\<l>p)(<l>p\ = I- -OO Furthermore, observe that i/>p(t) = e-itHo(j>p. Since the operator e~ltH° is unitary, {^p(t)} also forms a "complete orthonormal system" for each time t. We want to study the scattering of a particle stream on the real line under the influence of a time-dependent potential U = U{x,t). Formal Theorem 7. Let t > s. For the transition amplitude we get oo (g, t | S(t, s) | p, s) = S(q - p) + JjTfa | Sn(t, s)\p), (234) n=l where (q | Sn(t,s) \p):= J ^T(q \ U{h) \ Pl){Pl \ U{t2) \ p2)

400 5. Self-Adjoint Operators, the Friedrichs Extension, etc. ---(Pn-l\U(tn)\P), along with J := Js dt\ • • • dtn J^ dpi • • • dpn-\ and /oo 4>q(x)U(x,t))4>p(x)dx. -OO The chronological operator T was introduced in Section 5.24.1. The physical meaning of (g, t \ S(t, s) \ p, s) will be discussed in Remark 10. Remark 8 (The Feynman diagrams). Explicitly, we get (q\Si(t,8)\p) = -iJ dt1{q\U(t1)\p) and (q | S2(t, s)\p) = -\jr{q | U(h) | Pl)(Pl | U(t2) | p), where J := Js dtxdt2 J^ dpi. This is represented graphically in Figures 5.13(b) and (c) by means of the so-called Feynman diagrams. Intuitively, these diagrams show that A scattering process in quantum mechanics can be regarded as the superposition of infinitely many micro scattering processes. Observe the fundamental fact that the transition amplitude can be completely computed if the first approximation (q \ U(t) \ p) is known. In the language of Feynman diagrams, this means the following: The Feynman diagrams of higher order are obtained from the first-order Feynman diagrams (see Figure 5.13). In fact, Feynman diagrams provide us with a deep insight into the structure of scattering processes. Today they represent the most important tool in quantum field theory (elementary particle physics). In the quotation at the beginning of Section 5.23, Dyson pointed out that Feynman was able to compute complex physical effects on the blackboard in short time. The reason for this is the fact that the local language of Feynman diagrams is much closer to the physical effects than the global language of the Schrodinger equation. Formal Proof of Theorem 7. Set u(t) := S(t,s)u. By (228), iu'(t) = (H0 + U)u(t), u(s) = no. To get a simpler differential equation we define v(t):=eitHou{ty

5.24. The Importance of the Propagator in Quantum Physics 401 Then v'(t) = eitH°iH0u(t) + eitH°u\t) = eitH°(-iU)u(t). Hence iv'(t) = Vv(t), v(s) = e*sif°u0, where V := eitH°Ue-itH°. Letting wo := e~lsH°(j) and observing that v(s) = 0, it follows from the Formal Theorem 1 that v(t) = eitHoS(t,s)e-isH°(j) oo - »t (235) = </>+53 ^-t / ry(*i) • • • ^(*»)^*i • • • *»• n=l * n- J» Since ^>p = e~ltH°4>p, we obtain <«,t | 5(t,s) \p,a) = (4>q | e^5(t,S)e-^° | <j>p) = (<M <M + E ^ / r(<^ I ^(*i) • •' V(tn) | ^p>cfti • • • dtn. n=l * n* Js By the Dirac calculus, for example, /oo (</>, I V(t) I </>Pl)<0Pl I V(t) I 0p)dPl -oo = /<lMt) I C(*) I <M*)><lMr) | I7(r) | ^p(r))dpi. Furthermore, tyq{t) I ^W I ^pW> = cit(B(g)"£(p)) jdxJ^x)U(x,t)ct>p(x). This yields (234). □ Formal Standard Example 9. Suppose that the potential U — U(x) is independent of time t and vanishes outside a compact interval, say [c, d]. Let p > 0. Then, as t —> -foo and s —> — oo, the first approximation of the transition amplitude reads as follows: (q\S\p) := lim (<?, £ | S(t, s) | p, s) s—> —oo.c—»+oo (236) = %-p)a+(p)+%+p)a_(;p), where a+(p) := 1 - 27rmip~1/7F(0), a_(p) := 27rmip~1UF(-2p),

402 5. Self-Adjoint Operators, the Friedrichs Extension, etc. and Uf denotes the Fourier transform of U. In particular, if U(x) ~ 0, then a+{p) — 1 and a_(p) = 0. Formal Proof. By Remark 8, (q,t\S(t,s) \p,s) = 6(q-p) + (q\ Si(t,s) \ p) + • • • , where /OO POO dTeir(B(q)-B(p)) I <f,/x)U(X)4>p(x)dx -oo J —oo /OO e-*(«-P)*J7(x)da; -OO = -2m6(E(q)-E(p))UF(q-p). 2 Since i?(p) = ^, it follows from Problem 2.19 that S(E(q) - £(p)) = mp-1{6(q-p)+6(q + p)}. D The operator S<f>:= lim eitHoS(t,s)e-isHo<j> s—*• — oo,t—>+oo is called the Heisenberg S-operator. Since («,t | 5(«, a) | p, *> = <^g | eitH°S(t, s)e~isH° | </>„>, it follows from (236) that (q\S\p) = ((t>q\S\(t>p). Heisenberg introduced the S-operator (or the ^-matrix) in 1942 as a convenient way to describe scattering processes for elementary particles. Explicitly, it follows from (235) that S(j) °° I /»00 = <t> + J2 —\ / TV^ ' ' ' VitnWh • • • dtn, n=l % U' J~°° where V(t) := eitH°Ue-itH°. Remark 10 (Physical interpretation of (236) and the Born approximation). Consider a potential U = U(x) as in the Formal Standard Example 9. Let p > 0. Define Wq,p •= transition probability for a particle from momentum p to momentum q. Then, motivated by the Formal Example 6, it follows from (236) that

5.24. The Importance of the Propagator in Quantum Physics 403 u p c FIGURE 5.14. (i) WfllP = 0ifg^±p. (ii) W±P,P=\a±(p)\2. In particular, if U{x) = 0, then WP)P = 1 and Wq,p = 0 for q ^ p, as expected (no scattering). In terms of the function i/j, this means the following. We are given a stream of particles with mass m which moves from left to right with the velocity Vin — ^ and the particle density 9%n = 1. For large — t and —x, this stream is described by the function il>in(x,t) = e-itEWei*x. These particles either pass the potential barrier U or are reflected (cf. Figure 5.14). More precisely, we get two different particle streams near time t = +oo, namely: (i) a particle stream from left to right of velocity v^ut = v-m with the particle density Pout = Wp.pPinj (ii) and a reflected particle stream from right to left of velocity v~ut = —V[n with the particle density Pout = W_p)PPin. For large t and x, the particle streams in (i) and (ii) are described by the functions ^t{x,t)-=*±{p)e-UE{±p)e±ipx. Observe that Pout = lVdt|2 = |a±(p)|2- As we shall show, this result is identical to the Born approximation that follows from a completely different approach (the Lipman-Schwinger integral equation).

404 5. Self-Adjoint Operators, the Friedrichs Extension, etc. Naturally enough, (i) and (ii) above correspond to conservation of energy, E = 2-1mv2 + U(x). (237) In fact, since U(x) — 0 for large |x|, the incoming and outgoing particles have the energy E-m — 2~1mvfn and Eout = 2~1mvlut, respectively. It follows from E-m — Eout that vout — ±v-m. Observe that the structure of quantum scattering processes differs completely from the behavior of classical particles. For example, in classical mechanics, it follows from (237) and U(x) = 0 for all x $. [c, d] that a scattered particle V of initial velocity v-m has the energy E = 2~1mvfn. Thus, by (237), the particle V cannot pass the potential barrier if U{x) > 0 for some point x. 5.24-5 The Lipman-Schwinger Integral Equation in Time-Independent Scattering Theory The Lipman-Schwinger integral equation reads as follows: /oo eiplx-ylV(y)4>out(y)dy, (238) -OO where V(x) := imp~1U(x) and ft := 1. A physical interpretation of 0out will be given ahead. Let us assume that the time-independent potential U — U(x) vanishes outside the compact interval [c, d]. The integral equation (238) can be solved by using the following iteration method: /oo e^x-yW{y)<j>ttl\y)dy, n=l,2,..., -oo where 0oUt(x) :~ eWX- This way we get the first approximation /oo eip\x-y\+ipyv{y)dyt (239) -OO which is called the Born approximation. Formal Proposition 11 (Asymptotic behavior of (j>). If (j) is a solution of (238), then cj)out(x) = a+{p)eipx for all x > d and <l>out(x) = eipx + a.(p)e~ipx for all x < c, where /oo e-^V(y)4>out(y)dy -OO

5.24. The Importance of the Propagator in Quantum Physics 405 and /oo eipyV{y)^out{y)dy. -OO Formal Proof. By (238), 4>out{x) = tiv* - eip* f e-^Viy^oMdy J —OO /•OO -e-H* e^Viy^Mdy. J x □ (240) In particular, for the Born approximation (239) we get /OO e2ipyV{y)dy = -27rm*p-1l7j?(-2p), -OO /OO V(y)dy = 1 - 2Trmip-1UF(0), -OO where Up denotes the Fourier transform of the potential U. Formal Proposition 12 (Solution of the Schrodinger equation). Let </>out be a solution of the Lipman-Schwinger equation (238). Then, il>{x,t) := e-^Vout(z), x,t e R, (241) 2m* zs a solution of the Schrodinger equation ii\)t = Hijj. Here, E(p) := ^— Formal Proof. By Problem 5.1, (£-2+p2yipW=2ip6(x). Recall that H0 = ~^^i- Thus, it follows from (238) that /OO 6(y-x)U(y)<fiout(y)dy -OO = -U(x)(j)out(x). Hence H(j>out — E(p)^out. This implies ii/jt — H^. D Remark 13 (Physical interpretation). The solution i/j from (241) has the following asymptotic behavior: ijj(x, t) = ij)-m(x, t) + ijj~nt(x, t) for all x < c, V>(x, t) — ^oUt(x, *) f°r au x >d,

406 5. Self-Adjoint Operators, the Friedrichs Extension, etc. where 1M*.t) := e^e^, V&tfot) := ^(pJe^We^, for fixed p > 0. Here, V>in and ^J^. correspond to an incoming and outgoing particle stream of velocity v := ^ from left to right and particle density Pin = 1 and p+ut = |a+(p)|2, respectively. Furthermore, rf~ut corresponds to a reflected particle stream from right to left of velocity v = — ^ and particle density p~ut = |a_(p)|2 (cf. Figure 5.14). If we use the Born approximation (240) for a±(p), then this result coincides with Remark 10. 5.25 A Look at Solitons and Inverse Scattering Theory Waves are one of the most fundamental motions: waves on the water's surface and of earthquakes, waves along springs, light waves, radio waves, sound waves, waves of clouds, waves of crowds, brain waves, and so forth. Waves are recorded, and records are analyzed. In the case of sound waves and light waves, it is customary to analyze a wave as the sum of simple sinusoidal waves (Fourier series). This is the principle of linear superposition. However, when we observe water waves carefully, we see that the linear superposition principle cannot be applied in general, except for very small amplitudes. The study of water waves of finite amplitude was one of the main topics in nineteenth-century physics. In recent years, many nonlinear phenomena have become important. For example, strong laser beams and waves in gas plasmas exhibit nonlinear phenomena. The increasing importance of such phenomena has given rise to intensive study by means of high-speed computers, and it has been revealed that in many nonlinear waves, stable pulses are to be considered as fundamental entities. The stable pulses in nonlinear media are called solitons. The discovery of solitons has led to new developments in mathematics, which have made it possible to solve a variety of nonlinear evolution equations in the last 30 years. Morikazu Toda, 1989 5.25.1 Solitons The Korteweg-de Vries equation (KdV equation) is given by ut + 6uux + uxxx = 0, -oo < x, t < oo. (242)

5.25 A Look at Solitons and Inverse Scattering Theory 407 (a) soliton (solitary wave) C2 C2 (b) collision between two solitons FIGURE 5.15. An explicit calculation shows that this equation has the following solution:17 u(x, t) = 2fc2 sech2fc(x -ct- x0), (243) where c = 4fc2, k > 0, and xo £ R. This corresponds to a soliton (solitary wave) that moves with the velocity c from left to right (cf. Figure 5.15(a)). Proposition 1 (Two-soliton solutions). The KdV equation (242) has the solution d2 u(x,t)=2-^hi<p(x,t), (244) where <p(x,t) := 1 + Aie2m + A2e2ri2 + A3e2(?7l+772) a/on# with rjj := fcj(x - c^), Cj = 4fc2, j = 1,2, and A3 := (%*+%}) A\A2. Here, the numbers fci,fc2 > 0 and Au A2 > 0 are given. This follows from an explicit calculation. In particular, if A\ = 1, fci = fc, A2 = 0, then (244) is identical to the soliton (243) with xo = 0. Corollary 2. Let c\ < c2. Then as t —> ±oo, the solution (244) zs a superposition of the following two solitons: Uj(x, t) — 2fc2 sech2fcj(x - Cjt - xf0), j = 1,2. (245) 17Recall that sech x := ^^ = ^£=r.

408 5. Self-Adjoint Operators, the Friedrichs Extension, etc. Here, A\ = exp(—kix^0) and along with A2 = exp(-k2X20) and Al(lrt) =exp(-fci^o)- This shows that the solution (244) behaves like two solitons at time t = —oo and time t = +oo (cf. Figure 5.15(b)). It is quite remarkable that the two solitons are stable, i.e., they do not change their shape and velocity after collision. There appears only a phase shift x^0. In summary, we observe the following: Solitons behave like particles under collisions. This important fact was discovered by Kruskal and Zabusky via computer experiments in 1963. Proof. Let (x, t) G R2 be given such that \x - c\t\ < a for fixed a > 0. Then, as t —> +oo, rj2 = k2(x - c2t) = k2[x - c\t - (c2 - c\)t) —> -oo, since c2 > c\. Hence <p(x,t) ^1 + Aie2771, t^+oo. This yields (245) for j = 1. Now let (x,t) e M2 be given such that \x — c2t\ < a. Then, as t —> +oo, rji = ki(x - c\t) = ki[x — c2t + (c2 - c\)t) —> +oo. Hence ^(x,0 = ^ie277l+A3e2^1+?72) = Aie2r» (l + ^-e2A , t - +oo.

5.25 A Look at Solitons and Inverse Scattering Theory 409 This implies d2 u(x,t) = 2 g^2 WOM) = 2-* dx2 ^fr) which corresponds to (245) for j = 2. Similarly, we obtain the asymptotic behavior as t —> — oo. D 5.25.2 Summary of Inverse Scattering Theory and the Spectral Transform We consider the stationary Schrodinger equation -i>"(x) + u(x)ip(x) = k2i/j(x), -oo < x < oo. (246) We assume that the real C°°-potential u vanishes sufficiently fast at infinity, i.e., j — < |w(a;)|(l + \x\)dx < oo. We are looking for complex-valued eigenfunctions i\). The spectrum of (246) has the following structure: (i) Continuous spectrum. For each real k ^ 0, the number k2 is a double eigenvalue of (246) with the two linearly independent eigenfunctions ^l and V>2, which are uniquely characterized by the following asymptotic behavior: il>\{x) = e~ikx + o(l), * r -^, i/j2(x) = eikx + o(l), — lZ4° Additionally, we obtain ^i(ar) = a(fc)e"to + 6(fc)_eto + o(l) ^(ar) = b(k) e~ikx + a(fc) eto + o(l), X X 0. :i) —► ~* i 1, -00, -00. X —> X —J +00, ► +oo. (248) (ii) Discrete spectrum. Equation (246) has either no negative eigenvalues or a finite number of negative eigenvalues -oo < k2 < kl < • • • < k% < 0. All these eigenvalues are simple. Letting kj = iqj with qj > 0, the corresponding eigenfunctions ifo\ j = 1,..., iV, are characterized by the following asymptotic behavior: t/>W(x) = eq>x + o(eq>% x -► -oo. (249)

410 5. Self-Adjoint Operators, the Friedrichs Extension, etc. Additionally, we have i/jW(x) = Cje-q>x + o(e~q>x), x -> +00, where Cj is real. The mapping u V-+ (a(fc), 6(fc), ^-, Cj), (250) with fc G R and j = 1,..., iV, is called the spectral transform. In terms of quantum mechanics, (i) and (ii) correspond to scattered particles and to bound states of particles, respectively. Let all the scattering data a(fc), 6(fc), #j, and Cj be given. The main task of inverse scattering theory consists in constructing the corresponding potential u. To this end, we set _, , ^c3e-*>x , 1 f°° b(k) ikxj1 F(x) := > • #/■ x + 7T~ / ~{elkxdk, f^i «*'(*<&) 2tt 7.^ a(fc) and we consider the Gelfand-Levitan-Marchenko integral equation />oo K(x,y)+F(x + y)+ K(x,z)F(z + y)dz = 0. (251) ./CC If we know a solution K of this linear integral equation, then we obtain the unknown potential u by the relation j w(a;) = -2 —-K(x,x). ax 5.25.3 Construction of Solutions of the Korteweg-de Vries Equation via the Inverse Spectral Transform We consider the initial-value problem for the nonlinear Korteweg-de Vries equation18 ut — 6uux + uxxx = 0, —oo < x < oo, t > 0, (252) w(a;, 0) = uq together with the linear Schrodinger equation -i/j"(x)+u(x,t)ilj(x) = 0 for fixed, but otherwise arbitrary, time t. Then, we have the following important result: Replacing u with — u, we get (242).

5.25 A Look at Solitons and Inverse Scattering Theory 411 (R) Let u = u(x,t) be a solution of (252), which vanishes sufficiently fast as \x\ —» oo. Then the spectral transform of u satisfies the following simple linear equation: d(fc, t) = 0, 6(fc, t) = 8ik3b(k, t), (2SS) The dot denotes the ^-derivative. According to (R), we can use the following elegant procedure in order to solve the initial-value problem (252) for the Korteweg-de Vries equation: (a) For given initial values u = uq(x), we compute the spectral transform a(fc,0),6(fc,0),©(0),Ci(0). (b) We solve equation (253). This yields a(M)=a(fc,0), Qj(t)=Qj(0)> * > °> b(k,t) = b(k,0)e8ik3t, Cj(t) = Cj(0)e8^3^. (c) We obtain the solution of the original problem (252) by using the inverse spectral transform (a(k,t),b(k,t),qj(t),Cj(t)) *-+ u(x,t) from Section 5.25.2. Motivation of (R). In order to display the simple idea of proof as clearly as possible, we restrict ourselves to formal considerations. Step 1: The discrete spectrum. Let us introduce the following two differential operators: d2 L(t)ijj := -^2^(x) + u(x,^fa), A{t)il) := 4—t/>(x) - 3u{x,t)—i/j(x) - —{u{x,t)i/j(x)), where the fixed function u satisfies the KdV equation (252). From (252) we obtain the key relation Lt=LA- AL. (254) Hence L(t) = e-tAL(0)etA. Consequently, L(t)i/> = \il> iff L(0)(j> = \(j>, where (j> := etAi/j.

412 5. Self-Adjoint Operators, the Friedrichs Extension, etc. Thus, the operator L{i) has the same eigenvalues as the operator 1/(0), i.e., we obtain the crucial relation qj{t) = <7j(0) for all t, and hence qj = 0. The pair {L, A} with (254) is called a Lax pair. Let i/j — il)(x, t) and let L(t)il> = -q2i/j (255) for fixed £, where q = qj. By Section 5.25.2, for fixed t, the eigenfunction i\) can be characterized by the following asymptotic behavior: <ijj(x, t) = eqx + o(eqx), x -+ -oo. (256a) In addition, <ijj(x, t) = c(t)e~qx + o(e~qx), x -+ +oo. (256b) Differentiation of (255) with respect to time t yields Ltip + Lipt = ~q2^t' Note that q is independent of*. By (254) and (255), (L-\-q2)(i/jt-\-A^) = 0. Hence we obtain L(t)(j> = -q2cj>, where (j> := i\)t + Aijj. By (256a), $ = Aq*eqx + o(eqx), x -> -oo. Hence <\> = 4g3/0, i.e., we obtain the key equation tl>t + Atl> = 4tfty- (257) In this connection, note that u is rapidly vanishing as \x\ —> oo, i.e., we may put A = 4^ as \x\ —> oo. Finally, it follows from (256b) and (257) that c(t) = Sq3c(t). Step 3: Continuous spectrum. Let k > 0 be fixed, and let i/j denote the eigenfunction of the equation L(t)i/j = k2i/j (258) for fixed £, where i/> = i/>(x,t) \s characterized by the following asymptotic behavior: il>(x, t) = e~ikx + o(l), x -► -oo. (259a) In addition, i/>(x, t) = a(fc, *)e"to + b(k, 0eifca! + o(l), ar -+ +oo. (259b)

Problems 413 As above, differentiation of (258) with respect to time t yields L{t)(j) = k2(j), where (j> := ijjt + Ai/>. By (259a), cj) = 4ik3e'lkx + o(l), x -+ -oo. Hence 0 = 4zfc3/0, i.e., we obtain the key equation il>t + Ail) = Aik3^. Using (259b), we get ae~ikx + beikx = (-4-^g + 4ifc3 J (ae"ifca! + &eto) + o(l) = 8zfc36eto+o(l), x^+oo. Hence d = 0 and 6 = Sik3b. This finishes the motivation of (R). Remark 3 (Nonlinear Fourier transformation). The method (a) through (c) represents a nonlinear variant of the classical Fourier transformation. To explain this, consider the linearized Korteweg-de Vries equation ut + uxxx = 0. Using the Fourier transformation /oo b(k,t)eikxdk, -OO we get This implies J —c (6 - ik3b)eikxdk = 0. b = ik\ which corresponds to (253). A detailed discussion of this theory and its applications to the computation of special solutions (iV-soliton solutions) can be found in Novikov (1984) and Toda (1989). Problems 5.1. A special fundamental solution. Let p G R. Show that (t* + p2)eiPlxl = 2pisw for a11 x e R'

414 5. Self-Adjoint Operators, the Friedrichs Extension, etc. in the sense of generalized functions. Solution: Let /oo eiplx|4>(x)dx for all </> £ Cg°(R). -OO We have to show that C/(0/,)+p2/7(0)=2p^(O). In fact, using integration by parts, this follows from the decomposition /»oo pO U{(j>n) = / eipx<j>(x)dx + / e-ipx(j>(x)dx. JO J -oo 5.2. The nonhomogeneous stationary Schrodinger equation. Let f:R —> C be a continuous function that vanishes outside a compact interval. Set /°° ieip\x-y\ Show that, for each p eR with p^O, the function v is a C2-solution of -V -p2v^ f on R. Hint: Use the decomposition v(x) = f£° H ^/^oo * * * and an analogous argument as in the proof of Proposition 1 in Section 2.7. 5.3. Graph closed operators. Let A: D(A) C I -> I be a linear operator on the Hilbert space X over K such that D(A) is dense in X. The set G(A):={(u,Au):ueD(A)} is called the graph of A. The operator A is called #rap/z. c/osed iff G(A) is closed in X x X, i.e., un ^> u and A^n —> v in X as n —> oo imply Au = v. The linear operator B: D(B) C X —> X is called the closure of A iff A C 5 and19 G(A) = G(B). We write A instead of B. Show the following: (i) The adjoint operator A* is graph closed. 19Observe that (u,v) € G(A) iff there exists a sequence {(un,vn)} in G(A) such that un —> u and vn —* v in X as n —* oo.

Problems 415 (ii) The closure A exists iff it follows from un G D(A) for all n along with Aun —> v and un —> 0 as n —> oo that v = 0. (iii) If there exists a linear graph closed operator C: D(C) CI->I such that ACC, then the closure A exists and Acc. Hence the closure A is the smallest graph closed extension of A. In particular, A is uniquely determined by A. (iv) If A is symmetric, then the closure A exists and is symmetric. (v) If A exists, then (A)* = A*. (vi) If A is self-adjoint, then A — A. (vii) The operator A is graph closed iff D(A) is a Hilbert space over K equipped with the inner product {u | v)a := (u\v) + (u\ Av). 5.4. Symmetric operators. Let A: D(A) C X —> X be a linear symmetric operator on the complex Hilbert space X. For all A G C and uGl, show that (i) ||Au-Au||>|ImA|||tt||; (ii) (A -\I)*=A*- XI; (iii) X = N(A* - XI) 0 R(A - XI) (orthogonal direct sum), (iv) If A is graph closed, then R(A - XI) = R(A - XI), for all A 6 C with Im A ^ 0. (v) A** = A. Solution: Ad (i). By the Schwarz inequality and 2ab < a2 + b2, 2(Re X)(Au | u) < 2|Re A| ||Au|| ||u|| <||Att||2 + |ReA|2||«||2.

416 5. Self-Adjoint Operators, the Friedrichs Extension, etc. Hence \\Au - Xu\\2 = (Au - Xu | An - Xu) = \\Au\\2 - (Au | Xu) - (Xu | Au) + |A|2|M|2 = \\Au\\2 - (2Re X)(Au | u) + |A|2|M|2 > |Re A|2|M|2 - |A|2|M|2 = |Im A|2|M|2. Ad (ii). Use the definition of A*. Ad (iii). Let v G N(A* - XI). Then (Au -Xu\v) = (u\ A*v -Xv) = 0 for all u G D(A), and hence v G R(A — XI) . Conversely, if v G R(A — XI) , then (Au - Xu | v) = 0 for all u G D(A). This implies_v € D((A - XI)*) and A*v - Xv = (A - XI)*v = 0, i.e., veN(A* -XI). Ad (iv). Let vn :- (A - XI)u n —>• v as n —> oo. By Problem 5.4(i), ll^n — ^m|| > |Im A| \\un — wm||, and hence (wn) is Cauchy, i.e., iin —> u as n —> oo. Since A is graph closed, so is A — XL Hence (A — XI)u = v. Ad (v). Cf. Riesz and Nagy (1955), Section 117. 5.5. Self-adjoint operators. Let A:D(A) C X —> X be a linear symmetric operator on the complex Hilbert space X. Show the following: (i) If R(A - XI) = R(A - XI) - X for some fixed A G C, then A is self-adjoint. (ii) If A is self-adjoint, then all the points A G C with Im A^O belong to the resolvent set p(A) of A. (iii) The operator A is self-adjoint iff R(A ± ii) = X. (iv) The operator A is self-adjoint iff A is graph closed and N(A*±iI) = {0}. (v) A2 + I = (A-iI)(A + iI). (vi) If A is self-adjoint, then so is A2.

Problems 417 Solution: Ad (i). Since A is symmetric, A C A*. Thus, we have to show that A* C A. To this end, let v e D(A*). Since R(A - XI) = X, there is a w e D(A) such that Aw — Xw = A*v — Xv. Thus, for all u e D(A), (Au — Xu | v — w) = (u | A*v — Xv) — (u \ Aw — Xw) = 0. Since R(A — XI) — X, this implies v = w, i.e., v G D{A). Ad (ii). Let A G C be given with Im A ^ 0. We first show that N{A - XI) = {0}. (260) In fact, if Av — Xv = 0, then (Im A)|M|2 = lm{X(v \ v)} = lm(v \ Xv) = Im(t; | Av) = 0, since (Av \ v) = (v \ Av) = (Av \ v). Thus, the operator A - XI: D(A) —► X is injective. By Problem 5.4, ||(A - A/)"1™!! < |Im AI^MI f°r all w e R(A - A/), and R(A - XI) = X. In this connection, observe that A = A*, and hence A is graph closed. Thus, A e p(A). Ad (iii). Use (i) and (ii). Ad (iv). If A is self-adjoint, then A = A*. Hence A is graph closed and N(A* ± ii) = R(A ± ii)1- = {0}. Conversely, let A be graph closed and N(A* ± ii) = {0}. By Problem 5.4, R(A ± ii) = X. Hence A is self-adjoint, by (i). Ad (v). Observe that D(A2 +1) = D(A2) and D(A + iI) = D(A). Hence D((A - U)(A + ii)) = D(A2). Ad (vi). It follows from A2 + / = (A - U)(A + ii) and R(A ±%I) = X that R(A2 +1) = X. By Problem 5.5(i), A2 is self-adjoint. 5.6. The Kato perturbation theorem. Let A: D(A) C X —> X be a linear self-adjoint operator on the complex Hilbert space X, and let B: D(B) C X —> X be a linear symmetric operator such that D(A) C D(B) and ||Bu|| < a||i4u|| + b\\u\\ for all u e D(A), (261)

418 5. Self-Adjoint Operators, the Friedrichs Extension, etc. where a and b are fixed real numbers with 0 < a < 1 and b > 0. Show that A + B is self-adjoint. Solution: Let a G R with a ^ 0. Since ia G p(A), the operator (A - ial)"1: X —> X is linear and continuous. We shall show ahead that ||£(j4 - iaiy1]] < 1 for all a G R: |a| > a0, (262) provided qq is sufficiently large. Since {A + B - iaI)(A - iaiy1 =I + B(A- ial)~\ it follows from (262) that R(A + B - ial) = X for all a G R: |a| > a0 (cf. the Neumann series from Section 1.23). Thus, by Problem 5.5(i), A + B is self-adjoint. Proof of (262). By Problem 5.4(i), \\(A - ial)~lu\\ < lal^H for all ueX. Furthermore, ||yk>||2 + |a|2||v||2 = (Av — iav \ Av — iav) = \\Av - iav\\2 for all v G D(A). Letting v := (A ~ ial)~lu, this implies ||;4(j4 - ial^uf < \\u\\2 for all ueX. Thus, it follows from (261) that \\B(A - ial^uW < a\\A(A - ial)~lu\\ + b\\(A - ial)~lu\\ < (a + b^-^WuW for all u G X This yields (262). 5.7. The Hamiltonian. Set X = l|(R). Let H:=H0 + U, where the continuous function U: R —> R vanishes outside a compact interval. Let Hq := ^ be the free Hamiltonian, where P := 4 ^ with D(P) :={ugX:u'gX}. Show that H is self-adjoint. Solution: Since P:D(P) C X —> X is self-adjoint, by Example 8 in Section 5.2, so is P2. Moreover, /oo /»oo uU2udx < const / uudx < const |M|2. -oo J —oo

Problems 419 The assertion follows now from Problem 5.6. 5.8. The Friedrichs extension in complex Hilbert spaces. Let A: D(A) C X —> X be a symmetric operator on the complex Hilbert space X such that (Au | u) > c\\u\\2 for all u e D(A), where c is a real constant. For fixed AgM with A + c > 0, define (u | v)\ := (Au | v) + X(u \ v), and i \\u\\\ := (u | u)l for all u, v e D(A). Furthermore, let Xx be the set of all the points uGl such that there is an admissible sequence (un) for u, i.e., by definition, (a) un e D(A) for all n, (b) un —> u in X as n —» oo, and (c) (iin) is a Cauchy sequence with respect to || • ||a- Show that, for A,juGl with A + c > 0 and \i + c > 0, the following hold true: (i) Xx is a complex Hilbert space equipped with the inner product (u | v)x := lim (iin | vn)\ for all li, v G Xa, (263) n—*oo where (iin) and (vn) are admissible sequences for u and v, respectively. The limit (263) is independent of the chosen admissible sequences. (ii) Xx = XM; (iii) || • ||a is equivalent to || • H^. (iv) Set D(AF) := D(A*) D Xx and Apu := A*ii for all u e D(AF). Then, the operator Af- D(Ap) ^ X —> X is a self-adjoint extension of A. In addition, (AFu | u) > c|M|2 for all u e D(AF). The operators Ap is called the Friedrichs extension.

420 5. Self-Adjoint Operators, the Priedrichs Extension, etc. Hint: Use the same arguments as in Section 5.3ff. 5.9. A classical inequality. Show that, for all u G Co°(R3), / {u\ + ul + u2c)dx> [ ^dx, (264) Jr3 7r3 4rz where x = (£,77, C)- Solution: Let u G C£°(R3). Set v = r%u. Then u\ + u% + u\= r-l(v\ + v2 + v\) - r-2(v2)r + (4r3)~ V. Observe that, for sufficiently large R, /» /»7T /»27T /»/? / r-2(v2)rdx= / sin6d6dip (v2)rdr = 0, Jr3 Jo Jo Jo since v(x) = 0 for x = 0 and |x| = i?. 5.10. T/ie hydrogen atom and the Priedrichs extension. The Schrodinger equation for the motion of the electron in the hydrogen atom is given through ihi/j = Hi/j, where H := — ^A + U with the Coulomb potential for the electron 1 p1 Ulx) := — r-r on R3 - {0}. Aneo \x\ Here m = mass of the electron, e = electric charge of the electron, and £o = dielectricity constant. Let D(W):=Cg°(R3)c. Show that there is a real number c such that (Hu | u) > c(u | u) for all u G D(H). (265) Consequently, the self-adjoint Priedrichs extension H := Hf exists; H is called the Hamiltonian of the hydrogen atom. Solution: Let u G Co°(R). For each given a > 0, there is a b > 0 such that - < 4 + b for all r > 0. (266) Hence — / —dx > —a -wdx — b u2dx. 7r3 r JR3 rz JR3

Problems 421 Integration by parts yields <*"'">-jU=;«+'*+'S>-s;t}*- Using (264) this implies (Hu | u) > - — / u2dx for all u e C£°(R3), weo JR3 provided we choose the number a sufficiently small in (266). Finally, let w G D(H). Then, w = u + iv, where % v G Cq°(R3). Since u and v are real functions, (Hw | w) = (Hu | u) - i(Hv | u) + i(u \ Hv) + (Hv \ v) = (Hu | u) + (Hv | v) > c(u I u) + c(v I v) = c(w I w;). 5.11**. The spectrum of the hydrogen atom. Show that the spectrum o-(H) of the Hamiltonian H from Problem 5.10 consists of the eigenvalues En:=-^ n = l,2,..., 2 where 7 := ^r^, and of the essential spectrum aess(H) = [0,oo[. Hint: Study the proof in Triebel (1972). In terms of a classical picture, the eigenvalues En and E £ aess(H) correspond to the energy of bounded orbits (Figure 5.16(a)) and unbounded orbits (Figure 5.16(b)), respectively. 5.12. Trace class operators. Let A, B:X —» X be linear operators on the Hilbert space X over K with 0 < dim X < 00. Show the following: (i) The operator A is of trace class, (ii) tr(AB) = tr(BA). (iii) tr A* = trA (iv) tr(aA + 0B) = a tr A + /? tr B. Solution: Ad (i). Let {uj} and {vj} be two complete orthonormal systems in X. Then, by the Dirac calculus from Section 5.21,

422 5. Self-Adjoint Operators, the Friedrichs Extension, etc. electron (a) bound state (b) scattering FIGURE 5.16. and hence ^2(um | Aum) = ^2(um\ Vk)(vk | Avr)(vr I Um) m ra,/c,r = ]P(v/fc I Avr) ^(Vr | Um){Um \ Vk) k.r = ^2(vk | Avr)(vr | vk) = ^2(vr | Avr). k,r r Ad (ii). Observe that tr(AB) = ^2(uk | ABuk) k = Yl(Uk I Avm)(vm | Buk) k,m = Y2(vm | Buk)(uk | Ai;m) /c,ra = J^(t;m|5i4t;m>=tr(Bi4). 5.13. The extension of isometric operators. Let C:D(C) C I -^ I be a linear isometric operator on the Hilbert space X over K, i.e., ||Ct4|| = INI for all u € £>(C). Suppose that £>(C) is a closed linear subspace of X. Let dim £>(C)X < oo and dim jR(C)1- < oo, where "±" denotes the orthogonal complement. Show that the following two statements are equivalent:

Problems 423 (i) There exists a linear unitary operator U: X —» X with C C U. (ii) dim D(C)L= dim R(C)L. Hint: Let {ui,...,un} and {vi,...,vn} be an orthonormal basis of 0(C)1- and R(C)±, respectively. Set Uuj := Vj. 5.14. The Cayley transform. Let A: D(A) C I -> I be a linea^ symmetric operator on the complex Hilbert space X. The operator CA-iA-UHA + il)-1 is called the Cayley transform of A. Show that (i) D{CA) = i?(A + iJ) and £(CU) - R(A - ii). (ii) CU is graph closed iff A is graph closed, (iii) Ca is isometric, i.e., ||Caiz|| = ||ti|| for all u G D{Ca)> (iv) Ca is unitary iff A is self-adjoint. (v) Let B: D{B) C. X -> X be linear and symmetric. Then, ACB iff CUCCB. (vi) If A is graph closed, then D(Ca) and R(Ca) are closed linear sub- spaces of X. Hint: Use Problems 5.3 through 5.5. Cf. Riesz and Nagy (1955), Section 123. 5.15. The extension of symmetric operators to self-adjoint operators. Let A: D(A) C X —» X be a linear, symmetric, graph closed operator on the complex Hilbert space X. The two numbers n± := dim(A* ± ii) are called the defect indices of A. By Problem 5.4, n± := dim R(A±iI)±. Suppose that n± < oo. Show that the following two statements are equivalent: (i) The operator A can be extended to a self-adjoint operator, (ii) n+ = n_.

424 5. Self-Adjoint Operators, the Priedrichs Extension, etc. In particular, A has no proper self-adjoint extension iff n+ =n- = 0. Hint: By Problem 5.14, it is sufficient to study unitary extensions of the Cayley transform Ca- Observe that n+ = dim D(Ca)± and n_ = dim jR(CU)-1. Now use Problem 5.13. 5.16. Essentially self-adjoint operators. Let A: D(A) C X —» X be a symmetric operator on the complex Hilbert space X. Show that the following three statements are equivalent: (i) A is essentially self-adjoint, i.e., by definition, the closure A is self- adjoint. (ii) A has exactly one self-adjoint extension. (iii) N(A*±iI) = {0}. Solution: (i) => (ii). Let B be a self-adjoint extension of A, i.e., ACB and £* = B. Since B is graph closed and A = A**, by Problem 5.4(iv), we get A** C B. Hence B* C (A**)*. This implies 5 a**, since A is self-adjoint. Thus, B = A** = A (i) <^ (iity By Problem 5.5(iv), A is self-adjoint iff N((A)* ± iJ) = {0}. Note that (A)* = A*. (ii) & (iii). Cf. Problem 5.15. 5.17**. The Gelfand-Kostyuchenko theorem on generalized eigenvectors. Set X := L£(R). Let be a linear symmetric operator which can be extended to a self-adjoint operator A: D{A) C X —» X. Then, A has a complete system T = {F} of generalized eigenvectors, i.e., for all F £ T the following hold:20 (i) F £ S' and F ^ 0. (ii) F(Au) = XF(u) for all ueS, where A € R. (iii) F(u) = 0 for all F £ T implies u = 0. This result is the special case of a famous general theorem. Study the proof in Gelfand and Shilov (1964), Vol. 4, Chapter 1, §4. This proof is based on the theory of nuclear spaces and deep results from spectral theory. 'The space S was introduced in Section 3.7.

Epilogue If one does not sometimes think the illogical, one will never discover new ideas in science. Max Planck, 1945 Mathematics is not a deductive science—that's a cliche. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial-and-error, experimentation, and guesswork. Paul Halmos, 1985 The most vitally characteristic fact about mathematics, in my opinion, is its quite peculiar felationship to the natural sciences, or more generally, to any science which interprets experience on a higher more than on a purely descriptive level I think that this is a relatively good approximation to truth— which is much too complicated to allow anything but approximations —that mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure. But, once they are so conceived, the subject begins to live a peculiar life of its own and is better, compared to a creative one, governed by almost entirely aesthetic motivations, than to anything else and, in particular, to an empirical science But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will

426 Epilogue separate into a multitude of insignificant tributaries, and that the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical sources, or after much "abstract" inbreeding, a mathematical object is in danger of degeneration. At the inception, the style is usually classical; when it shows signs of becoming baroque, then the danger signal is up Whenever this stage is reached, then the only remedy seems to be a rejuvenating return to the source: the reinjection of more or less directly empirical ideas. I am convinced that this was a necessary condition to conserve the freshness and the vitality of the subject and that this will remain equally true in the future. John von Neumann, 1947 Mathematics is an ancient art, and from the outset it has been both the most highly esoteric and the most intensely practical of human endeavors. As long ago as 1800 B.C., the Babylonians investigated the abstract properties of numbers; and in Athenian Greece, geometry attained the highest intellectual status. Alongside this theoretical understanding, mathematics blossomed as a day-to-day tool for surveying lands, for navigation, and for the engineering of public works. The practical problems and the theoretical pursuits stimulated one another; it would be impossible to disentangle these two strands. Much the same is true today. In the twentieth century, mathematics has burgeoned in scope and in diversity and has been deepened in its complexity and abstraction. So profound has this explosion of research been that entire areas of mathematics may seem unintelligible to laymen—and frequently to mathematicians working in other subfields. Despite this trend towards—indeed because of it— mathematics has become more concrete and vital than ever before. In the past quarter of a century, mathematics and mathematical techniques have become an integral, pervasive, and essential component of science, technology, and business. In our technically oriented society, "innumeracy" has replaced illiteracy as our principal educational gap. One could compare the contributions of mathematics to our society with the necessity of air and food for life. In fact, we could say that we live in the age of mathematics—that our culture has been "mathematized." No reflection of mathematics around us is more striking than the omnipresent computer There is an exciting development taking place right now, reunification of mathematics with theoretical physics In the last ten or fifteen years mathematicians and physicists realized that modern geometry is in fact the natural framework for gauge theory (cf. Sections 2.20ff in AMS Vol. 109). The gauge potential or gauge theory is the connection of mathematics. The gauge

Epilogue 427 field is the mathematical curvature defined by the connection; certain "charges" in physics are the topological invariants studied by mathematicians. While the mathematicians and physicists worked separately on similar ideas, they did not just duplicate each other's efforts. The mathematicians produced general, far-reaching theories and investigated their ramifications. Physicists worked out details of certain examples which turned out to describe nature beautifully and elegantly. When the two met again, the results are more powerful than either anticipated In mathematics we now have a new motivation to use specific insights from the examples worked out by physicists. This signals the return to an ancient tradition Mathematical research should be as broad and as original as possible, with very long-range goals. We expect history to repeat itself: we expect that the most profound and useful future applications of mathematics cannot be predicted today, since they will arise from mathematics yet to be discovered. Arthur M. Jaflfe, 1984 Mathematics is an organ of knowledge and an infinite refinement of language. It grows from the usual language and world of intuition as does a plant from the soil, and its roots are the numbers and simple geometrical intuitions. We do not know which kind of content mathematics (as the only adequate language) requires; we cannot imagine into what depths and distances this spiritual eye (mathematics) will lead us. Erich Kahler, 1941

Appendix Almost all concepts, which relate to the modern measure and integration theory, go back to the works of Henri Lebesgue (1875-1941). The introduction of these concepts was the turning point in the transition from mathematics of the nineteenth century to mathematics of the twentieth century. Naum Jakovlevic Vilenkin, 1975 For the convenience of the reader we summarize a number of important results about the following topics: the Lebesgue measure; the Lebesgue integral; ordered sets and Zorn's lemma. The Lebesgue Measure Let us consider the space RN for fixed N = 1,2, By an N-cuboid we understand the set C :={&,...,£N) €RN:aj <£j <bj for j = 1,..., TV}, where aj and bj are fixed real numbers with aj < bj for all j. The volume of C is defined through N vol(C) :=£[&■-«*)•

430 Appendix The Lebesgue measure ji generalizes the classical volume of sufficiently regular sets in RN to certain "irregular" sets. More precisely, we have the following quite natural situation. There exists a collection A of subsets of RN which has the following properties: (i) Each open or closed subset of RN belongs to A. (ii) If A, B G A, then AUBgA, AdBgA, and A-BgA. (iii) If An G A for all n = 1,2,..., then oo oo (J An G A and f] An g A. n=l n=l (iv) To each set A in A there is assigned a number /x(A), where 0 < fJ,(A) < op. Here, ji{A) is called the (TV-dimensional) measure of A, and the sets A in A are called measurable (in RN). (v) If A, B G A and A n 5 - 0, then /z(Au£)=/z(.A)+/z(£). If yln € *4 for all n = 1,2,... and An n Am = 0 for all n, m with n ^ m, then (oo \ oo n=l / n=l Here, we use "oo -f- 0 = oo." (vi) If C is an TV-cuboid, then C G A and /x(C) = vol(C). (vii) The subset A of RN has the TV-dimensional measure zero, i.e., AG A and p(A) = 0 iflF, for each e > 0, there is a countable number of TV-cuboids Ci, C2, ... such that 00 00 AQ[)Cj and ^(C^Ke. j=i j=i

Appendix 431 (viii) If the set A has the TV-dimensional measure zero and B C A, then the set B also has the TV-dimensional measure zero. (ix) The collection A is minimal, i.e., if a collection Af satisfies conditions (i) through (viii), then A C A!. Then, the following hold true: The measure ji is unique on A. /j, is called the Lebesgue measure. As usual, we write "meas" instead of /z, i.e., meas(.A) := fj,(A) for all A £ A. Example. A finite or countable number of points in RN has the TV- dimensional measure zero. In particular, the set Q of rational numbers has the one-dimensional measure zero in R, and the set Q* = {(&,...,£*):& €Q for all j} has the TV-dimensional measure4n RN. Convention. By definition, a property P holds true "almost everywhere'''' iff P holds true for all points of RN with the exception of a set of TV- dimensional measure zero. One also uses "almost all." For example, almost all real numbers are irrational. Let M C RN. We write u(x) = lim un(x) for almost all x € M n-+oo iflF this limiting relation holds for all x € M — Z, where the set Z has the TV-dimensional measure zero. Approximation Property. The Lebesgue measure is regular, i.e., for each measurable set M in RN, we have meas(M) = inf meas(G), where the infimum is taken over all the open subsets G of RN with M C G. In particular, meas(RiV) = +00 and meas(0) = 0.

432 Appendix i •— 1— u —c 1 ». a b FIGURE A.l. Step Functions Recall that K = R or K = C. A function u:MCRN ->K is called a step function iff u is piecewise constant. To be precise, we suppose that the set M is measurable and that there exists a finite number of pairwise disjoint measurable subsets Mj of M such that meas(Mj) < oo for all j and , v _ (dj for x G Mj and all j ^ ' ~ \ 0 otherwise, where a,j € K for all j. The integral of a step function u is defined through / udx := yjmeas{M3)aj. Jm i Example. Let u: [a, 6] -> 1 be a step function as pictured in Figure A.l. Then the integral of u defined above is equal to the classic integral. Measurable Functions The function u:M CRN ->K is called measurable iff the following hold: (i) The domain of definition M is measurable. (ii) There exists a sequence (un) of step functions un: M —» K such that u(x) = lim un(x) for almost all x € M.

Appendix 433 Theorem of Luzin. Let M be a measurable subset of RN. Then, the function u:M ->K is measurable iff it is continuous up to small sets, i.e., for each 6 > 0, there is an open subset Ms of M such that the function u:M-M6->K is continuous and meas(Ms) < 6. Standard Example. The function /: M C RN —» K is measurable if it is almost everywhere continuous on the measurable set M (e.g., M is open or closed). Calculus. Linear combinations and limits of measurable functions are again measurable. More precisely, we set F(x) := a{x)u{x) -f- b(x)v(x), G(x) := \u(x)\, (L) H(x) := lim un(x), n—*oo and we assume that the functions a,b,u,v9Un:MCRN -> K are measurable for all n and the limit (L) exists for all x € M. Then, the functions F,G,H:M CRN ->K are also measurable. Modification of Measurable Functions. If we change a measurable function at the points of a set of measure zero, then the modified function is again measurable. For example, if the limit (L) exists only for almost all x G M, i.e., for all x € M—Z with meas(Z) = 0, and if we set H(x) := 0 for all x € Z, then the function H:M —> K is measurable provided all the functions un\M —» K are measurable. The Lebesgue Integral The definition of the Lebesgue integral is based on the very natural formula / udx := lim / undx (A) JM n-*°° Jm

434 Appendix together with the following two formulas u(x) = lim un(x) for almost all x £ M (B) 71—*00 and / \un(x) — um{x)\dx < e for all n, ra > no(e). (C) Jm Definition of the Lebesgue Integral. Let M be a nonempty measurable set. The function u:M C RN —» K is called integrable (over M) iff the following two conditions are satisfied: (i) There is a sequence (un) of step functions un: M —» K such that (B) holds. (ii) For each e > 0, there is a number no(e) such that (C) holds. If u is integrable, then we define the integral through (A). This definition makes sense since the limit exists in (A), and this limit is independent of the choice of the sequence (un). Obviously, each integrable function is measurable. For the empty set M = 0 we define f^udx = 0. We also use synonymously the following terminology: (a) fM u dx exists; (b) u is integrable (over M); (c) \!Mudx\ <OQ Standard Example 1. Let M be a bounded open or compact subset of R^, and suppose that the function u: M -> K is bounded and continuous almost everywhere, i.e., there is a set Z C M with meas(Z) = 0 such that u is continuous on the set M — Z and 1^0*01 < const for all x G M. Then, u is integrable over M. Standard Example 2. Let the function u:M C RN —» K be almost everywhere continuous on the measurable set M (e.g., M = RN). Suppose that const |u(aE)| < 7-—rnr- for all x e M (G)

Appendix 435 and fixed a > N. Then, u is integrable over M. Condition (G) controls the growth of the function u as \x\ —» oo. Standard Example 3. Let the function f:M C RN —» R be almost everywhere continuous on the bounded measurable set M (e.g., M is bounded and open or M is compact). Suppose that there is a point xq in M such that const M#)| ^ i ifl f°r all a; € M with x0 ^ x (H) | re — rcor and fixed (3:0 < (3 < N. Then, ia is integrable over M. Condition (H) controls the growth of the function u as x —» x$. Measure. Let M be a measurable subset of RN with meas(M) < oo. Then dx = meas(M), / JM IM where we write JM dx instead of JM u dx with u = 1. Linearity. Let the functions u, v:M —» K be integrable over M and let a, P £ K. Then, the function an + (3v is also integrable over M and / (au -f- f3v)dx = a udx + (3 vdx. JM JM JM Absolute Integrability. Let u:M C RN —» K be a measurable function. Then / w do; exists iff / |iz|drc exists. JM JM In addition, if one of these two integrals exists, then we have the generalized triangle inequality / udx\ < I \u\dx. JM JM Transformation rule. Let the function u: M C RN —» R be integrable over the nonempty open set M. Suppose that the function /: K —» M is a C1-diffeomorphism21 from the open subset K of RN onto M. Then / u(x)dx= [ u(f(y))detf'(y)dy. JM JK That is, / is bijective and both / and / x are C1.

436 Appendix Here, det f'(y) denotes the determinant of the first partial derivatives of the function / at the point y. Majorant Criterion. Let the function u: M C RN —» K be measurable, and suppose that there exists a function g:M —> R that is integrable over M such that \u>(x)\ < g{x) for almost all x G M. Then, the functions u and \u\ are also integrable over M and / udx < / \u\dx < gdx. \Jm I Jm Jm Vanishing Integrals. Let u:M C RN —» R be a measurable function such that u{x) > 0 for all x e M. Then / udx = 0 iff u(x) = 0 for almost all x € M. M Let the function v:M —» K be integrable. Then, the integral JMvdx remains unchanged if we change the function v at the points of a set of iV-dimensional measure zero. Additivity with respect to domains. Let M and K be two disjoint measurable subsets of RN, and suppose that the function u:M U K —>K is integrable over M and K. Then, u is also integrable over K U M, and / udx = I udx + I udx. Jkum Jk Jm Convergence with respect to domains. Let u: M C RN —» K be a function. Suppose that oo Mi C M2 C ... and M = (J Mn. n=l Then, u is integrable over M iff u is integrable over all sets Mn and supn JM \u\dx < oo. In this case, / udx = lim / udx. Jm n-^°° JMn Absolute Continuity. Let u: M C RN —> K be integrable. Then, for each e > 0, there is a S > 0 such that I f I / udx\ < e \Ja I

Appendix 437 holds true for all subsets A of M with meas(A) < S. Reduction to Bounded Sets. Let M be a nonempty unbounded measurable subset of RN, N = 1,2,..., and let the function u: M —» K be integrable. Then, for each e > 0, there is an open ball B in M.N such that / udx\< I \u\d: \jm-h I Jm-h X < £, where H := M DB. Hence idx \ I udx — ui \Jm Jh < e. Observe that the set H is bounded. p-Mean Continuity. Let u: M C RN —» K be a measurable function on the nonempty bounded measurable set M. Suppose that / |w|pGte < oo Jm for fixed p > 1. Set u(x) := 0 outside M. Then, for each e > 0, there is a 6(e) > 0 such that / Jm \u(x + h) - w(x)|pdo; < e for all h € RN with |h| < 5(e). Limits of Functions and Integrals Theorem on Dominated Convergence. We have lim / undx = / lim un(x)dx, n-^°° Jm Jm n-^°° where all the integrals and limits exist, provided the following two conditions are satisfied: (i) The functions un:M C RN —» K are measurable for all n and the limit lim un(x) exists for almost all x € M. n—>oo (ii) There is an integrable function g:M —» R such that \v>n(x)\ < g(x) for almost all x € M and all n.

438 Appendix Theorem on Monotone Convergence. Let (un) be a sequence of integrable functions un\ M C RN —» R such that 0 < ui(x) < u2(x) < ... for allx e M and / undx < C for all n and fixed C > 0. Jm Then, there exists an integrable function u:M —» R such that u{x) — lim un{x) for almost allx € M n—»-oo and / udx < C. Jm Lemma of Fatou. Le£ (wn) be a sequence of integrable functions un: M C RN -> R. Suppose that (a) wn(a:) > 0 /or all x £ M and all n. (b) JM undx < C for all n. Then / lim un(x)dx < lim / wnGte. More precisely, w(a:) := lim un(x) is finite for almost all x £ M. n—»-oo If we set u(x) := 0 for all the points x of M with limn_,00wn(o;) = oo, then the function u: M —» R is integrable and / udx < lim / wnGfo < C. Jm n->°° Jm Iterated Integration Our goal is the following fundamental formula: / u(x,y)dxdy= (/ u(x, y)dy ) dx Jm Jrn \Jrl J \ ( / u{x, y)dx ) dy. Jrl \Jrn (i)

Appendix 439 Here, we set u(x, y) = 0 outside M. Furthermore, let x £ RN, y GRL, and M C RN+L. Theorem of Fubini. Let u: M C RiV+L —» & 6e integrable. Then formula (I) /io/ds £rwe. 2b be precise, the inner integrals exist for almost all x £ RN (resp., for almost all y £ RL), and the outer integrals exist Theorem of Tonelli. Let u: M C RiV+L -> K 6e measurable. Then the following two conditions are equivalent: (i) The function u is integrable over M. (ii) There exists at least one of the iterated integrals from (I) if u is replaced by \u\, i.e., J (J \u\dy)dx exists or J(J \u\dx)dy exists. If condition (ii) is satisfied, then all the assertions of Fubini's theorem are valid. Special Case. Let M := {(x,y) £ R2:a < x < 6, c < y < d}, where —oo < a < b < oo and —oo < c < d < oo. Then, N = L = 1 and formula (I) reads as follows: / u(x,y)dxdy = / I / u(x, y)dy J dx = / I / u(x,y)dx\dy. Parameter Integrals We consider the function F(p) := / f{x,p)dx, JM for all parameters p £ P. We are given the function /: M x P -> K, where M is a measurable subset of RN and P is a subset of RL or CL. Continuity. The function F: P —» K is well-defined and continuous provided the following three conditions are satisfied: (i) The function x i—► f(x,p) is measurable on M for all parameters (ii) There exists an integrable function g: M —» R such that |/(£jP)| < ^(^) f°r all p € P and almost all x £ M.

440 Appendix (iii) The function p \-> f(x,p) is continuous on P for almost all x G M. Differentiability. Let P be a nonempty open subset of R or C. Then, the function F: P —» K is differentiate and F\p) = / fp(x,p)dx for all p £ P, Jm provided the following two conditions are satisfied: (i) The integral JM f(x,p)dx exists for all parameters p G P. (ii) There exists an integrable function g: M —» R such that |/p(#?p)| < #0*0 f°r all p € P and almost all x £ M. This condition tacitly includes the existence of the partial derivative fv{x,p) for all p e P and almost all x G M. Functions of Bounded Variation Let —oo<a<6<oo. The function g: [a, b] —» C is called of bounded variation iff V(g) := inf £ |5(4»>) - 5(4"_\)| < oo, (1) where the infimum is taken over all the possible finite decompositions V of the interval [a, 6], i.e., a = 4n) < 4n) < * * * < 4n) = & with n = 1,2,... . (2) The number V(g) is called the fofaZ variation of the function # on the interval [a, 6]. Theorem of Jordan. The function g:[a,b] —» C zs 0/ bounded variation iff there exist nondecreasing functions gy. [a, b] —» R, j = 1,2,3,4, 5^c/i £/ia£ #(z) = #i(a:) - g2(x) + z'(^(^) - #4(2)) for all x G [a, b]. (3) 77ie Classic Stieltjes Integral We are given the continuous function /: [a, 6] —» C and the function g: [a, 6] —» C of bounded variation, where —00 < a < b < 00. Then, there exists the limit f f(x)dg{x) := lim ^f^M^-g^)), (4)

Appendix 441 which is independent of the decomposition of the interval [a, b] from (2). Hence / f(x)dg(x)\<( max \f(x)\)v(g). Ja I \a<x<b J If the function /:R —» C is continuous and the function g:R —► C is of bounded variation on each compact interval, then we set /oo pb f{x)dg{x) := lim / f(x)dg(x), -oo fc-^+oo Ja a—► — oo provided this limit exists. The Lebesgue-Stieltjes Integral If the function / is not continuous, then one introduces the so-called Lebes- gues-Stieltjes integral which is identical to the Lebesgue integral in the special case where g(x) := x for all x G R. A summary of important properties of the Lebesgue-Stieltjes integral including measure theory can be found in Zeidler (1986), Vol. 2B, Appendix. Standard Example. Let — oo < a < b < oo. Then, the formula / f(x)dg(x) = f f(x)g'(x)dx (5) J a J a holds true provided the following assumptions are satisfied: (i) The functions /, h: ]a, 6[—> C are measurable, and the functions h and fh are integrable over ]a, 6[, in the sense of the Lebesgue integral. (ii) For all x G ]a, 6[, pX g(x) := / h(y)dy. Ja More precisely, under the assumptions (i) and (ii), the left-hand integral from (5) exists in the sense of a Lebesgue-Stieltjes integral, whereas the right-hand integral from (5) exists in the sense of a Lebesgue integral with g' = h. If, in addition, / is continuous on the closure of ]a,b[, then the left-hand integral from (5) exists in the sense of a classic Stieltjes integral. Ordered Sets and Zorn's Lemma The set C is called ordered iff there is a relation, written as u <v, among some pairs of elements of C such that the following hold:

442 Appendix (i) u <u for all u £ C. (ii) If u < v and v <w, then u <w. (iii) If u < v and v < u, then u = v. By a maximal element m of C we understand an element of C such that m <u and uGC imply m = u. A nonempty subset T of C is called totally ordered iflF, for all w,i; € T, we have w < w or v < u. Zorn's Lemma. Let C be a nonempty ordered set which has the property that each totally ordered subset TofC has an upper bound, i.e., there is an element b of C such that u <b for all u G T, where b depends on T. Then, there exists a maximal element in C. Example 1. Let S be a set, and let C be the collection of all the subsets of S. For u,v £C, we write u < v iflF u Cv. Then, C becomes an ordered set. Example 2. The set E of real numbers is totally ordered, but R does not have any maximal element. Zorn's lemma can be used in mathematics if the usual induction argument fails, since the set under consideration is not countable. In Section 1.1 of AMS Vol. 109 we use Zorn's lemma in order to prove the Hahn-Banach theorem.

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446 References Deuflhard, P. and Hohmann, A. (1993): Numerische Mathematik I. De Gruyter, Berlin. (English edition: Numerical Analysis: A First Course in Scientific Computation. De Gruyter, Berlin, 1994.) Deuflhard, P. and Bornemann, F. (1994): Numerische Mathematikll. Integration gewohnlicher Differentialgleichungen. De Gruyter, Berlin. (English edition in preparation.) DeVito, C. (1990): Functional Analysis and Linear Operator Theory. Addison-Wesley, Reading, MA. Dierkes, U., Hildebrandt, S., Kiister, A., and Wohlrab, O. (1992): Minimal Surfaces, Vols. 1, 2. Springer-Verlag, Berlin, Heidelberg. Dieudonne, J. (1969): Foundations of Modern Analysis. Academic Press, New York. Dieudonne, J. (1981): History of Functional Analysis. North-Holland, Amsterdam. Dieudonne, J. (1992): Mathematics—the Music of Reason. Springer-Verlag, Berlin, Heidelberg. Dittrich, W. and Reutter, M. (1994): Classical and Quantum Dynamics from Classical Paths to Path Integrals. Springer-Verlag, Berlin, Heidelberg. Donoghue, J., Golowich, E., and Holstein, B. (1992): The Dynamics of the Standard Model. Cambridge University Press, Cambridge, UK. Dunham, W. (1991): Journey Through Genius: The Great Theorems of Mathematics. Penguin Books, New York. Dunford, N. and Schwartz, J. (1988): Linear Operators, Vols. 1-3. Wiley, New York. Dyson, F. (1979): Disturbing the Universe. Harper & Row, New York. Economou, E. (1988): Green's Functions in Quantum Physics. Springer- Verlag, New York. Edwards, R. (1994): Functional Analysis. Dover, New York. Ekeland, I. and Temam, R. (1974): Analyse convex et problemes variationals. Dunod, Paris. (English edition: North-Holland, New York, 1976). Ekeland, I. (1990): Convexity Methods in Hamiltonian Mechanics. Springer- Verlag, New York. Esposito, G. (1993): Quantum Gravity, Quantum Cosmology, and Lorent- zian Geometries. Springer-Verlag, New York. Evans, L. (1994): Partial Differential Equations. Berkeley Mathematics Lecture Notes, Vols. 3A and 3B. University of Berkeley, CA. Fenyo, S. and Stolle, H. (1982): Theorie und Praxis der linearen Integral- gleichungen, Vols. 1-4. Deutscher Verlag der Wissenschaften, Berlin. Feynman, R., Leighton, R., and Sands, M. (1963): The Feynman Lectures in Physics. Addison-Wesley, Reading, MA.

References 447 Feynman, R. and Hibbs, A. (1965): Quantum Mechanics and Path Integrals. McGraw-Hill, New York. Finn, R. (1985): Equilibrium Capillary Surfaces. Springer-Verlag, Berlin, Heidelberg. Friedman, A. (1982): Variational Principles and Free Boundary-Value Problems. Wiley, New York. Friedman, A. (1989/94): Mathematics in Industrial Problems, Vols. 1-6. Springer-Verlag, New York. Gajewski, H., Groger, K., and Zacharias, K. (1974): Nichtlineare Operator- gleichungen. Akademie-Verlag, Berlin. Galdi, G. (1994): An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vols. 1-4. Springer-Verlag, Berlin, Heidelberg (Vols. 3 and 4 to appear). Gelfand, I. and Shilov, E. (1964): Generalized Functions, Vols. 1-5. Academic Press, New York. (Translated from the Russian.) Gell-Mann, M. (1994): The Quark and the Jaguar: Adventures in the Simple and the Complex. Freeman, San Francisco, CA. Giaquinta, M. (1993): Introduction to Regularity Theory for Nonlinear Elliptic Systems. Birkhauser, Basel. Giaquinta, M. and Hildebrandt, S. (1995): Calculus of Variations, Vols. 1, 2. Springer-Verlag, New York. Gilbarg, D. and Trudinger, N. (1994): Elliptic Partial Differential Equations of Second Order. 2nd edition. Springer-Verlag, New York. Gilkey, P. (1984): Invariance Theory, the Heat Equation, and the Atiyah- Singer Index Theorem. Publish or Perish, Boston, MA. Glimm, J. and Jaffe, A. (1981): Quantum Physics. Springer-Verlag, New York. Golub, G. and Ortega, J. (1993): Scientific Computing: An Introduction with Parallel Computing. Academic Press, New York. Green, M., Schwarz, J., and Witten, E. (1987): Superstrings, Vols. 1,2. University Press, Cambridge, UK. Greiner, W. (1994): Classical Physics, Vols. Iff. Springer-Verlag, New York. Greiner, W. (1993/94): Theoretical Physics, Vols. 1-6. Cf. the following titles. Greiner, W. (1994): Quantum Mechanics: An Introduction. Springer- Verlag, Berlin, Heidelberg. Greiner, W. and Mtiller, B. (1994): Quantum Mechanics: Symmetries. Springer-Verlag, Berlin, Heidelberg. Greiner, W. (1993): Relativistic Quantum Mechanics. Springer-Verlag, Berlin, Heidelberg.

448 References Greiner, W. (1993): Gauge Theory of Weak Interactions. Springer-Verlag, Berlin, Heidelberg. Greiner, W. and Reinhardt, J. (1994): Quantum Electrodynamics. Springer- Verlag, Berlin, Heidelberg. Greiner, W. and Schafer, A. (1994): Quantum Chromodynamics. Springer- Verlag, Berlin, Heidelberg. Grosche, C. and Steiner, F. (1995): A Table of Feynman Path Integrals. Springer-Verlag, Berlin, Heidelberg (to appear). Grosche, G., Ziegler, D., Ziegler, V., and Zeidler, E. (eds.) (1995): Teubner - Taschenbuch der Mathematik II. Teubner-Verlag, Stuttgart, Leipzig (English edition in preparation). Grosse, H. (1995): Models in Statistical Physics and Quantum Field Theory. Springer-Verlag, Berlin, Heidelberg (to appear). Gruber, P. and Wills, J. (1993): Handbook of Convex Geometry, Vols. 1, 2. North-Holland, Amsterdam. Guillemin, V. and Sternberg, S. (1990): Symplectic Techniques in Physics. Cambridge University Press, Cambridge, UK. Haag, R. (1993): Local Quantum Physics: Fields, Particles, Algebras. Springer-Verlag, Berlin, Heidelberg. Hale, J. and Kogak, H. (1991): Dynamics of Bifurcations. Springer-Verlag, Berlin, Heidelberg (cf. also Kogak (1989)). Hatfield, B. (1992): Quantum Field Theory of Point Particles and Strings. Addison-Wesley, Redwood City, CA. Heisenberg, W. (1989): Encounters with Einstein and Other Essays on People, Places, and Particles. Princeton University Press, Princeton, NJ. Henneaux, M. and Teitelboim, C. (1993): Quantization of Gauge Systems. Princeton University Press, Princeton, NJ. Henry, D. (1981): Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, Vol. 840. Springer-Verlag, New York. Hermann, C. and Sapoval, B. (1994): Physics of Semiconductors. Springer- Verlag, New York. Heuser, H. (1975): Funktionalanalysis. Teubner-Verlag, Stuttgart. (English edition: Functional Analysis, Wiley, New York, 1982.) Hilbert, D. (1912): Grundzuge einer allgemeinen Theorie der Integralglei- chungen. Teubner-Verlag, Leipzig. Hilbert, D. (1932): Gesammelte Werke (Collected Works), Vols. 1-3. Springer-Verlag, Berlin. Hildebrandt, S. and Tromba, T. (1985): Mathematics and Optimal Form. Scientific American Library, Freeman, New York. Hiriart-Urruty, J. and Lemarchal, C. (1993): Convex Analysis and Minimization Algorithms, Vols. 1, 2. Springer-Verlag, Berlin, Heidelberg.

References 449 Hirzebruch, F. and Scharlau, W. (1971): Einfilhrung in die Funktionalana- lysis. Bibliographisches Institut, Mannheim. Hofer, H. and Zehnder, E. (1994): Symplectic Invariants and Hamiltonian Dynamics. Birkhauser, Basel. Holmes, R. (1975): Geometrical Functional Analysis and Its Applications. Springer-Verlag, New York. Honerkamp, J. and Romer, H. (1993): Theoretical Physics: A Classical Approach. Springer-Verlag, New York. Hormander, L. (1983): The Analysis of Linear Partial Differential Operators; Vol. 1: Distribution Theory and Fourier Analysis; Vol. 2: Differential Operators with Constant Coefficients; Vol. 3: Pseudodifferen- tial Operators; Vol. 4: Fourier Integral Operators. Springer-Verlag, New York. Huang, K. (1992): Quarks, Leptons, and Gauge Fields. 2nd edition. World Scientific, Singapore. Iagolnitzer, D. (1993): Scattering in Quantum Field Theory. Princeton University Press, Princeton, NJ. Isham, C. (1989): Modern Differential Geometry for Physicists. World Scientific, Singapore. John, F. (1982): Partial Differential Equations. Springer-Verlag, New York. Jost, J. (1991): Two-Dimensional Geometric Variational Problems. Wiley, New York. Jost, J. (1994): Differentialgeometrie und Minimalflachen. Springer-Verlag, Berlin, Heidelberg. Jost, J. (1996): Postmodern Analysis. Springer-Verlag, Berlin, Heidelberg (to appear). Kac, M., Rota, G., and Schwartz, J. (1992): Discrete Thoughts: Essays on Mathematics, Science, and Philosophy. Birkhauser, Basel. Kadison, R. and Ringrose, J. (1983): Fundamentals of the Theory of Operator Algebras, Vols. 1-4. Academic Press, New York. Kaiser, G. (1994): A Friendly Guide to Wavelets. Birkhauser, Basel. Kaku, M. (1987): Introduction to Superstring Theory. Springer-Verlag, New York. Kaku, M. and Trainer, J. (1987): Beyond Einstein: The Cosmic Quest for the Theory of the Universe. Bantam Books, New York. Kaku, M. (1991): Strings, Conformal Fields, and Topology. Springer- Verlag, New York. Kaku, M. (1993): Quantum Field Theory. Oxford University Press, Oxford. Kantorovich, L. and Akilov, G. (1964): Functional Analysis in Normed Spaces. Pergamon Press, Oxford. (Translated from the Russian.)

450 References Kanwal, R. (1983): Generalized Functions. Academic Press, New York. Kato, T. (1976): Perturbation Theory for Linear Operators. Springer- Verlag, Berlin. Kevasan, S. (1989): Topics in Functional Analysis and Applications. Wiley, New York. Kevorkian, J. and Cole, J. (1995): Multiple Scale and Singular Perturbation Methods. Springer-Verlag, New York (to appear). Kirillov, A. and Gvishiani, A. (1982): Theory and Problems in Functional Analysis. Springer-Verlag, New York. Kogak, H. (1989): Differential and Difference Equations Through Computer Experiments. With Diskettes. Springer-Verlag, New York (cf. also Hale and Kogak (1991)). Kolmogorov, A., Fomin, S., and Silverman, R. (1975): Introductory Real Analysis. Dover, New York. (Enlarged translation from the Russian.) Kolmogorov, A. and Fomin, S. (1975): Reelle Funktionen und Funktional- analysis. Deutscher Verlag der Wissenschaften, Berlin. (Translated from the Russian.) Krasnoselskii, M. and Zabreiko, P. (1984): Geometrical Methods in Nonlinear Analysis. Springer-Verlag, New York. (Translated from the Russian.) Kress, R. (1989): Linear Integral Equations. Springer-Verlag, New York. Kreyszig, E. (1989): Introductory Functional Analysis with Applications. Wiley, New York. Kufner, A., John, O., and Fucik, S. (1977): Function Spaces. Academia, Prague. Kufner, A. and Fucik, S. (1980): Nonlinear Differential Equations. Elsevier, New York. Landau, L. and Lifsic, E. (1982): Course of Theoretical Physics, Vols. 1-10. Elsevier, New York. Lang, S. (1993): Real Analysis. 3rd edition. Springer-Verlag, New York. Lazutkin, V. (1993): KAM-Theory and Semiclassical Approximations to Eigenfunctions. Springer-Verlag, Berlin, Heidelberg. Leis, R. (1986): Initial-Boundary Value Problems in Mathematical Physics. Wiley, New York. Leung, A. (1989): Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering. Kluwer, Dordrecht. LeVeque, R. (1990): Numerical Methods for Conservation Laws. Birk- hauser, Basel. Levitan, B. and Sargsjan, I. (1991): Sturm-Liouville and Dirac Operators. Kluwer, Boston, MA. (Translated from the Russian.)

References 451 Lions, J. (1969): Quelques methodes de resolution des problemes aux limites nonlineaires. Dunod, Paris. Lions, J. (1971): Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, Berlin. (Translated from the French.) Lions, J. and Magenes, E. (1972): Inhomogeneous Boundary-Value Problems, Vols. 1-3. Springer-Verlag, New York. Louis, A. (1995): Inverse and Ill-Posed Problems. Springer-Verlag, New- York (to appear). Luenberger, D. (1969): Optimization by Vector Space Methods. Wiley, New York. Lust, D. and Theissen, S. (1989): Lectures on String Theory. Springer- Verlag, Berlin, Heidelberg. Lusztig, G. (1993): Introduction to Quantum Groups. Birkhauser, Boston, MA. Mackey, G. (1963): The Mathematical Foundations of Quantum Mechanics. Benjamin, New York. Mackey, G. (1992): The Scope and History of Commutative and Noncom- mutative Harmonic Analysis. American Mathematical Society, Providence, RI. Mandl, F. and Shaw, G. (1989): Quantum Field Theory. Wiley, New York. Marathe, K. and Martucci, G. (1992): The Mathematical Foundations of Gauge Theory. North-Holland, Amsterdam. Marchioro, C. and Pulvirenti, M. (1994): Mathematical Theory of Inviscid Fluids. Springer-Verlag, New York. Markowich, P. (1990): Semiconductor Equations. Springer-Verlag, Berlin, Heidelberg. Marsden, J. (1992): Lectures in Mechanics. Cambridge University Press, Cambridge, UK. Marsden, J. and Ratiu, T. (1994): Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems. Springer- Verlag, New York. Matveev, V. (1994): Algebro - Geometrical Approach to Nonlinear Evolution Equations. Springer-Verlag, New York. Mawhin, J. and Willem, M. (1987): Critical Point Theory and Hamiltonian Systems. Springer-Verlag, New York. Maurin, K. (1972): Methods of Hilbert Spaces. Polish Scientific Publishers, Warsaw. Meyer, K. and Hall, G. (1992): Introduction to Hamiltonian Dynamical Systems and the N-Body Problem. Springer-Verlag, New York.

452 References Mielke, A. (1991): Hamiltonian and Lagrangian Flows on Center Manifolds with Applications to Elliptic Variational Problems. Lecture Notes in Mathematics, Vol. 1489. Springer-Verlag, Berlin, Heidelberg. Monastirsky, M. (1993): Topology of Gauge Fields and Condensed Matter. Plenum Press, New York. Murray, J. (1989): Mathematical Biology. Springer-Verlag, Berlin, Heidelberg. Nakahara, M. (1990): Geometry, Topology, and Physics. Hilger, Bristol. Necas, J. (1967): Les methodes directes en theorie des equations elliptiques. Academia, Prague. Neumann, J.v. (1932): Mathematische Grundlagen der Quantenmechanik. Springer-Verlag, Berlin. (English edition: Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, NJ, 1955.) Newton, R. (1988): Scattering Theory of Waves and Particles. Springer- Verlag, Berlin, Heidelberg. Nikiforov, A. and Uvarov, V. (1987): Special Functions of Mathematical Physics. Birkhauser, Boston, MA. (Translated from the Russian.) Nishikawa, K. and Wakatani, M. (1993): Plasma Physics: Basic Theory with Fusion Applications. Springer-Verlag, Berlin, Heidelberg. Novikov, S. et al. (1984): Theory of Solitons. Plenum Press, New York. (Translated from the Russian.) Oberguggenberger, M. (1992): Multiplication of Distributions and Applications to Partial Differential Equations. Harlow, Longman, UK. Pazy, A. (1983): Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York. Peebles, P. (1991): Quantum Mechanics. Princeton University Press, Princeton, NJ. Peebles, P. (1993): Principles of Physical Cosmology. Princeton University Press, Princeton, NJ. Penrose, R. (1992): The Emperor's New Mind Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Oxford. Penrose, R. (1994): Shadows of the Mind: The Search for the Missing Science of Conciousness. Oxford University Press, Oxford. Polyakov, A. (1987): Gauge Fields and Strings. Academic Publishers, Har- wood, NJ. Prugovecki, E. (1981): Quantum Mechanics in Hilbert Space. Academic Press, New York. Quarteroni, A. and Valli, A. (1994): Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin, Heidelberg. Rabinowitz, P. (1986): Methods in Critical Point Theory with Applications. Amer. Math. Soc, Providence, RI.

References 453 Racke, R. (1992): Lectures on Evolution Equations. Vieweg, Braunschweig. Rauch, J. (1991). Partial Differential Equations. Springer-Verlag, New York. Reed, M. and Simon, B. (1972): Methods of Modern Mathematical Physics. Vol. 1: Functional Analysis; Vol. 2: Fourier Analysis, Self-Adjointness; Vol. 3: Scattering Theory; Vol. 4: Analysis of Operators. Academic Press, New York. Reid, C. (1970): Hilbert. Springer-Verlag, New York. Reid, C. (1976): Courant in Gottingen and New York. Springer-Verlag, New York. Renardy, M. and Rogers, R. (1993): Introduction to Partial Differential Equations. Springer-Verlag, New York. Riesz, F. and Nagy, B. (1955): Legons d'analyse fonctionelle. (English edition: Functional Analysis, Frederick Ungar, New York 1978.) Rivers, R. (1990): Path Integral Methods in Quantum Field Theory. University Press, Cambridge, UK. Rolnick, W. (1994): Fundamental Particles and Their Interactions. Addison-Wesley, Reading, MA. Royden, H. (1988): Real Analysis. Macmillan, New York. Rudin, W. (1966): Real and Complex Analysis. McGraw-Hill, New York. Rudin, W. (1973): Functional Analysis. McGraw-Hill, New York. Ruelle, D. (1993): Chance and Chaos. Princeton University Press, Princeton, NJ. Sakai, A. (1991): Operator Algebras. Cambridge University Press, Cambridge, UK. Sattinger, D. and Weaver, O. (1993): Lie Groups, Lie Algebras, and Their Representations. Springer-Verlag, New York. Schechter, M. (1971): Principles of Functional Analysis. Wiley, New York. Schechter, M. (1982): Operator Methods in Quantum Mechanics. North- Holland, Amsterdam. Schechter, M. (1986): Spectra of Partial Differential Operators. North - Holland, Amsterdam. Schmutzer, E. (1989): Grundlagen der theoretischen Physik, Vols. 1, 2. Deutscher Verlag der Wissenschaften, Berlin. Scott, G. and Davidson, K. (1994): Wrinkles in Time. Morrow, New York. Simon, B. (1993): The Statistical Mechanics of Lattice Gases. Princeton University Press, Princeton, NJ. Smoller, J. (1994): Shock Waves and Reaction-Diffusion Equations. 2nd enlarged edition. Springer-Verlag, New York. Spohn, H. (1991): Large Scale Dynamics of Interacting Particles. Springer- Verlag, Berlin, Heidelberg.

454 References Sterman, G. (1993): An Introduction to Quantum Field Theory. Cambridge University Press, Cambridge, UK. Strang, G. and Fix. G. (1973): An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs, NJ. Struwe, M. (1988): Plateau's Problem and the Calculus of Variations. Princeton University Press, Princeton, NJ. Struwe, M. (1990): Variational Methods. Springer-Verlag, New York. Sunder, V. (1987): An Invitation to von Neumann Algebras. Springer- Verlag, New York. Szabo, I. (1987): Geschichte der mechanischen Prinzipien und ihrer wich- tigsten Anwendungen. Birkhauser, Basel. Temam, R. (1988): Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer-Verlag, New York. Thaller, B. (1992): The Dirac Equation. Springer-Verlag, Berlin, Heidelberg. Thirring, W. (1991): A Course in Mathematical Physics. Vol. 1: Classical Dynamical Systems; Vol. 2: Classical Field Theory; Vol. 3: Quantum Mechanics of Atoms and Molecules; Vol. 4: Quantum Mechanics of Large Systems. Springer-Verlag, New York. Thorne, K. (1994): Black Holes and Time Warps: Einstein's Outrageous Legacy. Toda, M. (1989): Nonlinear Waves and Solitons. Kluwer, Dordrecht. Triebel, H. (1972). Hohere Analysis. Verlag der Wissenschaften, Berlin. Triebel, H. (1987): Analysis and Mathematical Physics. Kluwer, Dordrecht. Triebel, H. (1992): Theory of Function Spaces II. Birkhauser, Basel. Visintin, A. (1994): Differentiate Models of Hysteresis. Springer-Verlag, Berlin, Heidelberg. Weinberg, S. (1992): Dreams of a Final Theory. Pantheon Books, New York. Wendland, W. (1996): Integral Equation Methods for Boundary-Value Problems. Springer-Verlag, Berlin, Heidelberg (to appear). Wess, J. and Bagger, J. (1991): Supersymmetry and Supergravity. Second edition revised and expanded. Princeton University Press, Princeton, NJ. Wiggins, S. (1990): Introduction to Applied Dynamical Systems and Chaos. Springer-Verlag, New York. Yosida, K. (1988): Functional Analysis. 5th edition. Springer-Verlag, New York. Yosida, K. (1991): Lectures on Differential and Integral Equations. Dover, New York.

References 455 Zabczyk, J. (1992): Optimal Control Theory. Birkhauser, Basel. Zeidler, E. (1986): Nonlinear Functional Analysis and Its Applications. Vol. 1: Fixed-Point Theorems', Vol. 2A: Linear Monotone Operators; Vol. 2B: Nonlinear Monotone Operators; Vol. 3: Variational Methods and Optimization; Vols. 4, 5: Applications to Mathematical Physics. Springer- Verlag, New York. (Second enlarged edition of Vol. 1, 1992; second enlarged edition of Vol. 4, 1995, Vol. 5 in preparation.) Zuily, C. (1988): Problems in Distributions and Partial D<Jere, tial Equations. North-Holland, Amsterdam. Hints for Further Reading Comprehensive collection of exercises: Kirillov and Gvishiani (1982). History of functional analysis: Dieudonne (1981), Mackey (1992) (harmonic analysis). International mathematical congresses: Albers, Alexanderson, and Reid (1987). Biographies of Hilbert and Courant: Reid (1970), (1976). A summary of important material from linear functional analysis: appendices to Zeidler (1986), Vols. 1, 2B, and 3. Comprehensive bibliographies: Zeidler (1986), Vols. 1-5. Classical textbooks on linear functional analysis: Riesz and Nagy (1955), Schechter (1971), Rudin (1973), Kolmogorov and Fomin (1975), Kato (1976), Dunford and Schwartz (1988), Yosida (1988). Nonlinear functional analysis: Berger (1977), Aubin and Ekeland (1983), Deimling (1985), Zeidler (1986ff), Vols. 1-5, Ambrosetti (1993). Operator algebras: Kadison and Ringrose (1983), Vols. 1-4, Sunder (1987), Sakai (1991), Bellissard (1996). Generalized functions, pseudodifferential operators, and Fourier integral operators: Hormander (1983), Vols. 1-4, Kanwal (1983). Function spaces: Kufner, John, and Fucik (1980), Triebel (1992). Applications to partial differential equations: Leis (1986), Zeidler (1986), Vols. 1-5, Dautray and Lions (1990), Vols. 1-6, Alt (1992), Racke (1992), Giaquinta (1993), Renardy and Rogers (1993), Evans (1994), Smoller (1994), Amann (1995). Applications to the calculus of variations: Friedman (1982), Rabi- nowitz (1986), Zeidler (1986), Vol. 3, Mawhin and Willem (1987), Struwe (1990), Jost (1991), (1994), Bourguignon (1995), Guiaquinta and Hilde- brandt (1995), Chang (1996).

456 References Minimal surfaces: Dierkes, Hildebrandt, Ktister, and Wohlrab (1992). Applications to integral equations: Kress (1989), Dautray and Lions (1990), Vol. 4. Applications to optimization and mathematical economics: Lu- enberger (1969), Zeidler (1986), Vol. 3, Aubin (1993), Zabczyk (1993). Numerical functional analysis: Zeidler (1986), Vols. 2A, 2B, and 3, Dautray and Lions (1990), Vols. 1-6, Louis (1995). Scientific computing: Allgower and Georg (1990), LeVeque (1990), Golub and Ortega (1993), Deuflhard and Hohmann (1993), Deuflhard and Bornemann (1994), Quarteroni and Valli (1994). Applications to industrial problems: Friedman (1989/94), Vols. 1-6. Applications to the natural sciences: Zeidler (1986), Vols. 4 and 5, Dautray and Lions (1990), Vols. 1-6, Grosche, Ziegler, and Zeidler (1995) (handbook). Applications to mechanics: Marsden (1992). Applications to celestial mechanics: Meyer and Hall (1992), Am- brosetti and Coti-Zelati (1993). Applications to dynamical systems: Temam (1988), Amann (1990), Wiggins (1990), Hale and Kogak (1991), Mielke (1991), Hofer and Zehnder (1994), Marsden and Ratiu (1994). Manifolds: Abraham, Marsden, and Ratiu (1983), Zeidler (1986), Vol. 4, Isham (1989). Applications to mathematical biology: Murray (1989). Applications to nonlinear elasticity: Ciarlet (1983), Zeidler (1986), Vol. 4, Antman (1994). Applications to fluid mechanics: Zeidler (1986), Vol. 4, Galdi (1994), Marchioro and Pulvirenti (1994). Solitons: Novikov (1984), Toda (1989), Matveev (1994). Applications to capillarity: Finn (1985). Large scale dynamics of multi-particle systems: Spohn (1991), Cercigniani, Illner, and Pulvirenti (1995). Hysteresis effects: Visentin (1994). Semiconductors: Markowich (1990). Plasma physics and fusion: Nishikawa and Wakatani (1993). Symplectic techniques in physics: Guillemin and Sternberg (1990), Hofer and Zehnder (1994). Applications to quantum mechanics: Reed and Simon (1972), Vols. 1-4, Prugovecki (1981), Schechter (1982), Berezin and Shubin (1991). Quantum statistics: Bratelli and Robinson (1979), Haag (1993), Simon (1993), Grosse (1995), Bellissard (1996).

References 457 Quantum field theory: Glimm and Jaffe (1981), Reed and Simon (1972), Vol. 2 (the Garding-Wightman axioms), Mandl and Shaw (1989), Haag (1993), Kaku (1993), Schwarz (1993), Sterman (1993), Grosse (1995). Scattering theory: Reed and Simon (1972), Vol. 3, Newton (1988), Colton and Kress (1992), Iagolnitzer (1993). Elementary particles: Rolnick (1994). Standard model of elementary particles: Donoghue, Golowich, and Holstein (1992), Kaku (1993). Noncommutative geometry and the standard model of elementary particles: Connes (1994). The Feynman path integral: Albeverio (1975) as well as Albeverio and Brezniak (1993) (rigorous theory), Dittrich and Reutter (1994), Rivers (1990), Sterman (1994), Kaku (1993). Cosmology: Zeidler (1986), Vol. 4, Peebles (1993). Quantum cosmology: Esposito (1993). Supersymmetry: Berezin (1987), Wess and Bagger (1991). Superstring theory: Green, Schwartz, and Witten (1987), Kaku (1987), Lust and Theissen (1989), Hatfield (1992). Quantum groups: Lusztig (1993). Conformal field theory: Kaku (1991). Topology and physics: Nakahara (1990), Monastirsky (1993). Topology, partial differential equations, pseudodifferential operators, and the Atiyah-Singer index theorem: Gilkey (1984). Textbooks in physics: Feynman, Leighton, and Sands (1963), Schmut- zer (1989), Vols. 1, 2, Greiner (1993), Vols. 1-6, Honerkamp and Romer (1993). A survey on modern physics: Davies (1989). Essays on modern physics: Kaku and Trainer (1987), Heisenberg (1989), Weinberg (1992), Gell-Mann (1994), Scott and Davidson (1994), Thome (1994). Essays on modern mathematics: Cascuberta and Castellet (1992) (viewpoints of seven Fields medalists), Penrose (1992), (1994), Kac, Rota, and Schwartz (1994).

List of Symbols What's in a name? That which we call a rose By any other word would smell as sweet. William Shakespeare (1564-1616) Romeo and Juliet 2,2 General Notation A=> B A implies B iff if and only if A & B AiftB (i.e., A =» B and B => A) f(x):=2x f(x) = 2xby definition x e S # is an element of the set 5 x # S £ is not an element of the set S {x: ...} set of all elements x with the property ... S CT the set S is contained in the set T S CT S CT and S^T (the set S is properly contained in T SuT the union of the sets S and T (the set of all elements that live in S or T) S C\T the intersection of the sets S and T (the set of all elements that live in S and T) S — T the difference set (the set of all elements that live in S and not in T) 0 empty set 2^ set of all subsets of S (the power set of S)

460 List of Symbols SxT W *•> ^) ^£5 piV xiV KN Re z, Im z z \z\ [a,b] ]a,b[ ]a,b] [a,b[ sgn r $jk MS sup S min 5 max S lim an n—»oo limn^ooan product set {(x,y):x e S and y G T} set of the single point p set of the natural numbers 1,2,... set of the real, complex, rational, integer numbers lorC set of all real iV-tupels x = (xi,..., xn) (i.e., Xj G R for all j) set of all complex 7V-tupels (rci,..., xn) (i.e., Xj G C for all j) RN or CN real part of the complex number z = x -f- yi, imaginary part of z (i.e., Re z := #, Im z := ?/) conjugate complex number z := x — yi, absolute value of the complex number z, \z\ := \/x2 + y2 closed interval (the set {x eR: a < x < b}) open interval (the set {x G R: a < x < b}) half-open interval (the set {x G R: a < x < b}) half-open interval (the set {x G R: a < x < b}) signum of the real number r Kronecker symbol, 6jk := 1 if j = fc, and 6jk := 0 if j'^ k infimum of the set S of real numbers (the largest lower bound of S) supremum of the set S of real numbers (the smallest upper bound of S) the minimum of the set S of real numbers (the smallest element of S) the maximum of the set S of real numbers (the largest element of S) lower limit of the real sequence (an) upper limit of the real sequence (an) The Landau Symbols f(x) = 0(g(x)), 1/0*01 < const|<7(rc)| for all o; in a neighborhood x ► a f{x) = o{g{x)), lim of the point a m ■a g(X) 0

11*11 lim xn = n—kx) (or xn -» n —» oo) OO n=l {x\v) (x \y) - X x as List of Symbols 461 Norms and Inner Products norm of £ 7 the sequence (xn) converges to the point x 9 infinite series in a Banach space 76 inner product 105 N Euclidean inner product, (x | y) := V^ xnyn 109 n=l (yj conjugate complex number to yj) (N Y \x\ Euclidean norm, \x\ := (x \ x)* = I V^ \xn\2 J 109 \n=l / |rc|oo special norm, |a;|oo := sup |o;n| 11 n (ia | v)2 inner product on the Lebesgue spaces I/2(G) 114 and L%(G), (^ | v)2 := / u(x)v(x)dx Jg ||w||2 norm on the Lebesgue spaces I/2(G) and I/JjT(G), 114 ||^||2 := (^ | 1^)2 = ( / |^(o;)|2da; 1 (w | v)i,2 inner product on the Sobolev space W^G), , 120 (w | v)i,2 := / ( w +V] djUdjV I do; IM|i,2 norm on the Sobolev space W%{G), N 2 ||ti||i,2 := (ti I ^)i)2 = I / U2 + X](^) dx G v i=i (• I -)e energetic inner product 273 || • He energetic norm 273 Operators A:S C X —>Y operator from the set S into 17 the set y, where S CY D(A) (or dom A) domain of definition of the operator A 17 R(A) (or im A) range (or image) of the operator A 17 N(A) (or ker A) null space (or kernel) of the operator A, 70 N(A):={x:Ax = 0} I (or id) identical operator, Ix := x for all x 76

462 List of Symbols AB (or A o B) ACB A(S) image of the set S, A(S) := {Ax: x e S} A_1(T) preimage of the set T, A~l(T) := {x: Ax e T} 17 A'1 inverse operator to A 17 G(A) graph of the operator A, 414 G(A):={(x,Ax):xeD(A)} norm of the linear operator A 70 norm of the functional / 75 the product of the operators A and B, 28 (AB)(u) := A(5ti) _ the operator B is an extension of 260 the operator A A* adjoint operator adjoint operator to the linear operator A 263 AT dual operator dual operator to the linear operator A (see Section 3.10 of AMS Vol. 109) closure of the linear operator A 415 spectrum of the linear operator A 83 resolvent set of the linear operator A 83 spectral radius of the linear operator A 94 rank of the linear operator A, rank A := dim R(A) (see Section 3.9 of AMS Vol. 109) index of the linear operator A, ind A := dim N(A) - codim R(A) (see Section 5.4 of AMS Vol. 109) determinant of the matrix A trace of the (N x 7V)-matrix A = (a/fem), tr A := an H f- o,nn trace of the linear operator A 347 in a Hilbert space A a(A) P{A) r(A) rank A ind A detA tr A tr Special Sets S int S extS dS Ue{p) U(p) dimX Xc X/L codim L closure of the set S 31 interior of the set S 31 exterior of the set S 31 boundary of the set S 31 ^-neighborhood of the point p in a 15 normed space, U€(p) := {x € X: \\x — p\\ < s} neighborhood of the point p 15 dimension of the linear space X 6 complexification of the linear space X 98 factor space (see Section 3.9 of AMS Vol. 109) codimension of the linear subspace L, codim L := dim(X/L) (see Section 3.9 of AMS Vol. 109)

List of Symbols 463 V* aS S + T M@L X®Y X* Xe span S co 5 coS dist(p, S) diam S meas S S(x) 6 Derivatives At) djf daf \a\ d dn A/ SF{x\ h) orthogonal complement to the linear subspace L the product aS := {ax:x e S}, a eR,C the sum S+ T := {x + y:x e S and y eT} orthogonal direct sum (M 0 L, where L = ML), tensor product dual space energetic space linear hull of the set S convex hull of the set S closed convex hull of the set S distance of the point p from the set S diameter of the set S measure of the set S the Dirac delta function the delta distribution derivative of an operator function u — u(t) at time t partial derivative J^- d^d^ • • • d%Nf, where a = (au ..., aN) (the classical symbols are also used for the derivatives of generalized functions) the sum a\ H f- oln derivative in the direction of the exterior normal N Laplacian, A/ := V^djf n=l variation of the functional F at the 165 7 7 165 224 75 273 31 31 47 47 431 158 161 80 159 159 181 125 point x in direction of h (see Section 2.1 of AMS Vol. 109) SnF(x; h) nth variation of the functional F at the point x in the direction of h (see Section 2.1 of AMS Vol. 109) A!{x) (or dA{x)) Prechet-derivative of the operator A at the point x (see Section 4.2 of AMS Vol. 109) dnA(x)(hi,..., hn) nth Prechet-diflFerential of the operator A at the point x in the directions of hi,..., hn (see Section 4.2 of AMS Vol. 109)

464 List of Symbols Spaces of Continuous Functions C[a,b],C(G) 14,116 L(X,Y), Lim(X,Y) 73,79 Spaces of Holder Continuous Functions Ca[a,b], Ck'a[a,b], Ca(G), Ck<a{G) (Ca(G) = C°'a(G)) 95ff Spaces of Smooth Functions Ck[a, b], Ck(G), Ck(G), C°°(G), Ck(G)c (C°(G) := C(G)) 96, 116 C$°(G) (or V(G)), S 116, 214 Spaces of Integrable Functions (Lebesgue Spaces) L2(a, b), L2{G), Lf(G) (L2(G) := Lf(G) if K = K) 130, 114 Sobolev Spaces Wj(G), W\{G) 131, 132 Spaces of Sequences K00, &, If (h := If if K = R) 95, 177 Spaces of Distributions V'{G), S' 160, 219

List of Theorems A good memory does not recall everything, but forgets the unimportant. Folklore Theorem l.A (The Banach fixed-point theorem) 19 Theorem l.B (The Brouwer fixed-point theorem) 53 Theorem l.C (The Schauder fixed-point theorem) 61 Theorem l.D (The Leray-Schauder principle) 65 Theorem l.E (The method of sub- and supersolutions) 69 Theorem 2.A (Main theorem on quadratic minimum problems) 121 Theorem 2.B (The Dirichlet principle) 138 Theorem 2.C (The Ritz method) 141 Theorem 2.D (The perpendicular principle) 165 Theorem 2.E (The Riesz theorem) 167 Theorem 2.F (Dual quadratic variational problems) 170 Theorem 2.G (Nonlinear monotone operators) 173 Theorem 2.H (The nonlinear Lax-Milgram theorem) 175 Theorem 3.A (Complete orthonormal systems) 202 Theorem 4.A (Eigenvalues and eigenvectors of linear, symmetric, compact operators) 232 Theorem 4.B (The Predholm alternative for linear, symmetric, compact operators) 237 Theorem 5.A (The Priedrichs extension of symmetric operators) 280 Theorem 5.B (The abstract Dirichlet problem) 282

466 List of Theorems Theorem 5.C (The eigenvalue problem) 284 Theorem 5.D (The Fredholm alternative) 306 Theorem 5.E (The abstract heat equation) 310 Theorem 5.F (The abstract wave equation) 310 Theorem 5.G (The abstract Schrodinger equation) 323

List of the Most Important Definitions Intelligence consists of this; that we recognize the similarity of different things and the difference between similar ones. Baron de la Brede et de Montesquieu (1689-1755) Spaces linear space 7 dimension 7 linear subspace 30 Banach space 10 norm 7 separable 84 reflexive (see Section 2.8 of AMS Vol. 109) Hilbert space 107 inner product 105 orthogonal elements 105 orthogonal projection 165 complete orthonormal system 200 Fock space (bosons or fermions) 364 Lebesgue space 114 Sobolev space 273 energetic space 273 dual space 74

468 List of the Most Important Definitions metric space and topological space (see Chapter 1 of AMS Vol. 109) Convergence norm convergence 8 Cauchy sequence 10 weak convergence (see Section 2.4 of AMS Vol. 109) sequentially continuous 27 sequentially compact 33 relatively sequentially compact 33 Operators domain of definition range and preimage injective surjective bijective inverse operator linear symmetric the Friedrichs extension adjoint dual (cf. Section 3.10 of AMS Vol. 109) self-adjoint Hamiltonian orthogonal projection operator skew-adjoint unitary Fourier transformation trace class statistical state statistical operator Hilbert-Schmidt operator continuous /^-contraction Lipschitz continuous Holder continuous homeomorphism diffeomorphism compact strongly monotone monotone or coercive (see Section 2.18 of AMS Vol. 109) semigroup Green function (propagator) 17 17 17 17 17 17 70 264 280 263 264 328 270 264 212 216 347 348 350 347 26 19 27 97 28 436 39 273 298 386

List of the Most Important Definitions 469 one-parameter group 298 dynamics of a quantum system 328 Predholm alternative 237 linear Predholm operator and index (see Section 5.4 of AMS Vol. 109) nonlinear Predholm operator (see Section 5.15 of AMS Vol. 109) m-linear bounded (see Section 4.1 of AMS Vol. 109) Functional nonlinear linear convex bilinear form bounded symmetric distribution (generalized function) tempered distribution Fourier transformation generalized eigenfunction Dirac delta distribution Green function fundamental solution Palais-Smale condition (see Section 2.16 of AMS Vol. 109) Embedding continuous compact 17 74 29 120 120 120 160 219 220 344 161 160 183 * 261 261 Spectrum eigenvalue and eigenvector 83 generalized eigenvector 344 resolvent set 83 resolvent operator 83 essential spectrum 84 spectral family 333 measurements in quantum systems 343 Set open 15 neighborhood 15 interior 30

470 List of the Most Important Definitions closed 15 closure 30 boundary 31 compact or relatively compact 33 dense 83 convex 29 bounded 33 countable 84 Point fixed point 18 critical point (see Section 2.1 of AMS Vol. 109) saddle point (see Section 2.2 of AMS Vol. 109) bifurcation point (see Section 5.12 of AMS Vol. 109) Operator Algebras Banach algebra 76 von Neumann algebra 359 C*-algebra 357 observable 359 state 358 pure 358 mixed 358 KMS-state (thermodynamic equilibrium) 360 *-automorphism 358 dynamics of a quantum system 359 Derivative time derivative 80 generalized derivative of a function 129 derivative of a distribution 162 nth variation (see Section 2.1 of AMS Vol. 109) Prechet derivative (see Section 4.2 of AMS Vol. 109) Integral Lebesgue integral 434 Lebesgue measure 429 integration by parts 118 Lebesgue-Stieltjes integral 441 Feynman path integral 385

Subject Index a posteriori error estimate 19 a priori error estimate 19 a priori estimates 64 absolute continuity 437 absolute integrability 435 absolute temperature 352 absolutely convergent 76 abstract boundary-eigenvalue problem 255 abstract boundary-value problem 254 abstract Dirichlet problem 282 abstract Fourier series 200 abstract heat equation 255, 306 abstract Schrodinger equation 256, 323 abstract setting for quantum mechanics 327 abstract setting of quantum statistics 348 abstract wave equation 256, 310 action 393 addition theorems 305 adjoint operator 263 admissible paths 392 admissible sequence 274 algebraic approach to quantum statistics 357 almost all 431 almost everywhere 431 annihilation operator 366 anticommutation relations 367 antilinear 169 Appolonius' identity 178 Arzela-Ascoli theorem 35 asymptotically free 368 *-automorphism 358 balls 16, 94 Banach algebra 76 Banach fixed-point theorem 18 Banach space 10 barycenter 47 barycentric subdivision 49 basic equation of quantum statistics 351 basis 42 Bernstein polynomials 86 Bessel inequality 202 beauty of functional analysis 256

472 Subject Index Big Bang 357 bijective 17 bilinear form 120 black holes 340 Bolzano-Weierstrass theorem 34 Born approximation, 403, 404 Bose-Einstein statistics 355 bosonic Fock space 364 bosons 363 bound states 372, 394, 396 boundary 31 boundary point 31 boundary-eigenvalue problem 245, 285, 316 boundary-value problem 125 bounded 33 bounded bilinear form 120 bounded orbits 421 bounded sequence 9 Brouwer fixed-point theorem 53 Brownian motion 381 C*-algebra 357 calculus of variations 117, 126 Cauchy sequence 10 Cayley transform 423 characteristic equation 94 charge density 180 chemical potential 352 chronological operator 395 classical function spaces 95, 97 classical Schwarz inequality 12 closed 15 closed balls 16 closed linear subspace 30 closure 31 closure of an operator 414 closed convex hull 31 commutation relations 365 commutative C*-algebra 358 compact 90 compact embedding 96, 261 compact operator 39 compact set 33 compactness 33 complete 10 complete orthonormal system 200, 209, 210, 223, 232, 247, 374 complete system of generalized eigenfunctions 346 completeness relation 374 completeness theorem 222 complex linear space 4 complex normed space 7 complexification 98 complexification of real Hilbert spaces 178 composite states of elementary particles 227 conservation of energy 336, 404 continuity 26 continuous 27 continuous Dirac calculus 375 continuous embedding 261 continuous spectrum 409 contraction principle 19 convergence 9 convex 29 convex hull 31 convexity 29 convolution 182 Coulomb force 180 Coulomb potential 420 countable 84 creation operators 364 critical point 321 defect indices 423 deflection of a string 157 degenerate kernels 251 degrees of freedom 229 dense 84 density 84, 189, 222 derivative 80 diagonal sequence 36 diameter 47 dielectricity constant 420 diffeomorphism 435 differential operator 267

Subject Index 473 diffusion 379, 386 diffusion equation 379 dimension 6 Dirac calculus 222, 373 Dirac ^-distribution 161 Dirac £-function 158, 346 Dirac function 375 Dirichlet principle 101, 123, 138 Dirichlet problem 125 discrete Dirac calculus 374 discrete spectrum 409 disk 28 dispersion 328 dispersion of the energy 352 dispersion of the particle number 352 distance 7, 47 distributions 160 domain of definition 17 dominated convergence 437 dual maximum problem 169 dual space 75 duality map 169, 279 duality of quadratic variational problems 169 duality theory 142 dynamics 359 dynamical systems 303 dynamics of the harmonic oscillator 341 dynamics of quantum systems 328 dynamics of statistical systems 349 Dyson formula 395 eigenfunction 247, 340 eigenoscillations 198 eigenoscillations of the string 317 eigensolution 230, 241 eigenspace 230 eigenstate 329 eigenvalue 83, 230 eigenvalue problem 283 eigenvector 230 elastic energy 256 elasticity 145 electric field 180 electric field of a charged point 180 electrostatics 183 electrostatic potential 180 embedding theorems 262 energetic extension 279 energetic inner product 123, 144, 273 energetic norm 144, 273 energetic space 144, 154, 273 energy 309 energy of the harmonic oscillator 309 energy conservation 303, 310, 313 entropy 349, 353 equicontinuous 35 equivalent norms 42, 99 error estimate 19, 142 error estimates via duality 142 essential spectrum 84, 372 Euclidean norm 12, 109 Euclidean strategy in quantum physics 379 Euler-Lagrange equation 125 expansion of our universe 340, 357 exponential function 78 extension 261 extension principle 213 exterior 31 exterior point 31 Fatou lemma 110, 438 Fermi-Dirac statistics 356 fermions 363 fermionic Fock space 366 Feynman diagrams 400 Feynman formula 393 Feynman path integral 381, 385, 393

474 Subject Index Feynman relation for transition amplitudes 397 Feynman-Kac formula 391 finite-dimensional Banach spaces 42 finite-dimensional space 6 finite elements 151 finite £-net 38 finite multiplicity 230 fixed point 19 Fock space 363 force density 156 formal Hamiltonian 339 Fourier coefficients 195, 200 Fourier integral 198 Fourier method 315, 316 Fourier series 195, 203, 241, 247 Fourier transform 216, 376, 378, 413 Fourier transform of tempered generalized functions 219 Fredholm alternative 237, 245, 249, 284, 287 Fredholm integral operator 95 free Hamiltonian 371 Friedrichs extension 258, 280 Friedrichs extension in complex Hilbert spaces 419 Friedrichs' mollification 186 Fubini's theorem 439 function 17 functional calculus 293, 331 functions of bounded variation 440 functions of self-adjoint operators 293 fundamental solution 179, 182, 183, 186, 413 fundamental theorem of calculus 119 Gauss method of least squares 197 Gauss theorem 119 Gaussian functions 223 Gelfand-Kostyuchenko theorem 424 Gelfand-Levitan-Marchenko integral equation 410 general position 45 generalized boundary values 135, 138 generalized derivative 129 generalized diffusion equation 386 generalized Dirichlet problem 138 generalized eigenfunctions 343, 372, 377 generalized eigenvectors 424 generalized Fourier series 195 generalized functions 156, 160 generalized functions in mathematical physics 179 generalized initial-value problem 185 generalized plane wave 184 generalized problem 285, 306, 310 generalized solution 258 generalized triangle inequality 8 generator 298 geometric series 79 golden rule for the rate of convergence 143 golden rule of numerical analysis 143 graph 414 graph closed operators 414 Green function 147, 157, 164, 246, 251, 387, 392 group property 300 half-numberly spin 357 Hamiltonian 328, 371, 418 Hamiltonian of the harmonic oscillator 341

Subject Index 475 Hamiltonian of the hydrogen atom 420 harmonic oscillations 196, 303 harmonic oscillation in quantum mechanics 338 heat equation 182 Heisenberg 5-operator 402 Heisenberg uncertainty principle 331, 342 Hermitean functions 210, 339 Hermitean polynomials 210 Hilbert space 107 Hilbert-Schmidt operator 347 Hilbert-Schmidt theory 232 Holder continuous functions 96, 97 homeomorphic 28 homeomorphism 28 homogeneity in time 302 *-homomorphjsm 358 hydrogen atom and the Priedrichs extension 420 idea of orthogonality 124 ideal elements 127 identical operator 77 infinite series 76 infinite-dimensional space 6 initial-value problem 24 injective 17 inner product 105 integer spin 357 integrable 434 integration by parts 117, 119, 159 integral equations 22, 62, 240 integral operator 18, 40, 265 integral of a step function 432 iterated integration 438 interior 31 interior point 31 inverse Fourier transformation 216, 378 inverse operator 17 inverse scattering theory 406, 409 irreversible process in nature 300, 303 isometric operators 422 *-isomorphism 358 iteration method 18, 68, 395, 404 fc-contractive 19 Kato perturbation theorem 417 KdV equation 407 kinetic energy 321, 336 KMS-states 360 Knaster, Kuratowski, and Mazur- kiewicz lemma 58 Korteweg-de Vries equation 406 lack of classic solution 127 Lagrange multiplier rule 354 Laguerre functions 222 language of physicists 221, 373 Laplacian 125, 277, 285 Lax pair 412 Lax-Milgram theorem 174 least-squares method 201 Lebesgue integral 434 Lebesgue measure 429 Lebesgue spaces 114 Lebesgue-Stieltjes integral 441 Legendre polynomials 209 Leray-Schauder principle 64 linear combinations 2 linear continuous functional 74 linear hull 31 linear integral equation 23 linear Lax-Milgram theorem 175 linear operator 70 linear orthogonality principle 172 linear space 3 linear subspace 30 linearly independent 5 Lipman-Schwinger integral equation 404 Lipschitz continuous 27

476 Subject Index Luzin's theorem 432 major ant criterion 436 Malgrange-Ehrenpreis theorem 186 mapping degree 55 mass conservation 379 matrix 71 maximally skew-symmetric 267 maximally symmetric 267 maximum 37 Maxwell-Boltzmann statistics 356 mean continuity 437 measure 430 measurable functions 432 measurable set 430 measurements 328, 349 method of finite elements 145 microscattering processes 400 mild (generalized) solution 307, 311, 320, 324 minimal sequence 176 minimum 37 Minkowski functional 50 mixed state 358 models of quantum field theory 368 momentum operator 342, 345, 376 momentum vector 336 monotone convergence 437 monotone operators 173 multiindex 159 multiplication operator 269, 334 multiplicity 230 Navier-Stokes equations 65 neighborhood 15 Neumann series 79 Newtonian equation 336 nonexpansive semigroup 299 nonlinear Fourier transformation 413 nonlinear Lax-Milgram theorem 175 nonlinear mathematical physics 173 nonlinear orthogonality principle 174 norm 7 normal order cone 67 normed space 7 nuclear spaces 424 null space 70 observables 328, 359 one-dimensional wave 323 one-parameter group 299 one-parameter unitary group 301, 328 open 15 open neighborhood 15 operator 16 operator functions 76, 77, 257 operator norm 70 order cone 66 ordered Banach space 67 ordered normed space 67 ordered sets 441 ordinary differential equations 24,63 orthogonal 105 orthogonal complement 165, 178 orthogonal decomposition 165 orthogonal projection 165, 270 orthogonality 124 orthogonality principle 172, 175 orthonormal system 199 parallelogram identity 123, 178 parameter integrals 439 Parseval equation 203, 222 particle number 352 particle number operator 366 particle stream 344, 403 partial differential equations of mathematical physics 256

Subject Index 477 partition function 353 path integral 390 Pauli principle 356, 363 Payley-Wiener theorem 186, 223 Peano theorem 63 perpendicular principle 123, 165 phase space 393 photon 340 physical interpretation of the Green function 158 physical states 327 Picard-Lindelof theorem 24 Planck quantum action 329 Planck's radiation law 340, 357 plane wave 184 Poincare inequality 287 Poincare-Priedrichs inequality 136 Poisson equation 125, 258, 285 position operator 342, 346, 377 positive bilinear form 120 potential 182 potential barrier 403 potential energy 321, 336 potential equation 182 potential theory 95 pre-Hilbert space 105 preimage 17 principle of indistinguishability 363 principle of maximal entropy 353 principle of minimal potential energy 256 principle of minimal potential errors 142 principle of stationary action 321 principle of virtual power 142, 147 principle of virtual work 147 probability 336, 352 probability conservation of quantum processes 303 probability of measuring the position 343 product rule 107 propagation of probability 393 propagator 387 pure state 358 Pythagorean theorem 167 quadratic variational problems 121 quantization 336 quantization of action 393 quantum field theory 363 quantum hypothesis 340 quantum mechanics 327, 336 quantum statistics 348 quantum system 327 range 17 rate of convergence 19, 142 real normed space 7 real linear space 4 regularity theory 128 relatively compact 90 relatively sequentially compact 33 Rellich's compactness theorem 287 resolvent 83 resolvent set 83 resonance condition 249 restriction 259 retraction 55 reversibility 302 reversible processes in nature 302 Riesz theorem 167 Ritz equation 140, 152 Ritz method 140, 151, 179 S-matrix 402 scalar multiplication 3 scattering of a particle stream 399 scattering theory 368 Schauder fixed-point theorem 61 Schauder operator 41 Schmidt orthogonalization method 207

478 Subject Index Schrodinger equation 323, 336, 338, 381, 395 Schwarz inequality 105 self-adjoint operator 264, 416 semi-Predholm 83 semigroup 298 separable 84, 116 separability 191 sequentially compact 33 sequentially continuous 27 sharp energy 341 simple eigenvalue 247 simplex 46 skew-adjoint operator 264 skew-symmetric operator 264 smoothing of functions 186 smoothing technique 117 Sobolev embedding theorems 192 Sobolev space 130, 260. 277 solitons 406 spectral family 333 spectral radius 94 spectral transformation 410 spectrum 83, 94, 238 spectrum of the hydrogen atom 421 Sperner lemma 56 Sperner simplex 57 standard model in statistical physics 351, 360 states 327, 359 stationary particle states 337 stationary Schrodinger equation 337 statistical operator 350 statistical potential 353 statistical states 348 step function 432 Stieltjes integral 333, 440 string 158, 315 string energy 320 string equation 321 strong causality 302 strongly continuous semigroup 298 strongly monotone operators 173, 259, 273 strongly positive bilinear form 120 subsolution 69 superselection rules 328 superposition 196 superposition of eigenoscillations 317 supersolution 69 surjective 17 symmetric bilinear form 120 symmetric operator 230, 264, 415 temperature tempered delta distribution 221 tempered distributions 220 tensor product 224, 226 tensor product of functions 184 tensor product of generalized functions 185 thermodynamic equilibrium 360 thermodynamical quantities 353 time evolution 329 time evolution of quantum systems 330 time-dependent processes in nature 298 time-dependent scattering theory 398 time-independent scattering theory 404 Tonelli's theorem 439 total energy 321, 336 totally ordered 442 trace class operators 347, 421 transformation rule for integrals 435 transition amplitude 397 transition probabilities 397 Trefftz method 170 triangle inequality 7 triangulation 49

Subject Index 479 trigonometric polynomials 204, 206 two-soliton solutions 407 unbounded operator 300 unbounded orbits 421 uncertainty inequality 330 uncertainty principle 343 uniformly continuous 27 uniformly continuous one-parameter group 299 uniqueness implies existence 237 unitary operator 212, 219, 272, 330 variational equation 140 variational lemma 117 variational problem 125, 140, 281, 287 vertices 46 vibrating string 315 Volterra integral operator 95 volume potential 183 von Neumann algebra 359 waves 406 wave equation 182, 185, 309 wave operators 369 Weierstrass approximation theorem 84 Weierstrass classical counterexample 176 Weierstrass theorem 37 white dwarfs 357 Wiener path integral 381, 386 Zorn's lemma 442

This is the first part of an elementary textbook which combines linear functional analysis, nonlinear functional analysis, numerical functional analysis, and their substantial applications with each other. The book addresses undergraduate students and beginning graduate students of mathematics, physics, and engineering who want to learn how functional analysis elegantly solves mathematical problems which relate to our real world and which play an important role in the history of mathematics. The book's approach begins with the question "what are the most important applications" and proceeds to try to answer this question. The applications concern ordinary and partial differential equations, the method of finite elements, integral equations, special functions, both the Schrodinger approach and the Feynman approach to quantum physics, and quantum statistics. The presentation is self-contained. As for prerequisites, the reader should be familiar with some basic facts of calculus. The second part of this textbook has been published under the title Applied Functional Analysis: Main Principles and Their Applications. ISBN 0-387-94442-7 z ISBN 0-387-94442-7 9 780387"944425 >