FUNCTIONAL
ANALYSIS
Walter Rudin
Professor of Mathematics
University of Wisconsin
McGraw-Hill Book Company
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Library of Congress Cataloging in Publication Data
Rudin, Walter, 1921-
Functional analysis.
(Higher mathematics series)
1. Functional analysis. I. Title.
FUNCTIONAL ANALYSIS
Copyright © 1973 by McGraw-Hill, Inc. All rights reserved.
Printed in the United States of America. No part of this
publication may be reproduced, stored in a retrieval system,
or transmitted, in any form or by any means, electronic,
mechanical, photocopying, recording, or otherwise, without
the prior written permission of the publisher.
This book was set in Times New Roman. The editors were
Jack L. Farnsworth, Bradford Bayne, and M. E. Margolies;
and the production supervisor was Ted Agrillo.

CONTENTS
Preface xi
Part One—GENERAL THEORY
Chapter 1 Topological Vector Spaces 3
Introduction 3
Separation properties 9
Linear mappings 13
Finite-dimensional spaces 14
Metrization 17
Boundedness and continuity 21
Seminorms and local convexity 24
Quotient spaces 29
Examples 31
Exercises 36
1 Chapter 2 Completeness 41
Baire category 41
The Banach-Steinhaus theorem 43
The open mapping theorem 46
The closed graph theorem 49
Bilinear mappings 51
Exercises 52 ^
Chapter 3 Convexity 55
The Hahn-Banach theorems 55
Weak topologies 60
Compact convex sets 66

VI CONTENTS
Vector-valued integration 73
Holomorphic functions 78
Exercises 81
Chapter 4 Duality in Banach Spaces 87
The normed dual of a normed space 87
Adjoints 92
Compact operators 97
Exercises 105
Chapter 5 Some Applications 110
A continuity theorem 110
Closed subspaces of //-spaces 111
The range of a vector-valued measure 113
A generalized Stone-Weierstrass theorem 115
Two interpolation theorems 117
A fixed point theorem 120
Haar measure on compact groups 122
Uncomplemented subspaces 125
Exercises 130
Part Two—DISTRIBUTIONS AND FOURIER
TRANSFORMS
Chapter 6 Test Functions and Distributions 135
Introduction 135
Test function spaces 136
Calculus with distributions 142
Localization 147
Supports of distributions 149
Distributions as derivatives 152
Convolutions 155
Exercises 162

CONTENTS Vll
Chapter 7 Fourier Transforms 166
Basic properties 166
Tempered distributions 173
Paley-Wiener theorems 180
Sobolev's lemma 185
Exercises 187
Chapter 8 Applications to Differential Equations 192
Fundamental solutions 192
Elliptic equations 197
Exercises 204
Chapter 9 Tauberian Theory 208
Wiener's theorem 208
The prime number theorem 212
The renewal equation 218
Exercises 221
Part Three—BANACH ALGEBRAS AND
SPECTRAL THEORY
Chapter 10 Banach Algebras 227
Introduction *227
Complex homomorphisms 231
Basic properties of spectra 234
Symbolic calculus 240
Differentiation 248
The group of invertible elements 257
Exercises 259

Vlil CONTENTS
Chapter 11 Commutative Banach Algebras 263
Ideals and homomorphisms 263
Gelfand transforms 268
Involutions 275
Applications to noncommutative algebras 279
Positive functional 283
Exercises 288
Chapter 12 Bounded Operators on a Hilbert Space 292
Basic facts 292
Bounded operators 295
A commutativity theorem 300
Resolutions of the identity 301
The spectral theorem 305
Eigenvalues of normal operators 311
Positive operators and square roots 313
The group of invertible operators 316
A characterization of i?*-algebras 319
Exercises 323
Chapter 13 Unbounded Operators 329
Introduction 329
Graphs and symmetric operators 333
The Cayley transform 338
Resolutions of the identity 341
The spectral theorem 348
Semigroups of operators 355
Exercises 363

CONTENTS IX
Appendix A Compactness and Continuity 367
Appendix В Notes and Comments 372
Bibliography 384
List of Special Symbols 386
Index 389

PREFACE
Functional analysis is the study of certain topological-algebraic
structures and of the methods by which knowledge of these structures
can be applied to analytic problems.
A good introductory text on this subject should include a presen-
presentation of its axiomatics (i.e., of the general theory of topological
vector spaces), it should treat at least a few topics in some depth, and
it should contain some interesting applications to other branches of
mathematics. I hope that the present book meets these criteria.
The subject is huge and is growing rapidly. (The bibliography
in volume I of [4] contains 96 pages and goes only to 1957.) In order
to write a book of moderate size, it was therefore necessary to select
certain areas and to ignore others. I fully realize that almost any
expert who looks at the table of contents will find that some of his
(and my) favorite topics are missing, but this seems unavoidable. It
was not my intention to write an encyclopedic treatise. I wanted to
write a book that would open the way to further exploration.
This is the reason for omitting many of the more esoteric topics
that might have been included in the presentation of the general
theory of topological vector spaces. For instance, there is no dis-
discussion of uniform spaces, of Moore-Smith convergence, of nets, or
of filters. The notion of completeness occurs only in the context of
metric spaces. Bornological spaces are not mentioned, nor are
barreled ones. Duality is of course presented, but not in its utmost
generality. Integration of vector-valued functions is treated strictly as
a tool; attention is confined to continuous integrands, with values in
a Frechet space.
Nevertheless, the material of Part 1 is fully adequate for almost
all applications to concrete problems. And this is what ought to be
stressed in such a course: The close interplay between the abstract
and the concrete is not only the most useful aspect of the whole
subject but also the most fascinating one.
Here are some further features of the selected material. A fairly
large part of the general theory is presented without the assumption
of local convexity. The basic properties of compact operators are
derived from the duality theory in Banach spaces. The Krein-Milman
theorem on the existence of extreme points is used in several ways in

Xli PREFACE /
Chapter 5. The theory of distributions and Fourier transforms is
worked out in fair detail and is applied (in two very brief chapters)
to two problems in partial differential equations, as well as to
Wiener's tauberian theorem and two of its applications. The spectral
theorem is derived from the theory of Banach algebras (specifically,
from the Gelfand-Naimark characterization of commutative B*-
algebras); this is perhaps not the shortest way, but it is an easy one.
The symbolic calculus in Banach algebras is discussed in considerable
detail; so are involutions and positive functionals. Several fairly
recent results on Banach algebras that have not found their way into
other textbooks as yet are included.
I assume familiarity with the theory of measure and Lebesgue
integration (including such facts as the completeness of the LP-
spaces), with some basic properties of holomorphic functions (such
as the general form of Cauchy's theorem, and Runge's theorem), and
with the elementary topological background that goes with these two
analytic topics. Some other topological facts are briefly presented in
Appendix A. Almost no algebraic background is needed, beyond
the knowledge of what a homomorphism is.
Historical references are gathered in Appendix B. Some of
these refer to the original sources, and some to more recent books,
papers, or expository articles in which further references can be
found. There are, of course, many items that are not documented
at all. In no case does the absence of a specific reference imply any
claim to originality on my part.
Most of the applications are in Chapters 5, 8, and 9. Some are
in Chapter 11 and in the more than 250 exercises; many of these are
supplied with hints. The interdependence of the chapters is indi-
indicated in the following diagram.

PREFACE X1U
This book grew out of a course that I have taught at the Univer-
University of Wisconsin. I have had many fruitful conversations about
various topics in it with some of my colleagues, especially with
Patrick Ahern, Paul Rabinowitz, Daniel Shea, and Robert Turner.
It is a pleasure to record my thanks to them.
Walter Rudin

PART ONE
General Theory

1
TOPOLOGICAL VECTOR SPACES
Introduction
1.1 Many problems that analysts study are not primarily concerned with a single
object such as a function, a measure, or an operator, but they deal instead with large
classes of such objects. Most of the interesting classes that occur in this way turn out
to be vector spaces, either with real scalars or with complex ones. Since limit processes
play a role in every analytic problem (explicitly or implicitly), it should be no surprise
that these vector spaces are supplied with metrics, or at least with topologies, that
bear some natural relation to the objects of which the spaces are made up. The
simplest and most important way of doing this is to introduce a norm. The resulting
structure (defined below) is called a normed vector space, or a normed linear space,
or simply a normed space.
Throughout this book, the term vector space will refer to a vector space over the
complex field С or over the real field R. For the sake of completeness, detailed
definitions are given in Section 1.4.

4 GENERAL THEORY
't
1.2 Normed spaces A vector space X is said to be a normed space if to every
* е Jf there is associated a nonnegative real number ||*||, called the norm of *, in such
a way that
(a) ||* + j|| < ||*|| + |M| for all x and j in X,
(b) \\ctx\\ = |oc| ||*|| if *e A" and a is a scalar,
(c) ||*|| > 0 if *#0.
The word "norm" is also used to denote the function that maps * to ||*||.
Every normed space may be regarded as a metric space, in which the distance
d(x, y) between * and у is ||* — y\\. The relevant properties of d are:
@ 0 < d(x, y) < oo for all * and y,
(u) d(x, y) = 0 if and only if * = y\
(Hi) d(x, y) = d(y, x) for all * and y,
(iv) d(x, z) < d(x, y) + d(y, z) for all *, y9 z.
In any metric space, the open ball with center at * and radius r is the set
Br(x) = {y.d(X,y)<r}.
In particular, if X is a normed space, the sets ^
Bt(P) - {x: ||x|| < 1} and Бх@) = {x: \\x\\ < 1}
are the open unit ball and the closed unit ball of X, respectively.
By declaring a subset of a metric space to be open if and only if it is a (possibly
empty) union of open balls, a topology is obtained. (See Section 1.5.) It is quite easy
to verify that the vector space operations (addition and scalar multiplication) are
continuous in this topology, if the metric is derived from a norm, as above.
A Banach space is a normed space which is complete in the metric defined by its
norm; this means that every Cauchy sequence is required to converge.
1.3 Many of the best-known function spaces are Banach spaces. Let us mention
just a few types: spaces of continuous functions on compact spaces; the familiar
Lp-spaces that occur in integration theory; Hilbert spaces — the closest relatives of
euclidean spaces; certain spaces of differentiable functions; spaces of continuous linear
mappings from one Banach space into another; Banach algebras. All of these will
occur later on in the text.
But there are also many important spaces that do not fit into this framework.
Here are some examples:
(a) C(Q), the space of all continuous complex functions on some open set Q in a
euclidean space Rn.
(b) H(Q), the space of all holomorphic functions in some open set Q in the complex
plane.

TOPOLOGICAL VECTOR SPACES 5
(c) C£, the-space of all infinitely differentiable complex functions on Rn that vanish
outside some fixed compact set К with nonempty interior.
(d) The test function spaces used in the theory of distributions, and the distributions
themselves.
These spaces carry natural topologies that cannot be induced by norms, as we
shall see later. They, as well as the normed spaces, are examples of topological vector
spaces, a concept that pervades all of functional analysis.
After this brief attempt at motivation, here are the detailed definitions, followed
(in Section 1.9) by a preview of some of the results of Chapter 1.
1.4 Vector spaces The letters R and С will always denote the field of real numbers
and the field of complex numbers, respectively. For the moment, let Ф stand for either
R or С A scalar is a member of the scalar field Ф. A vector space over Ф is a set X,
whose elements are called vectors, and in which two operations, addition and scalar
multiplication, are defined, with the following familiar algebraic properties:
(a) To every pair of vectors x and у corresponds a vector x -f y, in such a way that
x + y = y + x and x + (y + z) = (x + y) + z;
X contains a unique vector 0 (the zero vector or origin of X) such that x -f 0 = x
for every xel; and to each xeX corresponds a unique vector — x such that
* + (-*) = ().
(b) To every pair (a, x) with a e Ф and x e X corresponds a vector ccx, in such a way
that
lx==x, a(px) = (aP)x,
and such that the two distributive laws
a(x 4- y) = ax 4- ay, (a + P)x = ocx + px
hold.
The symbol 0 will of course also be used for the zero element of the scalar field.
A real vector space is one for which Ф = R; a complex vector space is one for
which Ф = С Any statement about vector spaces in which the scalar field is not
explicitly mentioned is to be understood to apply to both of these cases.
If X is a vector space, A c= X, В c= X, x e X, and X e Ф, the following notations
will be used:
x 4- A = {x + a: a e A},
x — A — {x — a: a e A},
A + B={a + b: aeA, b e B},
XA — {Xa\ a e A).

6 GENERAL THEORY
In particular (taking A = — 1), — A denotes the set of all additive inverses of members
of A.
A word of warning: With these conventions, it may happen that 2A ф А + А
(Exercise 1).
A set Y c= X is called a subspace of X if Y is itself a vector space (with respect to
the same operations, of course). One checks easily that this happens if and only if
Oe У and
for all scalars a and /?.
A set С cz X is said to be convex if
tC + (\-t)Cc:C @<t< 1).
In other words, it is required that С should contain tx -f A — t)y if x e С, у е С, and
0<t <1.
A set В c= X is said to be balanced if аБ c= В for every a e Ф with | a | < 1.
A vector space X has dimension n (dim X = ri) if X has a &а?£? {щ,..., и„}.
This means that every x e X has a unique representation of the form
x = a1ui + •-• + <xnun (а.-еФ).
If dim X = л for some я, Zis said to have finite dimension. If X = {0}, then dim X = 0.
Example If Z = С (a one-dimensional vector space over the scalar field €),
the balanced sets are: C, the empty set 0, and every circular disc (open or
closed) centered at 0. If X = R2 (a two-dimensional vector space over the scalar
field R), there are many more balanced sets; any line segment with midpoint at
@, 0) will do. The point is that in spite of the well-known and obvious identifica-
identification of С with R2, these two are entirely different as far as their vector space
structure is concerned.
1.5 Topological spaces A topological space is a set S in which a collection т of
subsets (called open sets) has been specified, with the following properties: S is open,
0 is open, the intersection of any two open sets is open, and the union of every collec-
collection of open sets is open. Such a collection т is called a topology on S. When clarity
seems to demand it, the topological space corresponding to the topology t will be
written (S, t) rather than S.
Here is some of the standard vocabulary that will be used, if S and т are as above.
A set E c= S is closed if and only if its complement is open. The closure E of E
is the intersection of all closed sets that contain E. The interior E° of E is the union

TOPOLOGICAL VECTOR SPACES 7
of all open sets that are subsets of E. A neighborhood of a point p e S is any open set
that contains p. (S, т) is a Hausdorff space, and т is a Hausdorff'topology, if distinct
points of S have disjoint neighborhoods. A set К cz S is compact if every open cover
of К has a finite subcover. A collection т' cz т is a base for т if every member of т
(that is, every open set) is a union of members oft'. A collection у of neighborhoods
of a point p e S is a local base at p if every neighborhood of/? contains a member of y.
If E cz £ and if a is the collection of all intersections E n V, with Гет, then a
is a topology on E, as is easily verified; we call this the topology that E inherits from S.
If a topology т is induced by a metric d (see Section 1.2) we say that d and т
are compatible with each other.
A sequence {xj in a Hausdorff space X converges to a point jcel (or: limn_>oo
xn = x) if every neighborhood of x contains all but finitely many of the points xn.
1.6 Topological vector spaces Suppose т is a topology on a vector space Zsuch
that
(a) every point of X is a closed set, and
(b) the vector space operations are continuous with respect to т.
Under these conditions, т is said to be a vector topology on X, and X is a
topological vector space.
Here is a more precise way of stating (a): For every x e X, the set {x} Which has
x as its only member is a closed set.
In many texts, (a) is omitted from the definition of a topological vector space.
Since (a) is satisfied in almost every application, and since most theorems of interest
require (a) in their hypotheses, it seems best to include it in the axioms. [Theorem 1.12
will show that (a) and (b) together imply that risa Hausdorff topology.]
To say that addition is continuous means, by definition, that the mapping
(*, y) -> x + у
of the cartesian product X x X into X is continuous: If xt e X for /=1,2, and if V
is a neighborhood of xt + x2, there should exist neighborhoods Vt of xt such that
Similarly, the assumption that scalar multiplication is continuous means that the
mapping
(a, x) —> ax
of Ф x A" into A4s continuous: If x e X, a is a scalar, and Fis a neighborhood of ax,
then for some r > 0 and some neighborhood W of x we have pwcz V whenever
\p-oc\<r.

8 GENERAL THEORY
A subset £ of a topological vector space is said to be bounded if to every neighbor-
neighborhood V of 0 in X corresponds a number s > 0 such that E cz tVfor every t > s.
1.7 In variance Let X be a topological vector space. Associate to each a e X and
to each scalar X ф 0 the translation operator Ta and the multiplication operator Mx,
by the formulas
Ta(x) = a+x, Mx(x) = Xx (xe X).
The following simple proposition is very important:
Proposition Ta and Mx are homeomorphisms of X onto X.
proof The vector space axioms alone imply that Ta and Mx are one-to-one,
that they map Xonto X, and that their inverses are T_fl and Ml/X, respectively.
The assumed continuity of the vector space operations implies that these four
mappings are continuous. Hence each of them is a homeomorphism (a con-
continuous mapping whose inverse is also continuous). ////
One consequence of this proposition is that every vector topology т is translation-
invariant (or simply invariant, for brevity): A set E с X is open if and only if each of
its translates a -f E is open. Thus т is completely determined by any local base.
In the vector space context, the term local base will always mean a local base
at 0. A local base of a topological vector space X is thus a collection $ of neighbor-
neighborhoods of 0 such that every neighborhood of 0 contains a member of £$. The open sets
of X are then precisely those that are unions of translates of members of £%.
A metric Jona vector space X will be called invariant if
d(x + z,y + z) = d(x, y)
for all x, y, z in X.
1.8 . Types of topological vector spaces In the following definitions, X always
denotes a topological vector space, with topology т.
(a) X is locally convex if there is a local base 0& whose members are convex.
(b) X is locally bounded if 0 has a bounded neighborhood.
(c) X is locally compact if 0 has a neighborhood whose closure is compact.
(d) X is metrizable if т is compatible with some metric d.
(e) X is an F-space if its topology т is induced by a complete invariant metric d.
(Compare Section 1.25.)
(/) X is a Frechet space if X is a locally convex F-space.
(g) X is normable if a norm exists on Jfsuch that the metric induced by the norm is
compatible with т.

TOPOLOGICAL VECTOR SPACES 9
(h) Normed spaces and Banach spaces have already been defined (Section 1.2).
(/) X has the Heine-Borel property if every closed and bounded subset of X is
compact.
The terminology of (e) and (/) is not universally agreed upon: In some texts,
local convexity is omitted from the definition of a Frechet space, whereas others use
jF-space to describe what we have called Frechet space.
1.9 Here is a list of some relations between these properties of a topological vector
space X.
(a) If A" is locally bounded, then X has a countable local base [part (c) of Theorem
1.15].
(b) X is metrizable if and only if X has a countable local base (Theorem 1.24).
(c) X is normable if and only if X is locally convex and locally bounded (Theorem
1.39).
(d) X has finite dimension if and only if X is locally compact (Theorems 1.21, 1.22).
(e) If a locally bounded space X has the Heine-Borel property, then X has finite
dimension (Theorem 1.23).
The spaces H(Q) and CJ? mentioned in Section 1.3 are infinite-dimensional
Frechet spaces with the Heine-Borel property (Sections 1.45, 1.46). They are therefore
not locally bounded, hence not normable; they also show that the converse of (a) is
false.
On the other hand, there exist locally bounded F-spaces that are not locally
convex (Section 1.47).
Separation Properties
1.10 Theorem Suppose К and С are subsets of a topological vector space X, К is
compact, С is closed, and К n С — 0. Then 0 has a neighborhood V such that
Note that К -f К is a union of translates x + V of V (x e K). Thus К + V is an
open set that contains K. The theorem thus implies the existence of disjoint open sets
that contain К and C, respectively.
proof We begin with the following proposition, which will be useful in other
contexts as well:
If W is a neighborhood ofO in X, then there is a neighborhood UofO which
is symmetric (in the sense that U — —U) and which satisfies U 4- U <= W.

10 GENERAL THEORY }
! i
To see this, note that 0 + 0 = 0, that addition is continuous, and that 0
therefore has neighborhoods Vl9 V2 such that Vt + V2 с W. If
U=V1nV2n(-V1)n(-V2)9
then U has the required properties.
The proposition can now be applied to U in place of W and yields a new
symmetric neighborhood U of 0 such that
U+ C/+ C/+ Uc W.
It is clear how this can be continued.
If К = 0, then К -f V= 0, and the conclusion of the theorem is obvious.
We therefore assume that Кф 0, and consider a point x e iT. Since С is closed,
since x is not in C, and since the topology of Xis invariant under translations,
the preceding proposition shows that 0 has a symmetric neighborhood Vx such
that x + Vx + Vx + Vx does not intersect C; the symmetry of Vx then shows that
A) (x+Vx+ Vx) n(C+ Vx) = 0.
Since К is compact, there are finitely many points xlt ..., х„ in К such that
*<= (*i + FX1) и ••• и (*„+*;„).
Put K= KX1 n • • • n F^. Then
X + F c= 0 (x, + VXI + V) c= 0 (x, + VXi + И„),
and no term in this last union intersects С + V9 by A). This completes the proof.
////
Since С 4- К is open, it is even true that the closure of К 4- К does not intersect
С + F; in particular, the closure of К + К does not intersect C. The following special
case of this, obtained by taking K= {0}, is of considerable interest.
1.11 Theorem IfM is a local base for a topological vector space X then every member
of & contains the closure of some member of 0&.
So far we have not used the assumption that every point of X is a closed set.
We now use it and apply Theorem 1.10 to a pair of distinct points in place of i^and С
The conclusion is that these points have disjoint neighborhoods. In other words, the
Hausdorff separation axiom holds:
1.12 Theorem Every topological vector space is a Hausdorff space.
We now derive some simple properties of closures and interiors in a topological
vector space. See Section 1.5 for the notations Ё and E°. Observe that a point p
belongs to E if and only if every neighborhood of/? intersects E.

TOPOLOGICAL VECTOR SPACES 11
1ЛЗ Theorem Let X be a topological vector space.
(a) If Acz X then A = f] (A + V), where V runs through all neighborhoods ofO.
(b) If А с X and В с X, then A + В cz А + В.
(c) If Y is a subspace of X, so is Y.
(d) If С is a convex subset of X, so are С and C°.
(e) If В is a balanced subset of X, so is B; if also 0 e B° then B° is balanced.
(/) If E is a bounded subset of X, so is E.
proof (a) xeA if and only if (x + V) n А ф 0 for every neighborhood V
of 0, and this happens if and only if x e A — V for every such V. Since — V is
a neighborhood of 0 if and only if Fis one, the proof is complete.
(b) Take a e A, beB; let W be a neighborhood of a + b. There are
neighborhoods W1 and W2 of a and b such that Wx + W2 <= W. There exist
x e A n Wx and у е В n W2, since a e A and beB. Then x + у lies in
{A + B) n Jf, so that this intersection is not empty. Consequently, a + be A + B.
(c) Suppose a and /? are scalars. By the proposition in Section 1.7,
aY — aFif a ^ 0; if a = 0, these two sets are obviously equal. Hence it follows
* from (b) that
Y = aY+PY czaY+pYcz Y;
the assumption that У is a subspace was used in the last inclusion.
The proofs that convex sets have convex closures and that balanced sets
have balanced closures are so similar to this proof of (c) that we shall omit them
from (d) and (e).
(d) Since C° cz С and С is convex, we have
tC° + A - i)C° с С
if 0 < t < 1. The two sets on the left are open; hence so is their sum. Since
every open subset of С is a subset of C°, it follows that C° is convex.
(e) If 0 < | oc | < 1, then aB° = (a#)°, since x -> ax is a homeomorphism.
Hence ocB° cz aB cz B, since В is balanced. But аБ° is open. So aB° cz B°. If B°
contains the origin, then осЁ° cz B° even for а = 0.
(/) Let Fbe a neighborhood of 0. By Theorem 1.11, W cz F for some
neighborhood W of 0. Since E is bounded, £ cz tW fox all sufficiently large t.
For these f, we have EcztWcztV. ////
1.14 Theorem In a topological vector space X,
(a) every neighborhood ofO contains a balanced neighborhood o/O, and
(b) every convex neighborhood of 0 contains a balanced convex neighborhood of 0.

12 GENERAL THEORY
proof (a) Suppose U is a neighborhood of 0 in X. Since scalar multiplication
is continuous, there is a S > 0 and there is a neighborhood К of 0 in J^such that
aV cz U whenever |a| < 3. Let W be the union of all these sets aV. Then W is
a neighborhood of 0, W is balanced, and Жс[/.
(b) Suppose U is a convex neighborhood of 0 in X. Let A = P)a£/, where
a ranges over the scalars of absolute value 1. Choose W as in part (a). Since W
is balanced, a~1W= W when |a| = 1; hence Жса£Л Thus Wcz A, which
implies that the interior A° of ^4 is a neighborhood of 0. Clearly A° cz U. Being
an intersection of convex sets, A is convex; hence so is A°. To prove that A°
is a neighborhood with the desired properties, we have to show that A° is
balanced; for this it suffices to prove that A is balanced. Choose r and p so that
0<r< 1, \p\ = 1. Then
Since all is a convex set that contains 0, we have raU cz aU. Thus r/L4 cz A,
which completes the proof. ////
Theorem 1.14 can be restated in terms of local bases. Let us say that a local base
J1 is balanced if its members are balanced sets, and let us call & convex if its members
are convex sets.
Corollary
(a) Every topological vector space has a balanced local base.
(b) Every locally convex space has a balanced convex local base.
Recall also that Theorem 1.11 holds for each of these local bases.
1.15 Theorem Suppose V is a neighborhood ofO in a topological vector space X.
(a) I/O < r1 < r2 < " ' and rn -> oo as n -> oo, then
X=[)rnV.
(b) Every compact subset К of X is bounded.
(c) Ifd1>d2>'" and bn -> 0 as n -> oo, and if V is bounded, then the collection
{<5nF:/z= 1,2,3,...}
is a local base for X.
proof (a) Fix x e X. Since а -> ax is a continuous mapping of the scalar
field into X, the set of all a with ax e V is open, contains 0, hence contains
l/rn for all large n. Thus (I/O* e K, or x e rn V, for large n.

TOPOLOGICAL VECTOR SPACES 13
(bj Let W be a balanced neighborhood of 0 such that W с F. By (a),
Kcz \JnW.
Since К is compact, there are integers n1 < • - • < ns such that.
••unsW=nsW.
The equality holds because W is balanced. If t > ns, it follows that iT с
tWctV. '
(c) Let £/ be a neighborhood of 0 in X. If F is bounded, there exists
s > 0 such that Fez f£/for all f > s. If я is so large that s5n < 1, it follows that
Fez (l/Sn)U. Hence C/ actually contains all but finitely many of the sets Sn V.
Ill/
Linear Mappings
1.16 Definitions When X stud Fare sets, the symbol
f:X-+Y
will mean that/is a mapping of X into Y. If A cz J^and J5 cz Y, the image f (A) of A
and the inverse image or preimagef~1(B) of Б are defined by
ДЛ) = {Дх): хеЛ}, /"ЧД) = {*:/(*) б Я}.
Suppose now that Xand Fare vector spaces o^r the same scalar field. A map-
mapping Л: Х-+ Yis said to be //wear if
A(ax + py) = olAx + fiAy
for all x and у in X and all scalars a and /?. Note that one often writes Ax, rather
than Л(х), when Л is linear.
Linear mappings of Xinto its scalar field are called linear functionals.
For example, the multiplication operators Мл of Section 1.7 are linear, but the
translation operators Ta are not, except when a — 0.
Here are some properties of linear mappings Л: X -> Y whose proofs are so
easy that we omit them; it is assumed that A cz X and В cz Y:
(a) Л0 = 0.
(b) If A is a subspace (or a convex set, or a balanced set) the same is true of A(A).
(c) If Б is a subspace (or a convex set, or a balanced set) the same is true of Л-1(Б).
(d) In particular, the set
is a subspace of X, called the null space of Л.
We now turn to continuity properties of linear mappings.

14 GENERAL THEORY
1.17 Theorem Let X and Y be topological vector spaces. If A: X -> Y is linear
and continuous at 0, then A is continuous. In fact, A is uniformly continuous, in the
following sense: To each neighborhood WofO in Y corresponds a neighborhood VofO in
X such that
у — x e V implies Ay — Ax e W.
proof Once W is chosen, the continuity of Л at 0 shows that Л Fez W for
some neighborhood V of 0. If now у — x e F, the linearity of Л shows that
Ay — Ax = A(y — x) eW. Thus Л maps the neighborhood x + V of x into the
preassigned neighborhood Ax + W of Ax, which says that Л is continuous at x.
Ill/
1.18 Theorem Let A be a linear functional on a topological vector space X.
Assume Ax ф 0 for some x e X. Then each of the following four properties implies
the other three:
(a) A is continuous.
(b) The null space jV(A) is closed.
(c) jV(A) is not dense in X.
(d) A is bounded in some neighborhood V ofO.
proof Since jV{A) = A ^{O}) and {0} is a closed subset of the scalar field Ф,
(a) implies (b). By hypothesis, jV{A) ф X. Hence (b) implies (c).
Assume (c) holds; i.e., assume that the complement of JV{A) has non-
nonempty interior. By Theorem 1.14,
A) (x+V)n jV(A) = 0
for some x e X and some balanced neighborhood F of 0. Then AFis a balanced
subset of the field Ф. Thus either AF is bounded, in which case (d) holds, or
AF= Ф. In the latter case, there exists yeV such that Ay = -Ax, and so
x + у e JV(A), in contradiction to A). Thus (c) implies (d).
Finally, if (d) holds then | Ax\ < M for all x in F and for some M < oo.
If r > 0 and if PF= (r\M)V, then |Ax| < r for every jc in £F. Hence A is con-
continuous at the origin. By Theorem 1.17, this implies (a). ////
Finite-dimensional Spaces
1.19 Among the simplest Banach spaces are Rn and €n, the standards-dimensional
vector spaces over R and €, respectively, normed by means of the usual euclidean
metric: If, for example,

TOPOLOGICAL VECTOR SPACES 15
is a vector in (£", then
Other norms can be defined on Cn. For example,
N1 = к I + '" + kl or \\z\\ =max(|z£|: 1 <i<n).
These norms correspond, of course, to different metrics on €n (when n > 1) but one
can see very easily that they all induce the same topology on €n. Actually, more
is true:
If Xis a topological vector space over €, and dim X = n, then every basis of X
induces an isomorphism of X onto Cn. Theorem 1.21 will prove that this isomorphism
must be a homeomorphism. In other words, this says that the topology ofCn is the only
vector topology that an n-dimensional complex topological vector space can have.
We shall also see that finite-dimensional subspaces are always closed.
Everything in the preceding discussion remains true with real scalars in place of
complex ones.
We start with a lemma, which will be superseded by Theorems 1.21 and 1.22.
1.20 * Lemma Suppose Y is a subspace of a topological vector space X, and Y is
locally compact, in the topology inherited from X. Then Y is a closed subspace of X.
proof There is a compact set К cz Y whose interior (relative to Y) contains 0.
Hence there is a neighborhood U of 0 in X such that U n Y cz K. Choose a
symmetric neighborhood V of 0 in X such that V + V cz U. We claim that the
set
A) Yn(x+V)
is compact, for every xe X.
To see this, fix y0 in A). For any у in A),
У ~ Уо = (У - x) + (x - у о) е V + V с U.
Also, у — у0 е У, since Y is a subspace. Thus
YczK,
which implies that A) lies in the compact set y0 + K. But A) is also a closed
subset of 7, since x + V is closed in X and since Y inherits its topology from
X. Thus A) is a closed subset of a compact set and is therefore compact.
Now fix x e Y. Let $8 be the collection of all open sets Win Xsuch that
0 e W and Wcz V, and associate with each We $8 the set
Ew= Yn(x+W).

16 GENERAL THEORY
Since W 'czV9 each Ew is compact. Since x e Y, no Ew is empty. Since inter-
intersections of finitely many members of & belong to ^, it follows that {Ew : We Щ
is a collection of compact sets with the finite intersection property. Therefore
there exists z e f]Ew. This z lies in Y. On the other hand, z e x + W for every
We@. Thus z = x (Theorem 1.12). Hence x e Y. This proves that Y = Y,
and so Г is closed. ////
1.21 Theorem Suppose X is a complex topological vector space, Y is a subspace of
X, n is a positive integer, and dim Y = n. Then
(a) every isomorphism of €n onto Y is a homeomorphism, and
(b) Y is closed.
The term "homeomorphism" refers, of course, to the euclidean topology of
Cn on the one hand, and to the topology that Y inherits from X on the other. Since
Cn is locally compact, Lemma 1.20 shows that (b) follows from (a). The proof that
follows also yields the analogous theorem with real scalars in place of complex ones.
proof Let Pn be the theorem as stated. We first prove Pt. Let Л: € -> Y be
an isomorphism (i.e., a one-to-one linear mapping of € onto Y). Put и — Л1.
Then Ла = аи. The continuity of the vector space operations in У implies that Л
is continuous. Note that Л is a linear functional on Y with null space {0},
a closed set. By Theorem 1.18, Л is continuous. This proves Pt.
Assume next that n > 1 and Pn_± is true. Let Л: €n -> Y be an isomor-
isomorphism. Let {el9...9 en} be a basis of €n; the /:th coordinate of ek is 1; the others
are 0. Put uk = Aek, for к = 1,..., п. Then
Л(ах,..., ая) = ^щ + - - - + а„ ип,
and the continuity of the vector space operations in Y implies again that Л is
continuous. Since Л is an isomorphism, {щ, ..., un} is a basis of Y. Hence
there are linear functional yl9 ..., yn on Y such that every xe Y has a unique
representation of the form
х^у^щ + -" + yn(x)un.
Each уi has a null space in Y, of dimension n — 1, which is closed in Y9 by the
assumed truth of Pn-V Hence yt is continuous, by Theorem 1.18. Since
A* = (?,(*),..., у„(*)) (xeY),
it follows that Л is continuous. Hence Р„ is true, and the proof is complete.

TOPOLOGICAL VECTOR SPACES 17
1.22 Theorem Every locally compact topological vector space X has finite di-
dimension.
proof The origin of X has a neighborhood V whose closure is compact. By
Theorem 1.15, V is bounded, and the sets 2~nV(n = 1, 2, 3,...) form a local
base for X.
The compactness of V shows that there exist xu...9xmm X such that
Let Y be the vector space spanned by x1,...,xm. Then dim Y<m. By
Theorem 1.21, Fis a closed subspace of X.
Since Vcz Y+ ^Fand since AY— Г for every scalar Я ф 0, it follows that
so that
Vcz Y+iVa Y+ \ i
If we continue in this way, we see that
Fez f](Y+2~nV).
Since {2"~"V} is a local base, it now follows fromJ^Tof Theorem 1.13 that
VczY. ButF = Y. ThusFc Г, which implies that k Vcz Ffor к = 1, 2, 3, ....
Hence Y = Х,Ъу (a) of Theorem 1.15, and consequently dim X < m. ////
1.23 Theorem If X is a locally bounded topological vector space with the Heine-
Bor el property, then X has finite dimension.
proof By assumption, the origin of X has a bounded neighborhood V:
Statement (/) of Theorem 1.13 shows that V is also bounded. Thus V is com-
compact, by the Heine-Borel property. This says that X is locally compact, hence
finite-dimensional, by Theorem 1.22.
Metrization
We recall that a topology топа set X is said to be metrizable if there is a metric d on
X which is compatible with т. In that case, the balls with radius \\n centered at x form
a local base at x. This gives a necessary condition for metrizability which, for topo-
topological vector spaces, turns out to be also sufficient.

18 GENERAL THEORY
1.24 Theorem If X is a topo logical vector space with a countable local base, then
there is a metric d on X such that
(a) d is compatible with the topology of X,
(b) the open balls centered at 0 are balanced, and
(c) d is invariant: d(x + z, у + z) = d(x, y)for x, y, ze X.
If in addition, X is locally convex, then d can be chosen so as to satisfy (a), (b), (c),
and also
(d) all open balls are convex.
proof By Theorem 1.14, Zhas a balanced local base {Vn} such that
0) Vn+1 + Vn + 1cVH (n = 1,2,3,...);
when X is locally convex, this local base can be chosen so that each Vn is also
convex.
Let D be the set of all rational numbers r of the form
B) r = Ycn(rJ~",
where each of the "digits" ct(r) is 0 or 1 and only finitely many are 1. Thus
each r e D satisfies the inequalities 0 < r < 1.
Put A(r) = X if r > 1; for any r e D, define
C) Air) = <?1(гЖ + c2(r)V2 + c3(r)V3 + • • •.
Note that each of these sums is actually finite. Define
D) f{x) = inf {r: x e Air)} (x e X)
and
E) d(x9y)=f(x-y) ixeX,yeX).
The proof that this d has the desired properties depends on the inclusions
F) Air) + Ais) cz Air + s) ire D,se D).
Before proving F), let us see how the theorem follows from it. Since
every Ais) contains 0, F) implies
G) Air) с Air) 4- Ait - r) cz Ait) if r < t.
Thus {Air)} is totally ordered by set inclusion. We claim that
(8) fix + y) <fix) +Ду) ixeX,ye X).
In the proof of (8) we may, of course, assume that the right side is <1. Fix
e > 0. There exist r and s in D such that
fix) < r, fiy) <s, r + s <Дх) +/(y) + e.

TOPOLOGICAL VECTOR SPACES 19
. Thus xeA(r),yeA(s), and F) implies x + у е A(r + s). Now (8) follows,
because
fix + y)<r + s <f(x) +f(y) + e,
and г was arbitrary.
Since each A(r) is balanced, f(x) =/(-*). It is obvious that /@) = 0.
If x ф 0, then x ф Vn = ЛB~") for some я, and so/(x) > 2~rt > 0.
These properties of/show that E) defines a translation-invariant metric
d on A^ The open balls centered at 0 are the open sets
(9) Д,@) = {*:/(*) < 5} = U Л(г).
r<<5
If 5 < 2~", then 5,@) с ]/ . Hence {5,@)} is a local base for the topology of X.
This proves (a). Since each Л(>) is balanced, so is each Вдф). If each Vn is
convex, so is each A{r)\ and G) implies that the same is true of each Bd@), hence
also of each translate of Bd@).
The proof of F) will be by induction. Let PN be the statement:
Ifr + s<\and cn(r) = cn(s) = Ofor all n> N, then
A0) A(r) + A(s) с A(r + s).
Pt is true, by inspection. Assume PN-X is true, for some N > 1. Choose
r e D, s e D, so that r + s < I and с„(г) = cn(s) = 0 if л > N, and define r' and
s' by
A1) r = r' + cN(rJ-N, s = s' + cN(sJ~N.
Then
A2) A(r) = ^(r') + cw(r)Kw, ^(j) = ^(O + cN(s)VN.
ByPjv.^^rO + ^O^^' + j'). Hence
A3) A{r) + A(s) с ,4(r' + J') + ^W^iv + cN(s)VN.
If Cjv(r) = Cff(s) = 0, then r = r', 5- = 51', and A3) gives A0). If cN(r) = 0
and cN(s) = 1, the right side of A3) is
A(r' + s') +VN = A(r' + s' + 2~N) = A(r + s),
so that A0) holds again. The*case cN(r) = 1, cN(s) = 0 is handled the same way.
If cN(r) = cn(j) = 1, the right side of A3) is
A{r' + s') + VN + F,v cz ,4(r' + 5') + К„_!
= Л(г' + 5') + y4B~N+1) cz A(r' + s' + 2~* + 1) - ^(r + j).
The last inclusion depended on PN-t.
Thus PN_X implies P^. Hence F) is correct, and the proof is complete.

20 GENERAL THEORY
1.25 Cauchy sequences (a) Suppose d is a metric on a set X. A sequence {xj in
X is a Cauchy sequence if to every e > 0 there corresponds an integer N such that
d(xm, xn) < e whenever m> N and n > N. If every Cauchy sequence in X converges
to a point of X, then d is said to be a complete metric on X.
(b) Let т be the topology of a topological vector space X. The notion of Cauchy
sequence can be defined in this setting without reference to any metric: Fix a local
base J? for т. A sequence {х„} in X is then said to be a Cauchy sequence if to every
FgJ corresponds an N such that xn — xm e V if n > N and m> N.
It is clear that different local bases for the same т give rise to the same class of
Cauchy sequences.
(c) Suppose now that X is a topological vector space whose topology т is
compatible with an invariant metric d. Let us temporarily use the terms d-Cauchy
sequence and т-Cauchy sequence for the concepts defined in (a) and (&), respectively.
Since
d(x*>xm) = d(xn-xm90),
and since the d-balls centered at the origin form a local base for т, we conclude:
A sequence {xn} in Xis a d-Cauchy sequence if and only if it is a x-Cauchy sequence.
Consequently, any two invariant metrics on X that are compatible with т have
the same Cauchy sequences. They clearly also have the same convergent sequences
(namely, the т-convergent ones). These remarks prove the following theorem:
1.26 Theorem Ifd1 and d2 are invariant metrics on a vector space X which induce
the same topology on X, then
(a) di and d2 have the same Cauchy sequences, and
(b) dx is complete if and only if d2 is complete.
Invariance is needed in the hypothesis (Exercise 12).
The next theorem is an analogue of Lemma 1.20, with completeness in place of
local compactness. Note that the two proofs are quite similar.
1.27 Theorem Suppose Y is a subspace of a topological vector space X, and Y is
an F-space {in the topology inherited from X). Then Y is a closed subspace of X.
' proof Choose an invariant metric d on Y, compatible with- its topology. Let
let Un be a neighborhood of 0 in A" such that Y n Un = B1/n, and choose sym-
symmetric neighborhoods Vn of 0 in Xsuch that Vn+VnaUn.

TOPOLOGICAL VECTOR SPACES 21
Suppose x e Y, and define
En= Yn(x+Vn) (/i = l,2,3,...).
If jx e En and y2 e En, then j^ - j2 lies in Y and also in Vn + Vn cz £/n, hence
in Б1/п. The diameters of the sets En therefore tend to 0. Since each En is
nonempty and since Yis complete, it follows that the F-closures of the sets En
have exactly one point y0 in common.
Let W be a neighborhood of 0 in X, and define
Fn=Y n(x + WnVn).
The preceding argument shows that the Y-closures of the sets Fn have one
common point yw. But Fn cz En. Hence yw = y0. Since Fn cz х + Ж, it
follows that >>0 lies in the ^-closure of x + ИК, for every ЙК This implies y0 = x.
Thus x e Y. This proves that 7 = Y. ////
The following simple facts are sometimes useful.
1.28 Theorem
(a) If d is a translation-invariant metric on a vector space X then
d(nx, 0) < nd(x, 0)
for every xe X and for n = 1, 2, 3,
(b) If {xn} is a sequence in a metrizable topological vector space X and if xn -> 0
as n-+ oo, then there are positive scalars yn such that yn -> oo and yn xn -> 0.
proof Statement (a) follows from
d(nx, 0) < f d(kx, (k -l)x) = nd(x, 0).
To prove (b), let d be a metric as in (a), compatible with the topology of X.
Since ^(xrt, 0) ->0, there is an increasing sequence of positive integers nk such
that d(xn, 0) < k~2 if л > лл. Put ?„ = 1 if л < n±; put 7rt = к if яЛ < w < nk+1.
For such л,
d(yHXn90) = J(/cxrt, 0) < Ы(Х, 0) < Л.
Hence yn xn -> 0 as n -> oo. ^-; ////
Boundedness and Continuity
1.29 Bounded sets The notion of a bounded subset of a topological vector space X
was defined in Section 1.6 and has been encountered several times since then. When X
is metrizable, there is a possibility of misunderstanding, since another very familiar
notion of boundedness exists in metric spaces:

22 GENERAL THEORY
If d is a metric on a set X, a. set E cz X is said to be ^-bounded if there is a
number M < со such that <фс, ;>) < M for all x and у in £.
If JT is a topological vector space with a compatible metric d, the bounded sets
and the ^-bounded ones need not be the same, even if d is invariant. For instance, if d
is a metric such as the one constructed in Theorem 1.24, then X itself is d-bounded
(with M = 1) but, as we shall see presently, X cannot be bounded, unless X = {0}. If
A" is a normed space and d is the metric induced by the norm, then the two notions of
boundedness coincide; but if d is replaced by dx — d/(l + d) (an invariant metric
which induces the same topology) they do not.
Whenever bounded subsets of a topological vector space are discussed, it will be
understood that the definition is as in Section 1.6: A set E is bounded if, for every
neighborhood V of 0, we have E cz / V for all sufficiently large t.
We already saw (Theorem 1.15) that compact sets are bounded. To see another
type of example, let us prove that Cauchy sequences are bounded (hence convergent
sequences are bounded): If {xn} is a Cauchy sequence in X, and Fand Wave balanced
neighborhoods of 0 with V + V cz W, then [part (b) of Section 1.25] there exists N
such that xnexN + V for all n > N. Take s > 1 so that xN e sV. Then
xnesV+ VczsV + sVczsW (n >N).
Hence xn e tW for all n > 1, if t is sufficiently large.
Also, closures of bounded sets are bounded (Theorem 1.13).
On the other hand, if x ф 0 and E = {nx: n = 1, 2, 3, ...}, then E is not bounded,
because there is a neighborhood К of 0 that does not contain x; hence nx is not in nV;
it follows that no n V contains E.
Consequently, no subspace of X {other than {0}) can be bounded.
The next theorem characterizes boundedness in terms of sequences.
1.30 Theorem The following two properties of a set E in a topological vector
space are equivalent:
(a) E is bounded.
(b) If {xn} is a sequence in E and {ocn} is a sequence of scalars such that ccn -> 0 as
n -> oo, then <xn xn -> 0 as n -> oo.
proof Suppose E is bounded. Let V be a balanced neighborhood of 0 in X.
Then E cz t V for some t. If xn e E and ocn -> 0, there exists N such that | an \ t < 1
if n > N. Since t~xE cz V and V is balanced, anxn e V for all n > N. Thus
anxn->0.
Conversely, if E is not bounded, there is a neighborhood V of 0 and a
sequence rn -> oo such that no rn Vcontains E. Choose xn e E such that xn ф rn V.
Then no r~lxn is in F, so that {r~lxn} does not converge to 0. ////

TOPOLOGICAL VECTOR SPACES 23
1.31 Bounded linear transformations Suppose Xand У are topological vector
spaces and Л: X'-> У is linear. Л is said to be bounded if Л maps bounded sets into
bounded sets, i.e., if A(E) is a bounded subset of У for every bounded set E <= X.
This definition conflicts with the usual notion of a bounded function as being
one whose range is a bounded set. In that sense, no linear function (other than 0)
could ever be bounded. Thus when bounded linear mappings (or transformations) are
discussed, it is to be understood that the definition is in terms of bounded sets, as
above.
1.32 Theorem Suppose X and Y are topological vector spaces and A: X-+ Y is
linear. Among the following four properties of A, the implications
hold. If X is metrizable, then also
so that all four properties are equivalent.
(a) .■ Л is continuous.
(b) A is bounded.
(c) Ifxn-+0 then {Axn: n = 1, 2, 3, ...} is bounded.
(d) Ifxn-+0 then Axn -> 0.
Exercise 13 contains an example in which (b) holds but (a) does not.
proof Assume (a), let E be a bounded set in X, and let W be a neighborhood
of 0 in Y. Since Л is continuous (and Л0 = 0) there is a neighborhood V of 0 in X
such that A(V) с W. Since E is bounded, E с tV for all large r, so that
A(E) с A(tV) = tA(V) ctW.
This shows that Л(£) is a bounded set in Y.
Thus (a) -> (b). Since convergent sequences are bounded, (b) -> (c).
Assume now that A4s metrizable, that Л satisfies (c)9 and that xn ->0. By
Theorem 1.28, there are positive scalars yn -> oo such that ynxn-+Q. Hence
{A(ynxn)} is a bounded set in Y, and now Theorem 1.30 implies that
Axn=y;1A(ynxn)-^0 as n -> oo.
Finally, assume that (a) fails. Then there is a neighborhood W of 0 in Y
such that A~1(W) contains no neighborhood of 0 in X. If J^has a countable
local base, there is therefore a sequence {xn} in X so that xn -> 0 but Axn ^ W.
Thus (J) fails. ////

24 GENERAL THEORY
Seminorms and Local Convexity
1.33 Definitions A seminorm on a vector space Zis a real-valued function ;? on X
such that
(a) p(x + y)<p(x)+p(y)
(b) p(ax) = \*\p(x)
for all x and у in X and all scalars a.
Property (a) is called subadditivity. Theorem 1.34 will show that a seminorm^
is a norm if it satisfies
(c)
A family 0> of seminorms on X is said to be separating if to each x ф 0 corre-
corresponds at least one p e & with p{x) ф 0.
Next, consider a convex set А с X which is absorbing, in the sense that every
x e X lies in tA for some t = /(x) > 0. [For example, (a) of Theorem 1.15 implies that
every neighborhood of 0 in a topological vector space is absorbing. Every absorbing
set obviously contains 0.] The Minkowski functional jia of A is defined by
цА(х) = inf {t > 0: Гхх gA} (x e X).
Note that fiA(x) < oo for all x e X, since A is absorbing. The seminorms on X will
turn out to be precisely the Minkowski functionals of balanced convex absorbing sets.
Seminorms are closely related to local convexity, in two ways: In every locally
convex space there exists a separating family of continuous seminorms. Conversely, if
& is a separating family of seminorms on a vector space X, then 0* can be used to
define a locally convex topology on Zwith the property that every p e 0 is continuous.
This is a frequently used method of introducing a topology. The details are contained
in Theorems 1.36 and 1.37.
1.34 Theorem Suppose p is a seminorm on a vector space X. Then
(a) />@) = 0.
(b) \p(x)-p(y)\ <p{x-y).
(c) p(x)>0.
(d) {x: p(x) = 0} is a subspace of X.
(e) The set В = {x: p(x) < 1} is convex, balanced, absorbing, and p = [xB.
proof Statement (a) follows from p(ax) = \a\p(x), with a = 0. The subaddi-
subadditivity of p shows that
p(x) =p(x-y + y) <p{x - y) +p(y)
so that p(x) — p(y) < p(x — y). This also holds with x and у interchanged.

TOPOLOGICAL VECTOR SPACES 25
Since p(x - y) = p(y - x), (b) follows. With у = 0, (b) implies (c). If p(x) =
p{y) = 0 and a, /? are scalars, (c) implies
0<p((xx + py) < \a\p(x) + \p\p(y) = 0.
This proves (d).
As to (e), it is clear that В is balanced. If x e В, у e B, and 0 < / < 1, then
/?('* + A - t)y) < tp(x) + A - t)p(y) < 1.
Thus Б is convex. If x e J^ands >/?(•*) then^^) = s~ 1p(x) < 1. This shows
that Б is absorbing and also that juB(x) < s. Hence /лв <p. But if 0 < t <p(x)
then p(t~1x) > 1, and so t^ is not in Б. This implies p(x) < jnB(x) and com-
completes the proof. ////
1.35 Theorem Suppose Л is a convex absorbing set in a vector space X. Then
(a) fiA(x + y) < iijx) + fiA(y).
(b) fiA(tx) = tjnA(x)ift>0.
(c) juA is a seminorm if A is balanced.
(d) If В = {x: juA(x) < 1} and С = {x: цА(х) < 1}, then В а Л с С and juB = jia = juc.
proof Associate with each x e X the set
HA(x) = {t>0:t~lxeA}.
Suppose t e HA(x) and s > t. Since 0 e A and A is convex, it follows that
s e HA(x). Each HA(x) is a half line whose left end point is }iA(x).
Suppose fiA(x) < s, fiA(y) < t,u — s + t. Then s~1x e A, t~*y e A. Since A
is convex,
lies in A. Hence fiA(x -f y) <u. This gives (a). Properties (b) and (c) are now
obvious.
If iiA{x) < 1, then I e HA(x), and so x e A. Likewise, if x e A, then
цА(х) < 1. Thus BczAczC. This implies Нв(х) с НА(х) cz Hc(x), for every
x e X, so that
To prove that equality holds, suppose juc(x) < s < t. Then s~1xeC, hence
/^O.*) < 1, so that
Thus t~xx e B, fijsir^) < 1, /лв(х) < t. This completes the proof. ////

26 GENERAL THEORY
1.36 Theorem Suppose & is a convex balanced local base in a topological vector
space X. Associate to every V e $ its Minkow ski functional [xv. Then {jiv: УеЩ is a
separating family of continuous seminorms on X.
proof Since Fis convex, balanced, and absorbing, jiv is a seminorm. If x e X
and x ф 0, then x ф V for some Ve@. For this V we have jiv(x) > 1. Thus
{jiy} is a separating family. If x e V9 then tx e Ffor some t > 1, since Fis open.
Hence /zF < 1 in V. If r > 0, it follows from Theorem 1.34 that
Vv(x ~ У) < г
if x — у e rV. This proves that each jiv is continuous. ////
1.37 Theorem Suppose 0 is a separating family of seminorms on a vector space X.
Associate to each peg? and to each positive integer n the set
Let M be the collection of all finite intersections of the sets V(p, n). Then & is a convex
balanced local base for a topology % on X, which turns X into a locally convex space
such that
(a) every p e £P is continuous, and
(b) a set E с X is bounded if and only if every p e 0 is bounded on E.
proof Declare a set А с X to be open if and only if A is a (possibly empty)
union of translates of members of 0&. This clearly defines a translation-invariant
topology т on X; each member of M is convex and balanced, and ^ is a local
base for т.
Suppose xe X, x ф 0. Then p(x) > 0 for some p e 0*. Since x is not in
V(p, n) if np(x) > 1, we see that 0 is not in the neighborhood x — V(p, n) of x,
so that x is not in the closure of {0}. Thus {0} is a closed set, and since т is
translation-invariant, every point of X is a closed set.
Next we show that addition and scalar multiplication are continuous.
Let U be a neighborhood of 0 in X. Then
A) U^V{Pl,ni)n---^V{pm,nm)
for some pl9 ... ,pmeP and some positive integers nl9..., nm. Put
B) K=F(p1>2«1)n-"n К(ри, 2иш).
Since every p e 0 is subadditive, V + V <= U. This proves that addition is
continuous.

TOPOLOGICAL VECTOR SPACES 27
Suppose now that x e X, a is a scalar, and U and V are as above. Then
xesVfov some s> 0. Put / = j/(l + \oc\s). If у ex + tV and \fl — a\ < 1/s,
then
py-(xx = p(y -x) + {p- *)x
which lies in
\P\tV+ \p-oc\sVcz V+VczU
since |/?|f < 1 and К is balanced. This proves that scalar multiplication is
continuous.
Thus X is a locally convex space. The definition of V(p, n) shows that
every pe 0>is continuous at 0. Hence p is continuous on X, by (b) of Theorem
1.34.
Finally, suppose E cz X is bounded. Fix p e &. Since F(p, 1) is a neigh-
neighborhood of 0, £ cz kV(p, 1) for some к < со. Hence />(jt) < к f°r every xe E.
It follows that every /7 e ^ is bounded on E.
Conversely, suppose E satisfies this condition, U is a neighborhood of 0,
and A) holds. There are numbers Mt < oo such that pt < Mt on E A < / < m).
.■ If л > Л//Л,- for 1 < / < m, it follows that E <=: nU, so that £ is bounded. ////
1.38 Remarks (a) It was necessary to take finite intersections of the sets V(p, n)
in Theorem 1.37; the sets F(p, n) themselves need not form a local base. (They do form
what is usually called a subbase for the constructed topology.) To see an example of
this, take X = R2, and let 0* consist of the seminorms p± and p2 defined by Pi(x) =
|x;|; here x = (xlf x2). Exercise 8 develops this comment further.
(b) Theorems 1.36 and 1.37 raise a natural problem: If & is a convex balanced
local base for the topology т of a locally convex space X, then <% generates a separating
family 0* of continuous seminorms on X, as in Theorem 1.36. This 0 in turn induces a
topology тх on X, by the process described in Theorem 1.37. Is т = тх ?
The answer is affirmative. To see this, note that every p e 0 is т-continuous, so
that the sets V(p, n) of Theorem 1.37 are in т. Hence тх cz т. Conversely, if We 3&
and p = /%, then
W={x:nw{x)<\}=V{p,\).
Thus Wexx for every We$&\ this implies that тс^.
(c) If ^ = {/?,-: / = 1, 2, 3,...} is a countable separating family of seminorms on
X, Theorem 1.37 shows that 0> induces a topology т with a countable local base. By
Theorem 1.24, т is metrizable. In the present situation, a compatible translation-
invariant metric can be defined directly in terms of {pt}. Define
M 1 + Pt(x - y)

28 GENERAL ^THEORY
] j
It is easy to verify that d is a metric on X. To prove that d is compatible with т, we
show that the balls
B) Br = {x:d(x,0)<r} (r>0)
form a local base for т.
Since each pt is continuous (Theorem 1.37) and since the series A) converges
uniformly onlx X, dis continuous; hence each Br is open. If W is a neighborhood
of 0, then W contains the intersection of appropriately chosen sets
C) V(Pi, n,) = (*: Pi(x) < 1} A < / < k).
IfxeBr, then
D) ЙПП<' ('=1.2,3,...).
If r is small enough, D) forces Pt(x),.. .,pk(x) to be so small that Br lies in each of the
sets C); hence Br с W.
This proves that d is compatible with т.
Formula A) has considerable advantages over the more complicated construc-
construction of Theorem 1.24. Of course, A) is applicable only in locally convex spaces, and
it has a flaw even there: The balls which it defines need not be convex. An example of
this is given in Exercise 18.
1.39 Theorem A topological vector space X is normable if and only if its origin has
a convex bounded neighborhood.
proof If Xis normable, and if ||-|| is a norm that is compatible with the topol-
topology of X, then the open unit ball {x: \\x\\ < 1} is convex and bounded.
For the converse, assume К is a convex bounded neighborhood of 0. By
Theorem 1.14, К contains a convex balanced neighborhood U of 0; of course, U
is also bounded. Define
A) ||*||= M*) (xeX)
where ft is the Minkowski functional of U.
By (c) of Theorem 1.15, the sets rU (r > 0) form a local base for the topol-
topology of X. If x ф 0, then x ф rU for some r > 0; hence ||x|| > r. It now follows
from Theorem 1.35 that A) defines a norm. The definition of the Minkowski
functional, together with the fact that U is open, implies that
B) {x:\\x\\<r} = rU
for every r > 0. The norm topology coincides therefore with the given one.

TOPOLOGICAL VECTOR SPACES 29
Quotient Spaces
1.40 Definitions Let TV be a subspace of a vector space X. For every xe X, let
n(x) be the coset of TV that contains x; thus
n(x) = x + N.
These cosets are the elements of a vector space X/N, called the quotient space of X
modulo TV, in which addition and scalar multiplication are defined by
A) n(x).+ n(y) = n(x + y), an(x) = n(ocx).
[Note that now an(x) = TV when a — 0. This differs from the usual notation, as
introduced in Section 1.4.] Since TV is a vector space, the operations A) are well
defined. This means that if n(x) = n(xf) (that is, x' — x e TV) and n(y) = n(y') then
B) n(x) + n(y) = n(x') + л(у'), an(xf) = an(x).
The origin of X/N is n@) = N. By A), n is a linear mapping of X onto X/N with
7Vas its null space; n is often called the quotient map of J^onto X/N.
Suppose now that т is a vector topology on J^and that TV is a closed subspace of
X. • Let tn be the collection of all sets E с X/JV for which n~1(E) e т. Then tn turns
out to be a topology on JT/iV, called the quotient topology. Some of its properties are
listed in the next theorem. Recall that an open mapping is one that maps open sets to
open sets.
1.41 Theorem Let N be a closed subspace of a topological vector space X. Let т
be the topology of X and define xN as above.
(a) xN is a vector topology on X/N; the quotient map n: X-+X/N is linear, con-
continuous, and open.
(b) If ^ is a local base for т, then the collection of all sets n(V) with Ve 88 is a local
base for tn .
(c) Each of the following properties of X is inherited by X/N: local convexity, local
. boundedness, metrizability, normability.
(d) If X is an F-space, or a Frechet space, or a Banach space, so is X/N.
proof Since n~l(A n Bj^= n~\A) n n~\B) and
rN is a topology. A set F с X/N is Тдг-closed if and only if n~l(F) is т-closed.
In particular, every point of X/N is closed, since
and TV was assumed to be closed.

30 GENERAL THEORY
The continuity of n follows directly from the definition of rN. Next,
suppose Vex. Since
and N + V e т, it follows that n(V) e xN . Thus n is an open mapping.
If now W is a neighborhood of 0 in A77V, there is a neighborhood К of
0 in X such that
V+ V^7i\
Hence 7r(K) + n(V) с Ж. Since 7г is open, 7r(F) is a neighborhood of 0 in X/N.
Addition is therefore continuous in X/N.
The continuity of scalar multiplication in X/Nis proved in the same manner.
This establishes (a).
It is clear that (a) implies (b). With the aid of Theorems 1.32, 1.24, and
1.39, it is just as easy to see that (b) implies (c).
Suppose next that d is an invariant metric on X, compatible with т.
Define p by
p(n(x), n(y)) = M{d(x -y,z):ze N}.
This may be interpreted as the distance from x — у to N. We omit the veri-
verifications that are now needed to show that p is well defined and that it is an
invariant metric on X/N. Since
n({x:d(x,0)<r})={u:p(u,0)<r},
it follows from (b) that p is compatible with %.
If X is normed, this definition of p specializes to yield what is usually
called the quotient norm of X/N:
To prove (d) we have to show that p is a complete metric whenever d is
complete.
Suppose {un} is a Cauchy sequence in X/N, relative to p. There is a
subsequence {un.} with p(un., ua. + l) <2~\ One can then inductively choose
xt £ X such that n(x^) = un. and d(xt, xi+1) <2~\ If d is complete, the Cauchy
sequence {xt} converges to some xeX. The continuity of я implies that un. -> n(x)
as /-> oo. But if a Cauchy sequence has a convergent subsequence then the full
sequence must converge. Hence p is complete, and so is the proof of Theorem
1.41. ////

TOPOLOGICAL VECTOR SPACES 31
Here is*an easy application of these concepts:
1.42 Theorem Suppose N and F are subspaces of a topological vector space X,
N is closed and F has finite dimension. Then N -f F is closed.
proof Let n be the quotient map of X onto X/N, and give X/N its quotient
topology. Then n(F) is a finite-dimensional subspace of X/N; since X/N is a
topological' vector space, Theorem 1.21 implies that n(F) is closed in X/N. Since
N + F = 7C1Gt(F)) and n is continuous, we conclude that N + Fis closed. (Com-
(Compare Exercise 20). . ////
1.43 Seminorms and quotient spaces Suppose p is a seminorm on a vector space
Zand
. N={x:p(x) = 0}.
Then JVis a subspace of X (Theorem 1.34). Let n be the quotient map of X onto X/N,
and define
p(n(x)) = p(x).
If it(x) = n(y), then p(x — y) = 0, and since
<Р(х- У)
it follows that p{n(x)) = p(n(y)). Thus p is well defined on X/N, and it is now easy to
verify that p is a norm on X/N.
Here is a familiar example of this. Fix r, 1 < r < oo; let П be the space of all
Lebesgue measurable functions on [0, 1] for which
This defines a seminorm on П, not a norm, since ||/||r = 0 whenever /=0 almost
everywhere. Let N be the set of these "null functions." Then П/N is the Banach
space that is usually called П. The norm of П is obtained by the above passage from
ptop.
Examples
1.44 The spaces C(fl) If Q is a nonempty open set in some euclidean space, then
Q is the union of countably many compact sets Kn Ф 0 which can be chosen so that
Kn lies in the interior of Kn + 1 (n = 1, 2, 3, ...). C(Q) is the vector space of all complex-
valued continuous functions on Q, topologized by the separating family of seminorms
A) Pnif) = su

32 GENERAL THEORY
j
in accordance with Theorem 1.37. Since p1 < p2 < • • •, the sets
B) Vn ={/e C(U):pn(f) < I) (л = 1, 2, 3, ...)
form a convex local base for C(Q). According to remark (c) of Section 1.38, the
topology of C(Q) is compatible with the metric
C)
pn(
If {/J is a Cauchy sequence relative to this metric, then pn(ft —/}) -> 0 for every n, as
h j-+ °°, so tnat {/;} converges uniformly on In, to a function fe C(Q). An easy
computation then shows d(ff) ->0. Thus J is a complete metric. We have now
proved that C(Q) is a Frechet space.
By (b) of Theorem 1.37, a set £"cz C(Q) is bounded if and only if there are
numbers Mn < oo such that pn(f) < Mn for all/e E; explicitly,
D) \f(x) | < Mn if /e £ and jc e A,.
Since every Fn contains an/for which pn+1(f) is as large as we please, it follows that
no Vn is bounded. Thus C(Q) is not locally bounded, hence is not normable.
1.45 The spaces H(Q) Let п now be a nonempty open subset of the complex
plane, define C(Q) as in Section 1.44, and let H(Q) be the subspace of C(Q) that con-
consists of the holomorphic functions in Q. Since sequences of holomorphic functions
that converge uniformly on compact sets have holomorphic limits, H(Q) is a closed
subspace of C(Q). Hence H(Q) is a Frechet space.
We shall now prove that H(Q) has the Heine-Borel property. It will then follow
from Theorem 1.23 that H(Q) is not locally bounded, hence is not normable.
Let £ be a closed and bounded subset of #(£>). Then E satisfies inequalities
such as D) of Section 1.44. Montel's classical theorem about normal families (Th. 14.6
of [23]x) implies therefore that every sequence {/)} cz E has a subsequence that con-
converges uniformly on compact subsets of Q [hence in the topology of H(Q)] to some
fe H(Q). Since E is closed, fe E. This proves that E is compact.
1.46 The spaces C°°(Q) and Qjk We begin this section by introducing some
terminology that will be used in our later work with distributions.
'In any discussion of functions of n variables, the term multi-index denotes an
ordered «-tuple
A) a = (ax, ...,а„)
1 Numbers in brackets refer to sources listed in the Bibliography.

TOPOLOGICAL VECTOR SPACES 33
of nonnegative integers a,-. With each multi-index a is associated the differential
operator
/ д V1 id \*n
B) "-У -У
whose order is
C) , " |a| =<*!+••• + «„.
If |a| =0,/)-/=/.
A complex function/defined in some nonempty open set Q cz R" is said to belong
to C°°(Q) if £>*/e C(Q) for every multi-index a.
The support of a complex function/(on any topological space) is the closure of
{х:/(х)Ф0}. -
If К is a compact set in /?", then ^x denotes the space of all/e C°°(jR") whose
support lies in j£. (The letter ^ has been used for these spaces ever since Schwartz
published his work on distributions.) If KczQ, then <3K may be identified with a
subspace of C°°(Q).
•' We now define a topology on С°°(Ц) which makes C°°(Q) into a Frechet space with
the Heine-Borel property, such that Q)K is a closed subspace ofC°°(Q) whenever К cz Q.
To do this, choose compact sets K{ {i— 1, 2, 3, ...) such that Kv lies in the
interior of Ki + 1 and Q — \jK-t. Define seminorms pN on С°°(П), N — 1, 2, 3, ..., by
setting
D) pN(f) = max{\D7(x)\:xeKN, \oc\<N}.
They define a metrizable locally convex topology on C°°(Q); see Theorem 1.37 and
remark (c) of Section 1.38. For each xeQ, the functional/->/(x) is continuous in
this topology. Since <3)K is the intersection of the null spaces of these functional, as x
ranges over the complement of K, it follows that Q)K is closed in C°°(Q).
A local base is given by the sets
E) VN = (/6 С"(П): Ы/) < ^} (JV = 1, 2, 3,.. .)•
^ *
If {/;.} is a Cauchy sequence in C°°(O) (see Section 1.25) and if A4s fixed, then/ -/} e FN
if i and у are sufficiently large. Thus \D*fi - D%\ <l/N on KN, if \oc\ < N. It
follows that each D% converges (uniformly on compact subsets of Q) to a function^.
In particular, ft(x) -+go(x). It is now evident that g0 e C°°(Q), that ga = D"g0, and
that/ -+g in the topology of C°°(Q).
Thus C°°(Q) is a Frechet space. The same- is true of each of its closed sub-
spaces @K.

34 GENERAL THEORY
Suppose next that E а С°°(П) is closed and bounded. By Theorem 1.37, the
boundedness of E is equivalent to the existence of numbers MN < oo such that
pN(f) < MN for N = 1, 2, 3, ... and for all fe E. The inequalities | DJ\ < MN, valid
on A^ when | oc | < N, imply the equicontinuity of {Dpf:fe E}on KN-l9if \fi\ <N— 1.
It now follows from Ascoli's theorem (proved in Appendix A) and Cantor's diagonal
process that every sequence in E contains a subsequence {/)} for which {Dpft} con-
converges, uniformly on compact subsets of Q, for each multi-index /?. Hence {/J con-
converges in the topology of С°°(£У). This proves that E is compact.
Hence C°°(Q) has the Heine-Borel property. It follows from Theorem 1.23 that
С°°(£У) is not locally bounded, hence not normable. The same conclusion holds for Q)K
whenever К has nonempty interior (otherwise Q)K = {0}), because dim $)K = oo in that
case. This last statement is a consequence of the following proposition:
If Вг and В2 are concentric closed balls in Rn, with B1 in the interior of B2, then
there exists ф е С°°(Яп) such that ф(х) = 1 for every x e B^ and ф(х) = 0 for every x
outside B2.
To find such а ф, we construct g e C00(R1) such that g(x) = 0 for x < a, g(x) = 1
for x > b (where 0 < a < b < oo are preassigned) and put
F) ф(хи...,хп) = 1-д(х21 + ---+х2п).
The following construction of g has the advantage that suitable choices of {<5f} can lead
to functions with other desired properties.
Suppose 0 < a < b < oo. Choose positive numbers So, dl9 d2, ---, with Sc); =
b — a; put
G) mn = ^_ (й = 1,2,3,...);
let /0 be a continuous monotonic function such that fo(x) = 0 when x < a, fo(x) = 1
when x > a + <50; and define
(8) /„(*) = if* fn_,(t)dt (n = 1,2,3,...).
OnJx-6n
Differentiation of this integral shows, by induction, that/, has n continuous derivatives
and that | Dnfn \<mn. If n > r, then
(9) %W = ~ f V/«-1)(* - 0 dt,
°nJ0
so that
A0) ^ \DJn\<mr (n>r)f

TOPOLOGICAL VECTOR SPACES 35
again by induction on n. The mean value theorem, applied to (9), shows that
A1) | D% - ///„-i I < i"r+i »n (V>r + 2).
Since ZSn < oo, each {Drfn} converges, uniformly on (— oo, oo), as n -> oo. Hence {/J
converges to a function g, with | Drg\ < mr for r = 1, 2, 3, ..., such that #(X) = 0 for
x < a and g(x) = 1 for x > b.
1.47 The spaces Lp with 0 < p < 1 Consider a fixed/? in this range. The elements
of IF are those Lebesgue measurable functions / on [0, 1 ] for which
О) А(/)= Jo |/@|'A<oo,
with the usual identification of functions that coincide almost everywhere. Since
0 <p < 1, the inequality
B) (a + b)p <ap + bp
holds when a > 0 and b > 0. This gives
C) ■
so that
D)
defines an invariant metric on IF. That this d is complete is proved in the same way as
in the familiar case p>\. The balls
E) Br = {feU:A(f)<r}
form a local base for the topology of II. Since B^ = r~1/pBr, for all r > 0, Bt is
bounded.
Thus IF is a locally bounded F-space.
We claim that IF contains no convex open sets, other than 0 and IF.
To prove this, suppose V ф 0 is open and convex in IF. Assume 0 e F, without
loss of generality. Then V^> Br9 for some r > 0. Pick/e II. Since p < 1, there is a
positive integer n such that np~^A(f) < r. By the continuity of the indefinite integral
of |/|p, there are points
0 = x0 < X] < • • • < xn = 1
such that
F) J" |/@l'Л = 11-

36 GENERAL THEORY
}
Define g^t) = nf(t) if Х;_х < t < xt, #,@ = 0 otherwise. Then gt e V, since F) shows
G) A(gi)=np~1 A(f)<r (\<i<n)
and K=> Br. Since Fis convex and
(8) /=;;tei+ -"+Д.),
it follows that/e К Hence F= Lp.
This lack of convex open sets has a curious consequence.
Suppose Л : IF -> Y is a continuous linear mapping of IP into some locally con-
convex space Y. Let ^ be a convex local base for У. If И^е ^, then A^) is convex,
open, not empty. Hence A~l(W) = IF. Consequently, A(IF) <= W for every We $8.
We conclude that Л/= О for every/e IF.
Thus 0 is the only continuous linear mapping of IF into any locally convex space Y,
ifO <p < 1. In particular, 0 is the only continuous linear functional on these IF-spaces.
This is, of course, in violent contrast to the familiar case p > 1.
Exercises
1 Suppose Xis a vector space. All sets mentioned below are understood to be subsets of X.
Prove the following statements from the axioms as given in Section 1.4. (Some of these
are tacitly used in the text.)
(a) If xe Xand yeXthere is a unique ze Xsuch that x + z = y.
(b) Ox = 0 = aO if x e X and a is a scalar.
(c) 2A <=■ A + A; it may happen that 2A Ф A + A.
(d) A is convex if and only if (s + t)A =sA-\- tA for all positive scalars s and t.
(e) Every union (and intersection) of balanced sets is balanced.
(/) Every intersection of convex sets is convex.
(g) If Г is a collection of convex sets that is totally ordered by set inclusion, then the
union of all members of Г is convex.
(h) If A and В are convex, so is A + B.
(/) If A and В are balanced, so is A + B.
(j) Show that parts (/), (g), and (h) hold with subspaces in place of convex sets.
2 The convex hull of a set A in a vector space X is the set of all convex combinations of
members of A, that is, the set of all sums
tlxl -(-•••-(- tn xn
in which Xi e A, tt > 0, 2 * * = 1; «is arbitrary. Prove that the convex hull of Л is convex
and that it is the intersection of all convex sets that contain A.
3 Let ХЫ a topological vector space. All sets mentioned below are understood to be sub-
subsets of X. Prove the following statements.
(a) The convex hull of every open set is open.

TOPOLOGICAL VECTOR SPACES 37
(b) If A4s locally convex then the convex hull of every bounded set is bounded. (This is
false without local convexity; see Section 1.47.)
(c) If A and В are bounded, so is A + B.
(d) If A and В are compact, so is A -f B.
(e) If A is compact and В is closed, then A-\- В is closed.
(/) The sum of two closed sets may fail to be closed. [The inclusion in (b) of Theorem
1.13 may therefore be strict.]
Let B = {(zl9z2)'e <Z2: |z,|<|z2|}. Show that В is balanced but that its interior is not.
[Compare with (e) of Theorem 1.13.]
Consider the definition of "bounded set" given in Section 1.6. Would the content of
this definition be altered if it were required merely that to every neighborhood V of 0
corresponds some t > 0 such that E^ tVI
Prove that a set E in a topological vector space is bounded if and only if every countable
subset of E is bounded.
Let ХЫ the vector space of all complex functions on the unit interval [0, 1], topologized
by the family of seminorms
This topology is called the topology of pointwise convergence. Justify this terminology.
Show that there is a sequence {/,} in X such that (a) {/„} converges to 0 as и -> oo,
but (b) if {у„} is any sequence of scalars such that yn -> oo then {ynfn} does not converge
to 0. (Use the fact that the collection of all complex sequences converging to 0 has the
same cardinality as [0, 1].)
This shows that metrizability cannot be omitted in (b) of Theorem 1.28.
(a) Suppose УР is a separating family of seminorms on a vector space X. Let & be the
smallest family of seminorms on X that contains 0* and is closed under max. [ This
means: If px e£l,p2e Д and p = max (pi,p2), then p e й.] If the construction of
Theorem 1.37 is applied to 0* and to % show that the two resulting topologies coin-
coincide. The main difference is that ^ leads directly to a base, rather than to a subbase.
[See Remark (a) of Section 1.38.]
(b) Suppose 2, is as in part (a) and Л is a linear functional on X. Show that Л is contin-
continuous if and only if there exists a/? e 2, such that | Ax\ < Mp(x) for all x e Jifand some
constant M < oo.
Suppose
(a) X and Y are topological vector spaces,
li
(b) Л: A^y
(c) N is a closed subspace of X, .
(d) it: X->X/N is the quotient map, and
(e) Ax = 0 for every x e N.
Prove that there is a unique/: X/N-> У which satisfies Л =/о тг, that is, Ax =/(тг(х))
for all x e X. Prove that this /is linear and that Л is continuous if and only if /is contin-
continuous. Also, Л is open if and only if/is open.

38 GENERAL THEORY
10 Suppose Jifand Гаге topological vector spaces, dim Y<oo, A: X-+Y is linear, and
A(X) = Y.
(a) Prove that Л is an open mapping.
(b) Assume, in addition, that the null space of Л is closed, and prove that Л is then
continuous.
11 If N is a subspace of a vector space X, the codimension of N in X is, by definition, the
dimension of the quotient space X/N.
Suppose 0 <p < 1 and prove that every subspace of finite codimension is dense in
Lp. (See Section 1.47.)
12 Suppose A(jc, y) = | x - у |, d2(x, у) = | ф(х) - ф(у) |, where ф(х) = x/(l + | x|). Prove
that di and d2 are metrics on R which induce the same topology, although dt is complete
and d2 is not.
13 Let С be the vector space of all complex continuous functions on [0, 1]. Define
Let (C, cr) be С with the topology induced by this metric. Let (C, r) be the topological
vector space defined by the seminorms
**(/) = !/(*) I (o<x<i),
in accordance with Theorem 1.37.
(a) Prove that every т-bounded set in С is also cr-bounded and that the identity map
id: (С, т)-+(С, a) therefore carries bounded sets into bounded sets.
(b) Prove that id: (C, r) -> (C, a) is nevertheless not continuous, although it is sequen-
sequentially continuous (by Lebesgue's dominated convergence theorem). Hence (C, r) is
not metrizable. (See Appendix A6, or Theorem 1.32.) Show also directly that
(C, r) has no countable local base.
(c) Prove that every continuous linear functional on (C, r) is of the form
for some choice of xu ..., xn in [0, 1] and some ct e С
{d) Prove that (C, a) contains no convex open sets other than 0 and С
(e) Prove that id: (C, a) ->(C, r) is not continuous.
Put K= [0, 1] and define ^K as in Section 1.46. Show that the following three families
of seminorms (where n = 0, 1, 2,...) define the same topology on $)K, if D = d/dx:
(a) \\Dnf\U = sup {| Dnf{x) |: - <x> < x < oo}.
.i
(b) ||Z>-/||i= |Z>"/(jc)| Л:.
(с) 1№п/112 = tt \Dnf(x)\2dx) '\
15 Prove that the spaces C(O) (Section 1.44) do not have the Heine-Borel property.
16 Prove that the topology of C(O) does not depend on the particular choice of {Kn}, as
long as this sequence" satisfies the conditions specified in Section 1.44. Do the same for
С°°(а) (Section 1.46).

TOPOLOGICAL VECTOR SPACES 39
17 In the setting of Section 1.46, prove that/-> D*/is a continuous mapping of С°°(Ц) into
C°°(O) and also of <3K into @K, for every multi-index a.
18 The seminorms
pn(f) = sup {\f(x) \:-n<x<n}
induce the metric
oo 2""/? (f— a)
««1 1 +/>„(/-#)
in the space C(i?); compare Section 1.46 and remark (c) of Section 1.38. Define
fix) = max @, 1 - | x|), #(x) = 100/(x - 2), 2Л =/+ g,
and compute that
rf(/.<0=5- ^'°>=Ж' ^Н + Ш"
The balls with radius £ are therefore not convex, although d is compatible with the
usual locally convex topology of CiR).
Is there any r < 1 for which the balls of radius r are convex?
19 Suppose M is a dense subspace of a topological vector space X, Г is an F-space, and
Л: M-> Fis continuous (relative to the topology that M inherits from X) and linear.
Prove that Л has a continuous linear extension &: X-+Y.
Suggestion: Let Vn be balanced neighborhoods of 0 in Xsuch that Vn + Vn <=• Fn_i
and such that d@, Ax) <2~n if xeM n Vn. If x e X and xn e (x + Fn) n M, show that
{Axn} is a Cauchy sequence in Y, and define Ax to be its limit. Show that Л is well de-
defined, that Ax = Лх if x e M, and that A is linear and continuous.
20 For each real number t and each integer n, define еп{1) = eint, and define
fn=e-n + nen in -1,2,3,...).
Regard these functions as members of L\~тг, тг). Let Xx be the smallest closed sub-
space of L2 that contains e0, eu e2,..., and let X2 be the smallest closed subspace of
L2 that contains fufi ,/з ? Show that Jfi + X2 is dense in L2 but not closed. For
instance, the vector
is in L2 but not in Xt + X2. (Compare with Theorem 1.42.)
2/ Let F be a neighborhood of 0 in a topological vector space X Prove that there is a real
continuous function / on X such that /@) = 0 and fix) — 1 outside V. (Thus X is a
completely regular topological space.) Suggestion: Let Fn be balanced neighborhoods
of 0 such that Fi + Fi с F and Fn+i + Fn+i с Fn. Construct / as in the proof of
Theorem 1.24. Show that/is continuous at 0 and that
\fix)~fiy)\<fix~y).

40 GENERAL THEORY
22 If/is a complex function defined on the compact interval / = [0,1] с R, define
cod(f)=sup{\f(x)-f(y)\: \x~-y\<S,xeI9yeI}.
If 0 < a < 1, the corresponding Lipschitz space Lip a consists of all/for which
ll/ll = l/@)| + sup {8-*wd(f): 8 > 0}
is finite. Define
lip a = {/e Lip a: lim S~aa>5(/) = 0}.
Prove that Lip a is a Banach space and that lip a is a closed subspace of Lip a.
23 Let X be the vector space of all continuous functions on the open segment @, 1). For
fe X and r > 0, let V(f9 r) consist of all g e Xsuch that \g(x) - fix) \ < r for all jc e @, 1).
Let т be the topology on X that these sets V(f9 r) generate. Show that addition is r-
continuous but scalar multiplication is not.
24 Show that the set W that occurs in the proof of Theorem 1.14 need not be convex, and
that A need not be balanced unless U is convex.

COMPLETENESS
The validity of many important theorems of analysis depends on the completeness of
the systems with which they deal. This accounts for the inadequacy of the rational
number system and of the Riemann integral (to mention just the two best-known
examples) and for the success encountered by their replacements, the real numbers and
the Lebesgue integral. Baire's theorem about complete metric spaces (often called
the category theorem) is the basic tool in this area. In order to emphasize the role
played by the concept of category, some theorems of this chapter (for instance,
Theorems 2.7 and 2.11) are stated in a little more generality than is usually needed.
When this is done, simpler versions (more easily remembered but sufficient for most
applications) are also given.
Baire Category
2.1 Definition Let S be a topological space. A set E a S is said to be nowhere
dense if its closure Ё has empty interior. The sets of the first category in S are those
that are countable unions of nowhere dense sets. Any subset of S that is not of the
first category is said to be of the second category in S.
This terminology (due to Baire) is admittedly rather bland and unsuggestive.

42 GENERAL THEORY
Meager and nonmeager have been used instead in some texts. But "category argu-
arguments " are so entrenched in the mathematical literature and are so well known that it
seems pointless to insist on a change.
Here are some obvious properties of category that will be freely used in the
sequel:
(a) If A cz В and В is of the first category in S, so is A.
(b) Any countable union of sets of the first category is of the first category.
(c) Any closed set E с S whose interior is empty is of the first category in S.
(d) If h is a homeomorphism of S onto S and if E a S, then E and h(E) have the
same category in S.
2.2 Baire's theorem If S is either
(a) a complete metric space, or
(b) a locally compact Hausdorff space,
then the intersection of every countable collection of dense open subsets of S is dense
in S.
This is often called the category theorem, for the following reason.
If {Ei} is a countable collection of nowhere dense subsets of S, and if Vt is the
complement of Ei9 then each Vt is dense, and the conclusion of Baire's theorem is that
(\У{Ф0. Hence S^{jEt.
Therefore, complete metric spaces, as well as locally compact Hausdorff spaces,
are of the second category in themselves.
proof Suppose Vl9 V2, V3, ... are dense open subsets of S. Let Bo be an
arbitrary nonempty open set in S. If n > 1 and an open Вп_г ф 0 has been
chosen, then (because Vn is dense) there exists an open Bn Ф 0 with
In case {a), Bn may be taken to be a ball of radius < l/n; in case (b) the choice can
be made so that Bn is compact. Put
In case (a), the centers of the nested balls Bn form a Cauchy sequence which
converges to some point of K, and so К Ф 0. In case (b), К Ф 0 by compactness.
Our construction shows that К cz Bo and К cz Vn for each n. Hence Bo intersects
C\vn. ' . mi

COMPLETENESS 43
The Banach-Steinhaus Theorem
2.3 Equicontinuity Suppose X and Y are topological vector spaces and Г is a
collection of linear mappings from X into Y. We say that Г is equicontinuous if to
every neighborhood J^of 0 in Г there corresponds a neighborhood Fof 0 in Xsuch
that A(K).cz Wfor all ЛеГ.
If Г contains only one Л, equicontinuity is, of course, the same as continuity
(Theorem 1.17). We already saw (Theorem 1.32) that continuous linear mappings are
bounded. Equicontinuous collections have this boundedness property in a uniform
manner (Theorem 2.4). It is for this reason that the Banach-Steinhaus theorem B.5)
is often referred to as the uniform boundedness principle.
2.4 Theorem Suppose X and Y are topological vector spaces, Г is an equicon-
equicontinuous collection of linear mappings from Xinto Y, and E is a bounded subset of X. Then
Y has a bounded subset F such that A(E) a F for every ЛеГ.
proof Let Fbe the union of the sets A(E), for ЛеГ. Let W be a neighborhood
of 0 in Y. Since Г is equicontinuous, there is a neighborhood V oi 0 in JJfsuch
that A(V) <= W for all ЛеГ. Since E is bounded, E a tV for all sufficiently
large t. For these /,
A(E) с A(tV) = tA(V) с tW,
so that F a tW. Hence F'\s bounded. ////
2.5 Theorem (Banach-Steinhaus) Suppose X and Y are topological vector
spaces, Г is a collection of continuous linear mappings from X into Y, and В is the set
of all x e X whose orbits
Г(х)={Ах:АеГ}
are bounded in Y. * *
If В is of the second category in X, then В = X and Г is equicontinuous.
proof Pick balanced neighborhoods W and U of 0 in Fsuch that U + U cz W.
Put
>' E=[)A~\V).
ЛеГ
If x e B, then Г(х) cz nil for some n, so that x e nE. Consequently,
В a 0 nE.
At least one nE is of the second category in X, since this is true of B. Since
x -> nx is a homeomorphism of X onto X, E is itself of the second category in X.

44 GENERAL THEORY
But E is closed because each Л is continuous. Therefore E has an interior point
x. Then x — £ contains a neighborhood Fof 0 in X, and
A(V) a Ax- A(E) czU -U czW
for every ЛеГ.
This proves that Г is equicontinuous. By Theorem 2.4, Г is uniformly
bounded; in particular, each T(x) is bounded in 7. Hence В = X. ////
In many applications, the hypothesis that В is of the second category is a con-
consequence of Baire's theorem. For example, jF-spaces are of the second category.
This gives the following corollary of the Banach-Steinhaus theorem:
2.6 Theorem If Y is a collection of continuous linear mappings from an F-space X
into a topological vector space 7, and if the sets
are bounded in Y,for every x e X, then Г is equicontinuous.
Briefly, pointwise boundedness implies uniform boundedness (Theorem 2.4.)
As a special case of Theorem 2.6, let X and Y be Banach spaces, and suppose
that
A) sup ||Лх|| < oo for every x e X.
ЛеГ
The conclusion is that there exists M < oo such that
B) ||Лх|| < M if |M| < 1 and Л е Г.
Hence
C) ||Ajc|| < M\\x\\ if x e X and Л е Г.
The following theorem establishes the continuity of limits of sequences of
continuous linear mappings.
2.7 Theorem Suppose X and Y are topological vector spaces, and {An} is a sequence
of continuous linear mappings of X into Y.
(a) If С is the set of all x e X for which {An x} is a Cauchy sequence in 7, and if С
is of the second category in X, then С — X.
(b) IfL is the set of all x e X at which
Ax = lim Л„ x
n~*oo

COMPLETENESS 45
exists, if L is of the second category in X, and if Y is an F-space, then L = X
and A: X-> Yis continuous.
proof (a) Since Cauchy sequences are bounded (Section 1.29) the Banach-
Steinhaus theorem asserts that {AJ is equicontinuous.
One checks easily that С is a subspace of X. Hence С is dense. (Other-
(Otherwise, С is a proper subspace of X; proper subspaces have empty interior; thus
С would be of the first category.)
Fix x'E X; let W be a neighborhood of 0 in Y. Since {An} is equicontinuous,
there is a neighborhood V of 0 in X such that An(V) с W for n = 1, 2, 3,
Since С is dense, there exists x' e С n (x + V). If n and m are so large that
Anx'-Amx'eW,
the identity
(А„ - Л Jx = An(x - *') + (Ля - Am)x' + Am (xf - x)
shows that Anx — Amx e W + W + W. Consequently, {Anx} is a Cauchy
sequence in Y, and x e С
(b) The completeness of Y implies that L — C. Hence L — X, by (a).
If Fand W are as above, the inclusion An(V) с W, valid for all n, implies now
that A(V) с W. Thus Л is continuous. " ////
The hypotheses of (b) of Theorem 2.7 can be modified in various ways. Here is
an easily remembered version:
2.8 Theorem If{An} is a sequence of continuous linear mappings from an F-space
X into a topological vector space Y, and if
Ax = lim An x
n-+oo
exists for every x e X, then A is continuous.
proof Theorem 2.6 implies that {Л„} is equicontinuous. Therefore if W is a
neighborhood of 0 in У, we have An(V) a JPFfor all n and for some neighborhood
V of 0 in X. It follows that A(V) с W; hence (being obviously linear) Л is
continuous. ////
In the following variant of the Banach-Steinhaus theorem the category argument
is applied to a compact set, rather than to a complete metric one. Convexity also
enters here in an essential way (Exercise 8).

46 GENERAL THEORY
) *
2.9 Theorem Suppose X and Y are topological vector spaces, К is a compact
convex set in X, Г is a collection of continuous linear mappings of X into Y9 and the
orbits
T(x) = {Ax:AeT}
are bounded subsets of Y, for every x e K.
Then there is a bounded set В с Y such that A(K) с В for every A e Г.
proof Let В be the union of all sets Г(х), for x e K. Pick balanced neighbor-
neighborhoods W and U of 0 in Y such that U + E7 cz W. Put
A) E=(]A-l(U).
ЛеГ
If x e K, then T(x) cz nU for some n, so that x e nE. Consequently,
B) K= [J(KnnE).
Since E is closed, Baire's theorem shows that К n nE has nonempty interior
(relative to K) for at least one n.
We fix such an n, we fix an interior point jc0 of К n nE, we fix a balanced
neighborhood V of 0 in X such that
C) К n (x0 + V) с л£,
and we fix a /? > 1 such that
D) tfcXo+^K.
Such a /? exists since К is compact.
If now x is any point of К and
E) z^O-^Xo+i?-1*,
then z e K, since j£ is convex. Also,
F) z~xo=p~1(x~xo)e V,
by D). Hence zenE, by C). Since A(nE)cznU for every ЛеГ and since
x =pz — (p — l)x0, we have
Ax epnU - О - l)nU cpn(V + D)
Thus В czpnW, which proves that В is bounded. . ////
The Open Mapping Theorem
2.10 Open mappings Suppose /maps S into Г, where 5 and T are topological
spaces. We say that/is open at a point p eS if f(V) contains a neighborhood of

COMPLETENESS 47
/(/?) whenever* К is a neighborhood of p. We say that/is open if f(U) is open in T
whenever U is open in S.
It is clear that/is open if and only if/ is open at every point of S. Because
of the invariance of vector topologies, it follows that a linear mapping of one topo-
logical vector space into another is open if and only if it is open at the origin.
Let us also note that a one-to-one continuous mapping/of S onto Г is a homeo-
morphism precisely when/is open.
2.11 The open mapping theorem Suppose
(a) X is an F-space,
(b) Y is a topological vector space,
(c) А: Х-+ Yis continuous and linear, and
(d) A(X) is of the second category in Y.
Then
(i) A(X)=Y,
(ii) Л is an open mapping, and
(in) Y is an F-space.
proof Note that (ii) implies (/), since Y is the only open subspace of Y. To
prove (ii), let V be a neighborhood of 0 in X. We have to show that A(V)
contains a neighborhood of 0 in Y.
Let d be an invariant metric on X that is compatible with the topology of
X. Define
A) Vn = {x: d(x, 0) < 2~nr} (n = 0, 1, 2, ...),
where r > 0 is so small that Vo c: V. We will prove that some neighborhood W
of 0 in Y satisfies
B) Wcz A(V^ cz A(V).
Since Vx zd V2 — V2 , statement (b) of Theorem 1.13 implies
C) A(VX) =d A{V2) - A(V2) => A(V2) - A(V2).
The first part of B) will therefore be proved if we can show that A(V2) has
nonempty interior. But
D) A(X) = 0 kA{V2),
fc=l

48 GENERAL THEORY
because V2 is a neighborhood of 0. At least one kA(V2) is therefore of the second
category in Y. Since у -» ку is a homeomorphism of Y onto У, A( F2) is of the
second category in Y. Its closure therefore has nonempty interior.
To prove the second inclusion in B), fix y\ e A(VX). Assume n > 1 and
yn has been chosen in A( Vn). What was just proved for Vx holds equally well for
Vn+i, so that A(Vn + l) contains a neighborhood of 0. Hence
E) Ov
This says that there exists xn e Vn such that
F) Axneyn-A(Vn
Put yn+l = у„ ~ Ах„. Then yn + l eA(Vn + l), and the construction proceeds.
Since d(xn, 0) < 2" V, for я = 1, 2, 3, ..., the sums xt + • • • + xn form a
Cauchy sequence which converges (by the completeness of X) to some x e X,
with d(x, 0) < r. Hence x e V. Since
G) Z Axn Ё
and since ^m + 1 -> 0 as m -> oo (by the continuity of A), we conclude that yY —
Ax e A(V). This gives the second part of B), and (//) is proved.
Theorem 1.41 shows that X/N is an F-space, if N is the null space of A.
Hence (Hi) will follow as soon as we exhibit an isomorphism/of X/N onto Y
which is also a homeomorphism. This can be done by defining
(8) f(x -{-N) = Ax (xe X).
It is trivial that this/is an isomorphism and that Ax =f(n(x)), where n is the
quotient map described in Section 1.40. If V is open in Y, then
(9) f-\V) = n(A
is open, since A is continuous and n is open. Hence /is continuous. If E is
open in X/N, then
A0) Л^) = Л(тс-1(^))
is open, since n is continuous and A is open. Consequently, / is a homeomor-
homeomorphism. . ////
2.12 Corollaries
(a) If A is a continuous linear mapping of an F-space X onto an F-space Y, then A is
open.

COMPLETENESS 49
(b) If К satisfies (a) and is one-to-one, then Л l: Y'-» X is continuous.
(c) If X and Y are Banach spaces, and if A: X -* Y is continuous, linear, one-to-one,
and onto, then there exist positive real numbers a and b such that
a\\x\\ < \\\x\\ < Ъ\\х\\
for every x e X.
(d) IfT1czT2 are vector topologies on a vector space Xandifboth{X,x^) and(X, т2)
are F-spaces, then Tt = r2 .
proof Statement (a) follows from Theorem 2.11 and Baire's theorem, since Y
is now of the second category in itself. Statement (b) is an immediate con-
consequence of (a), and (c) follows from (b). The two inequalities in (c) simply
express the continuity of Л and of Л. Statement (d) is obtained by applying
(b) to the identity mapping of (X, т2) onto (X Ti)- ////
The Closed Graph Theorem
2.13 Graphs If X and Y are sets and/maps Xinto У, the graph of/is the set of
all points (x,f(x)) in the cartesian product X x Y. If A^and У are topological spaces,
if X x У is given the usual product topology (the smallest topology that contains all
sets U x К with £/and К open in Xand Y, respectively), and if/: X-* У is continuous,
one would expect the graph of/ to be closed in X x Y (Proposition 2.14)/ For
linear mappings between F-spaces this trivial necessary condition is also sufficient to
assure continuity. This important fact is proved in Theorem 2.15.
2.14 Proposition If X is a topological space, Y is a Hausdorff space, and
/: X-> Y is continuous, then the graph G offis closed.
proof Let Q. be the complement of С in X x Y; fix (x0, y0) e Q. Then;
y0 Ф/(х0). Thus y0 and f(x0) have disjoint neighborhoods V and W in Y.
Since / is continuous, x0 has a neighborhood U such that f(U) a W. The
neighborhood U x К of (x0, y0) lies therefore in Q. This proves that Q is open.
Note: One cannot omit the hypothesis that У is a Hausdorff space. To see this,
consider an arbitrary topological space X, and \etf:X-*X be the identity. Its
graph is the diagonal
D={(x,x):xeX}czXx X.
The statement " D is closed inlx X" is just a rewording of the Hausdorff separation
axiom.

50 GENERAL THEORY
2.15 The closed graph theorem Suppose
(a) X and Y are F-spaces,
(b) Л: Х-» Y is linear,
(c) G = {(*, Ax): x e X} is closed in X x Y.
Then A is continuous.
proof X x У is a vector space if addition and scalar multiplication are defined
componentwise:
<*(*i» У\) + P(X2 , У2) = (a*i + Pxi , ayl + fiy2).
There are complete invariant metrics dx and dY on X and 7, respectively,
which induce their topologies. If
4(*i, У1), (x2, y2)) = dx{xu x2) + dY(yl9 y2),
then <i is an invariant metric on X x Y which is compatible with its product
topology and which makes X x Y into an F-space. (The easy but tedious
verifications that are needed here are left as an exercise.)
Since Л is linear, G is a subspace of X x Y. Closed subsets of complete
metric spaces are complete. Therefore G is an F-space.
Define nt: G-> Хапйл2: Хх Y-> У by
щ(х, Ax) = x, тг2(х, у) = у.
Now 7cx is a continuous linear one-to-one mapping of the F-space G onto the
F-space X. It follows from the open mapping theorem that
is continuous. But Л = п2 о щ1 and n2 is continuous. Hence Л is continuous.
////
Remark The crucial hypothesis (c), that G is closed, is often verified in
applications by showing that Л satisfies property (cf) below:
{c) If{xn} is a sequence in X such that the limits
x = lim xn and у = lim Axn
n-+ 00 n-*oo
exist, then у = Ax.
Let us prove that (cf) implies (c). Pick a limit point (x, y) of G. Since
X x Y is metrizable,
(x,y)= Um(xn,Axn)

COMPLETENESS 51
for some- sequence {xn}. It follows from the definition of the product topology
that xn -> x and Axn -+y. Hence у = Ax, by (c), and so (x, у) е G, and G is
closed.
It is just as easy to prove that (c) implies (cf).
Bilinear Mappings
2.16 Definitions Suppose X, Y, Z are vector spaces and В maps X x Y into Z.
Associate to each x e X and to each у е Y the mappings
Bx: Y->Z and By: X->Z
by defining
Bx(y) = B(x,y) = B\x).
В is said to be bilinear if every i^ and every By are linear.
If X, Y, Z are topological vector spaces and if every Bx and every By is con-
continuous, then В is said to be separately continuous. If В is continuous (relative to the
product topology of X x Y) then В is obviously separately continuous. In certain
situations, the converse can be proved with the aid of the Banach-Steinhaus theorem.
2.17 Theorem Suppose B: X x Y-+Z is bilinear and separately continuous, X is
an F-space, and Y and Z are topological vector spaces. Then
A) B(xn,yn)->B(xo,yo)inZ
whenever xn -» x0 in X and yn -> y0 in Y. If Y is metrizable, it follows that В is con-
continuous.
proof Let U and W be neighborhoods of 0 in Z such that U + U a W.
Define
bn(x) = B(x, yn) (xeX,n = l,2,3,...).
Since В is continuous as a function of y,
limbn(x) = B(x,y0) (xeX).
n~* oo
Thus {bn(x)} is a bounded subset of Z, for each x e X. Since each bn is a con-
continuous linear mapping of the F-space X, the Banach-Steinhaus theorem 2.6
implies that {bn} is equicontinuous. Hence there is a neighborhood К of 0 in X
such that
bn(V)czU (n = 1,2,3,...).
Note that
B{xn, yn) - £(x0 , y0) = bn(xn - *0) + B(x0, yn - jo)-

52 GENERAL THEORY
If n is sufficiently large, then (/) xn e x0 -f V, so that bn(xn — x0) e U, and (ii)
B(xo > Уп — Уо) e U> since В is continuous in у and B(x0, 0) = 0. Hence
B(xn,yn)-B{xo,yo)eU+UczW
for all large n. This gives A).
If Y is metrizable, so is X x У, and the continuity of В then follows
from(l). (See Appendix A6.) ////
Exercises
1 If X is an infinite-dimensional topological vector space which is the union of countably
many finite-dimensional subspaces, prove that X is of the first category in itself. Prove
that therefore no infinite-dimensional F-space has a countable Hamel basis.
(A set j8 is a Hamel basis for a vector space X\f ]3 is a maximal linearly independent
subset of X. Alternatively, j8 is a Hamel basis if every x e X has a unique representation
as a finite linear combination of elements of j8.)
2 Sets of first and second category are " small" and " large " in a topological sense. These
notions are different when " small" and " large" are understood in the sense of measure,
even when the measure is intimately related to the topology. To see this, construct a
subset of the unit interval which is of the first category but whose Lebesgue measure is 1.
3 Put К = [— 1, 1]; define <3K as in Section 1.46 (with R in place of Rn). Suppose {/„} is a
sequence of Lebesgue integrable functions such that
=lim f
exists for every фе@к. Show that Л is a continuous linear functional on $)K. Show that
there is a positive integer p and a number M < oo such that
for all n. For example, iffn{t) =n2t on [—1//?, l/n] and 0 elsewhere, show that this can be
done with p = 1. Construct an example where it can be done with p = 2 but not with
p=\.
4 Let L1 and L2 be the usual Lebesgue spaces on the unit interval. Prove that L2 is of the
first category in L1, in three ways:
(a) Show that {/: J |/|2 < n} is closed in L1 but has empty interior.
(b) Put gn = n on [0, n~3], and show that
for every /el2 but not for every /el1.
(c) Note that the inclusion map of L2 into L1 is continuous but not onto.
Do the same for Lp and Lq if p < q.

COMPLETENESS 53
5 Prove results analogous to those of Exercise 4 for the spaces Л, where Л is the Banach
space of all complex functions x on {0,1, 2,...} whose norm
{« \llp
I \x(n)\p)
0 j
is finite.
6 Define the Fourier coefficients f(n) of a function fe L2(T) (Tis the unit circle) by
for all n eZ (the integers)/ Put
= 2 f(k).
Prove that {feL2(T): limn_>со Kf exists} is a dense subspace of £2(Г) of the first
category,
7 Let C(T) be the set of all continuous complex functions on the unit circle T. Suppose
{yn} (n e Z) is a complex sequence that associates to each /e С(Г) a function Л/е С(Г)
whose Fourier coefficients are
(The notation is as in Exercise 6.) Prove that {yn} has this multiplier property if and
only if there is a complex Borel measure [л on T such that
yn = \е~1пв dfx(9) (neZ).
Suggestion: With the supremum norm, C(T) is a Banach space. Apply the closed
graph theorem. Then consider the functional
and apply the Riesz representation theorem ([23], Th. 6.19). (The above series may
not converge; use it only for trigonometric polynomials.)
8 Define functional Лт on P (see Exercise 5) by
Amx = 2 n2x(n) (m = 1, 2, 3,...).
я=1
Define xn e «if2 by х„(п) — 1/n, xn(r) = 0 if / Ф n. Let К <= /2 consist of 0, xb x2, x3,... .
Prove that К is compact. Compute Amxn. Show that {Amx} is bounded for each xe К
but {Am*m} is not. Convexity can therefore not be omitted from the hypotheses of
Theorem 2.9.
Choose с„ > 0 so that ]?с„ = 1, ^пс„== со. Take x = 2 с„х„. Show that x lies
in the closed convex hull of К (by definition, this is the closure of the convex hull) and
that {Лт х} is not bounded.
Show that the convex hull of К is not closed.

54 GENERAL THEORY
i '
9 Suppose X, У, Z are Banach spaces and
B:Xx Y-+Z
is bilinear and continuous. Prove that there exists M < oo such that
(хеХууе Y).
Is completeness needed here?
10 Prove that a bilinear mapping is continuous if it is continuous at the origin @, 0).
11 Define B(xux2 \y) = (xiy, x2y). Show that В is a bilinear continuous mapping of
R2 X R onto R2 which is not open at A, 1; 0). Find all points where this В is open.
12 Let X be the normed space of all real polynomials in one variable, with
ll/ll = f 1/@1 dt.
•>0
Put B(f, g) = Jo f(t)g(t) dt, and show that В is a bilinear functional on X x X which is
separately continuous but is not continuous.
13 Suppose X is a topological vector space which is of the second category in itself. Let К
be a closed, convex, absorbing subset of X. Prove that К contains a neighborhood of 0.
Suggestion: Show first that H = Kc\(~K) is absorbing. By a category argument, H
has interior. Then use
Show that the result is false without convexity of K, even if X = Я2. Show that the
result is false if X is L2 topologized by the L1-norm (as in Exercise 4).
14 (a) Suppose X and Y are topological vector spaces, {Л„} is an equicontinuous sequence
of linear mappings of X into Y, and С is the set of all x at which {An(x)} is a Cauchy
sequence in Y. Prove that С is a closed subspace of X.
(b) Assume, in addition to the hypotheses of (a), that Y is an F-space and that {&n(x)}
converges in some dense subset of X. Prove that then
A(x) = lim An(x)
n-+ oo
exists for every x e X and that Л is continuous.

3
CONVEXITY
This chapter deals primarily (though not exclusively) with the most important class of
topological vector spaces, namely, the locally convex ones. The highlights, from the
theoretical as well as the applied standpoints, are (a) the Hahn-Banach theorems
(assuring a supply of continuous linear functionals that is adequate for a highly
developed duality theory), (b) the Banach-Alaoglu compactness theorem in dual
spaces, and (c) the Krein-Milman theorem about extreme points. Applications to
various problems in analysis are postponed to Chapter 5.
The Hahn-Banach Theorems
The plural is used here because the term "Hahn-Banach theorem" is customarily
applied to several closely related results. Among these are the dominated extension
theorems 3.2 and 3.3 (in which no topology is involved), the separation theorem 3.4,
and the continuous extension theorem 3.6. Another separation theorem (which
implies 3.4) is stated as Exercise 3.
3.1 Definitions The dual space of a topological vector space X is the vector
space X* whose elements are the continuous linear functionals on X.

56 GENERAL THEORY
Note that addition and scalar multiplication are defined in X* by
(A1 + Л2)х = AjX + A2 x, {ocA)x = a • Ax.
It is clear that these operations do indeed make X* into a vector space.
It will be necessary to use the obvious fact that every complex vector space is
also a real vector space, and it will be convenient to use the following (temporary)
terminology: An additive functional Л on a complex vector space X is called real-
linear {complex-linear) if Л(ах) = аЛх for every x e X and for every real (complex)
scalar a. Our standing rule that any statement about vector spaces in which no scalar
field is mentioned applies to both cases is unaffected by this temporary terminology
and is still in force.
If и is the real part of a complex-linear functional/on X, then и is real-linear and
A) f{x) = u{x) - iu{ix) {x e X)
because z = Re z — i Re (iz) for every z sG.
Conversely, if u: X-* R is real-linear on a complex vector space X and if/ is
defined by A), a straightforward computation shows that/is complex-linear.
Suppose now that X is a complex topological vector space. The above facts
imply that a complex-linear functional on X is in X* if and only if its real part is
continuous, and that every continuous real-linear u: X -> R is the real part of a unique
/el*.
3.2 Theorem Suppose
(a) M is a subspace of a real vector space X,
(b) p: X-+ R satisfies
p{x + y)< p{x) + p{y) and p{tx) = tp{x)
ifxeX,yeX, t > 0,
(c) /: M->R is linear and fix) < pix) on M.
Then there exists a linear A: X-+ R such that
Ax = f[x) (xeM)
and
~p(~x)<Ax<p(x) {xeX).
proof If M ф X, choose xx e X, xx ф М, and define
M1 = {x + txt: x e M, t e R}.
It is clear that Mx is a vector space. Since
/(*) + Л» = Л* + y) < pix + y)< p(x - Xl) + p(xx + y\
we have
A) Дх) - p(x - Xl) <p(y + Xl) - f{y) (x, у 6 M).

CONVEXITY 57
Let a be the least upper bound of the left side of A), as x ranges over M. Then
B) f(x)-oc<p(x~x1) (xeM)
and
C) f(y) + o^<p(y + x1) (yeM).
Define yi on M1 by
D) ' fi(x + txj = f(x) + ta (xeM,teR).
Then/ =fonM, and/ is linear on Mv
Take / > 0, replace x by t~1x in B), replace у by t~1y in C), and multiply
the resulting inequalities by t. In combination with D), this proves that
/ <p on Afj.
The second part of the proof can be done by whatever one's favorite
method of transfinite induction is; one can use well-ordering, or Zorn's lemma,
or Hausdorff's maximality theorem:
Let <P be the collection of all ordered pairs (Af',/'), where M' is a subspace
of X that contains M and /' is a linear functional on M' that extends / and
satisfies f <p on M'. Partially order & by declaring (M',/') < (M",f") to
mean that M' <= M" and/" =/' on M'. By Hausdorff's maximality theorem
there exists a maximal totally ordered subcollection п of 0>.
Let Ф be the collection of all M' such that (M',f) e £1 Then Ф is totally
ordered by set inclusion, and the union M of all members of Ф is therefore a
subspace of X. If xeM then x e M' for some М'еФ; define Лх=/'(х),
where/' is the function which occurs in the pair (M',/') e Q.
It is now easy to check that Л is well defined on M, that Л is linear, and
that Л < p. If M were a proper subspace of X, the first part of the proof would
give a further extension of Л, and this would contradict the maximality of Q.
Thus M = X.
Finally, the inequality Л<р implies that
-p(-x) < - Л(-х) = Ax
for all x e X. This completes the proof. ////
3.3 Theorem Suppose M is a subspace of a vector space X, p is a seminorm on X,
and f is a linear functional on M such that
\f(x)\ <p(x) (xeM).
Then f extends to a linear functional A on X that satisfies
\Ax\ <p(x) (xeX).

58 GENERAL THEORY
proof If the scalar field is R, this is contained in Theorem 3.2, since p now
satisfies /?( — *) = p(x).
Assume that the scalar field is @. Put и = Re/. By Theorem 3.2 there is
a real-linear U on X such that U=u on M and U < p on X. Let Л be the
complex-linear functional on X whose real part is U. The discussion in Section
3.1 implies that Л =/on M.
Finally, to every x e X corresponds an a e <Z, \a\ =1, such that ocAx =
|Ax|. Hence
| Ax | = Л(а*) = Щах) < р(ах) = p(x). ////
Corollary If X is a normed space and x0 e X, there exists AgI* swc/г that
Ax0 - ||*01| л/и/ |Л*| < ||*|| for all xeX.
proof If x0 = 0, take Л = 0. If x0 ф 0, apply Theorem 3.3, with p(x) = ||*||,
M the one-dimensional space generated by x0, and/(a*0) = a||*0|| on M. ////
3.4 Theorem Suppose A and В are disjoint, nonempty, convex sets in a topological
vector space X.
(a) If A is open there exist ЛеР and у eR such that
Re Лд: < у < Re Ay
for every x e A and for every у е В.
(b) If A is compact, В is closed, and X is locally convex, then there exist Л е X*,
7i G ^» li E R> sucn that
Re Ax <yx < 72 < Re Лу
for every x e A and for every у е В.
Note that this is stated without specifying the scalar field; if it is R, then Re Л =
Л, of course.
proof It is enough to prove this for real scalars. For if the scalar field is <Z
and the real case has been proved, then there is a continuous real-linear A1 on
Xthat gives the required separation; if Л is the unique complex-linear functional
on X whose real part is Лъ then Л e X*. (See Section 3.1.) Assume real scalars.
(a) Fix a0 eA, b0 e B. Put x0 = b0 — a0; put С = A — В + x0. Then
С is a convex neighborhood of 0 in X. Let p be the Minkowski functional
of С By Theorem 1.35, p satisfies hypothesis (b) of Theorem 3.2. Since
A n В = 0, x0 ф C, and so p(x0) > 1.
Define f(tx0) = / on the subspace MofI generated by лг0. If t > 0 then
/0*o)= t<tp(xo)=p(txo);

CONVEXTIY 59
if t < 0 then/(te0) < 0 <p(tx0). Thus f<p on M. By Theorem 3.2,/extends
to a linear functional Л on X that also satisfies Л < p. In particular, Л < 1 on C,
hence Л > - 1 on — C, so that |Л| < 1 on the neighborhood С п (-С) of 0.
By Theorem 1.18, Л e Z*.
If now a e л( and b e B, we have
-• Ля - Л6 + 1 = A(a - b 4 x0) < p(a - b 4 x0) < 1
since Лл:0 =J,«-HioeC, and С is open. Thus Aa <Ab.
It follows that A(A) and A(B) are disjoint convex subsets of R, with
Л(/1) to the left of A(B). Also, A(A) is an open set since A is open and since
every nonconstant linear functional on X is an open mapping. Let у be the
right end point of A(A) to get the conclusion of part (a).
(b) By Theorem 1.10 there is a convex neighborhood V of 0 in X such
that (A + V)nB = 0. Part (a), with A 4 К in place of >4, shows that there
exists AeX* such that A(A 4 K) and A(B) are disjoint convex subsets of R,
with A(A 4- V) open and to the left of A(B). Since Л(Л) is a compact subset of
A(A 4- F), we obtain the conclusion of F). ////
Corollary If X is a locally convex space then X* separates points on X.
proof If xt e X, x2 e X, and ххФх2, apply (b) of Theorem 3.4 with A =
3.5 Theorem Suppose M is a subspace of a locally convex space X, and x0 e X.
If x0 is not in the closure ofM, then there exists A e X* such that Ax0 = 1 but Ax = 0
for every x e M.
proof By (b) of Theorem 3.4, with A = {x0} and В — M, there exists AeX*
such that Лл:0 and A(M) are disjoint. Thus A(M) is a proper subspace of the
scalar field. This forces A(M) = {0} and Лх0 Ф 0. The desired functional is
obtained by dividing Л by Лх0. ////
Remark This theorem is the basis of a standard method of treating certain
approximation problems: In order to prove that an x0 e X lies in the closure of
some subspace M of X it suffices (if X is locally convex) to show that Ax0 — 0
for every continuous linear functional AonI that vanishes on M.
3.6 Theorem Iff is a continuous linear functional on a subspace M of a locally
convex space X, then there exists AeX* such that A=fon M.
Remark For normed spaces this is an immediate corollary of Theorem 3.3.
The general case could also be obtained from 3.3, by relating the continuity

60 GENERAL THEORY
of linear functionals to seminorms (see Exercise 8, Chapter 1). The proof
given below shows that Theorem 3.6 depends only on the separation property
of Theorem 3.5.
proof Assume, without loss of generality, that / is not identically 0 on M.
Put
and pick x0 e M such that f(x0) = 1. Since / is continuous, x0 is not in the
M-closure of Mo, and since M inherits its topology from X, it follows that
x0 is not in the Z-closure of Mo.
Theorem 3.5 therefore assures the existence of aAel* such that
Лх0 = 1 and Л = 0 on M0.
If x e M, then x -f(x)x0 e Mo, since f(x0) = 1. Hence
Ax - f(x) = Ax- f(x)Ax0 = A(x - /(*)*0) = О.
Thus Л=/оп М. ////
We conclude this discussion with another useful corollary of the separation
theorem.
3.7 Theorem Suppose В is a convex, balanced, closed set in a locally convex
space X, x0 e X, but x0 ф В. Then there exists Ael* such that \ Ax \ < 1 for all
x e B, but Ax0 > 1.
proof Apply (b) of Theorem 3.4, with A = {x0}, note that A(B) is convex and
balanced, and multiply the corresponding Л by an appropriate scalar. ////
Weak Topologies
3.8 Topological preliminaries The purpose of this section is to explain and
illustrate some of the phenomena that occur when a set is topologized in several ways.
Let xx and т2 be two topologies on a set X, and assume ^ ст2; that is, every
reopen set is also T2-open. Then we say that xx is Weaker than т2, or that т2 is stronger
than xx. [Note that (in accordance with the meaning of the inclusion symbol cz) the_
terms "weaker" and "stronger" do not exclude equality.] In this situation, the
identity mapping on X is continuous from (X, т2) to (X, Tj) and is an open mapping
from (X, Tt) to (X, т2).
As a first illustration, let us prove that the topology of a compact Hausdorff
space has a certain rigidity, in the sense that it cannot be weakened without losing the
Hausdorff separation axiom and cannot be strengthened without losing compactness:

CONVEXITY 61
.(a) If x1 c=42 are topologies on a set X, if тх is a Hausdorff topology, and if т2 is
compact, then xx = т2.
To see this, let Fc-X be T2-closed. Since X is T2-compact, so is F. Since
тх cz r2, it follows that F is тх-compact. (Every тх-ореп cover of F is also a T2-open
cover.) Since тх is a Hausdorff topology, it follows that Fis -^-closed.
As another .illustration, consider the quotient topology xN of X/N, as defined in
Section 1.40, and the quotient map n: X-+ X/N. By its very definition, xN is the
strongest topology on X/N that makes n continuous, and it is the weakest one that
makes n an open mapping. Explicitly, if т' and т" are topologies on X/N, and if n is
continuous relative to т' and open relative to т", then т' cz т^ cz т".
Suppose next that X is a set and IF is a nonempty family of mappings/: X-> Yft
where each Yf is a topological space. (In many important cases, Yf is the same for
all/e IF.) Let т be the collection of all unions of finite intersections of sets/^),
with/e !F and К open in Yf. Then т is a topology on X, and it is in fact the weakest
topology on X that makes every fe !F continuous: If % is any other topology with
that property, then т cz т'. This т is called the weak topology on X induced by #", or,
more succinctly, the IF-topology of X.
• The best-known example of this situation is undoubtedly the usual way in which
one topologizes the cartesian product X of a collection of topological spaces Xa.
If na(x) denotes the ath coordinate of a point x e X, then rca maps X onto Xa, and the
product topology т of X is, by definition, its {rcj-topology, the weakest one that makes
every па continuous. Assume now that every Xa is a compact Hausdorff space. Then
т is a compact topology on X (by Tychonoff's theorem), and proposition (a) implies
that т cannot be strengthened without spoiling Tychonoff's theorem.
In the last sentence a special case of the following proposition was tacitly used:
(b) If ^ is a family of mappings f\ X -> Yf, where X is a set and each Yf is a Haus-
Hausdorff space, and if ^ separates points on X, then the fF-topology of X is a Haus-
Hausdorff topology.
For \i p Ф q are points of X, then f(p) Ф f(q) for some/e#"; the points f(p)
and f{q) have disjoint neighborhoods in Yf whose inverse images under/are open (by
definition) and disjoint.
Here is an application of these ideas to a metrization theorem.
(c) If X is a compact topological space and if some sequence {/,} of continuous real-
valued functions separates points on X, then X is metrizable.
Let т be the given topology of X. Suppose, without loss of generality, that
I/, | < 1 for all n, and let xd be the topology induced on X by the metric

62 GENERAL THEORY
This is indeed a metric, since {fn} separates points. Since eachy^ is т-continuous and the
series converges uniformly on X x X, d is a т-continuous function on X x X. The
balls
Br(p) = {qeX:d(p,q)<r}
are therefore т-ореп. Thus xd с т. Since rd is induced by a metric, rd is a HausdorrT
topology, and now (a) implies that т = xd.
The following lemma has applications in the study of vector topologies. In fact,
the case n = 1 was needed (and proved) at the end of Theorem 3.6.
3.9 Lemma Suppose Л1?..., Л„ and A are linear functionals on a vector space X.
Let
The following three properties are then equivalent:
(a) There are scalars a1? ..., an such that
A = a1A1 + ... + аиЛи.
(b) There exists у < оо such that
|Ajc|<7 max |A,.jc| (x e X).
l<i<n
(c) Ajc = Ofor every x e N.
proof It is clear that (a) implies (b) and that (b) implies (c). Assume (c)
holds. Let Ф be the scalar field. Define n: X-> Фп by
If n(x) = n(x') then (c) implies Ax = Ax'. Hence A = F ° n, for some function
F on Ф". This F is a linear functional on Ф". Hence there exist at e Ф such that
Thus
n
Ax = F«x)) = ДЛхд:, ..., Anx) = ^ a£A^,
which is (a). ////
3.10 Theorem Suppose X is a vector space and X' is a separating vector space of
linear functionals on X. Then the X'-topology %' makes X into a locally convex space
whose dual space is X'.

CONVEXITY 63
The assumptions on X' are, more explicitly, that X' is closed under addition and
scalar multiplication and that Axt ф Ax2 for some Ae X' whenever xt and x2 are
distinct points of X.
proof Since R and С are Hausdorif spaces, (b) of Section 3.8 shows that т' is a
Hausdorff topology. The linearity of the members of X' shows that т' is trans-
translation-invariant. If Л1э ..., An e X', if rt > 0, and if
A) ' V={x:\AiX\ <rt for l</</7},
then V is convex, balanced, and V e т'. In fact, the collection of all V of the
form A) is a local base for т'. Thus т' is a locally convex topology on X.
If A) holds, then \V + iV= V. This proves that addition is continuous.
Suppose x e X and a is a scalar. Then x e sV for some s > 0. If |/? — a| < r
and j — л: е rK then
fiy-ax= W- a)y + ф - x)
lies in V, provided that r is so small that
Hence scalar multiplication is continuous.
We have now proved that т' is a locally convex vector topology. Every
Л e I' is т'-continuous. Conversely, suppose Л is a r'-continuous linear func-
functional on X. Then |Ajc| < 1 for all x in some set Fof the form A). Condition
(b) of Lemma 3.9 therefore holds; hence so does (а): Л^^а^Л^. Since
Л- g X' and X' is a vector space, Л е X'. This completes the proof. ////
Note: The first part of this proof could have been based on Theorem 1.37
and the separating family of seminorms pA(A e X') given by pA(x) — \Ax\.
3.11 The weak topology of a topological vector space Suppose X is a
topological vector space (with ^topology т) whose dual X* separates points on X.
(We know that this happens in every locally convex X. It also happens in some others;
see Exercise 5.) The X*-topology of A" is called the weak topology of X.
We shall let Xw denote X topologized by this weak topology tw. Theorem 3.10
implies that Xw is a locally convex space whose dual is also X*.
Since every Л e X* is т-continuous and since tw is the weakest topology on X
with that property, we have tw c= t. In this context, the given topology т will often be
called the original topology of X.

64 GENERAL THEORY
Self-explanatory expressions such as original neighborhood, weak neighborhood,
original closure, weak closure, originally bounded, weakly bounded, etc., will be used
to make it clear with respect to which topology these terms are to be understood.1
For instance, let {xn} be a sequence in X. To say that xn -> 0 originally means
that every original neighborhood of 0 contains all xn with sufficiently large n. To say
that xn->0 weakly means that every weak neighborhood of 0 contains all xn with
sufficiently large n. Since every weak neighborhood of 0 contains a neighborhood of
the form
A) V={x:\Atx\ <rt for 1 <i<n},
where AjeI* and rt > 0, it is easy to see that xn -> 0 weakly if and only if Axn -> 0
for every Ael*.
Hence every originally convergent sequence converges weakly. (The converse
is usually false; see Exercises 5 and 6.)
Similarly, a set E <= X is weakly bounded (that is, E is a bounded subset of Xw) if
and only if every V as in A) contains tE for some / = t(V) > 0. This happens if and
only if there corresponds to each AePa number y(A) < oo such that \Ax\ < у(Л)
for every x e E. In other words, a set E a X is weakly bounded if and only if every
A e X* is a bounded function on E.
Let V again be as in A), and put
Since x -> (AjX, ..., An x) maps X into Cn, with nullspace N, we see that dim X <
n + dim N. Since N a V, this leads to the following conclusion.
If X is infinite-dimensional then every weak neighborhood of '0 contains an infinite-
dimensional sub space; hence Xw is not locally bounded.
This implies in many cases that the weak topology is strictly weaker than the
original one. Of course, the two may coincide: Theorem 3.10 implies that (Xw)w — Xw.
We now come to a more interesting result.
3.12 Theorem Suppose E is a convex subset of a locally convex space X. Then
the weak closure Ew of E is equal to its original closure Ё.
1 When Xh a Frechet space (hence, in particular, when Xh a Banach space) the original
topology of X is usually called its strong topology. In that context, the terms " strong "
and "strongly" will be used in place of "original" and "originally." For locally
convex spaces in general, the term "strong topology" has been given a specific
technical meaning. See [15], pp. 256-268; also [14], p. 104. It seems therefore
advisable to use "original" in the present general discussion.

CONVEXITY 65
proof E$ is weakly closed, hence originally closed, so that E c= Ew. To obtain
the opposite inclusion, choose д:0 e X, x0 ф Ё. Part (b) of the separation theorem
3.4 shows that there exist Л e X* and у e R such that, for every x e Ё,
Re Лх0 < у < Re Ax.
The set {x: Re Л* < у} is therefore a weak neighborhood of x0 that does not
intersect E. TJius x0 is not in Ew. This proves Ew c= E. ////
Corollaries 'Let X be a locally convex space.
(a) A sub space of X is originally closed if and only if it is weakly closed.
(b) A convex subset of X is originally dense if and only if it is weakly dense.
The proofs are obvious. Here is another noteworthy consequence of Theorem
3.12:
3.13 Theorem Suppose X is a metrizable locally convex space. If {xn} is a sequence
in X that converges weakly to some x e X, then there is a sequence {yL} in X such that
(a) each y{ is a convex combination of finitely many xn, and
(b) yt-> x originally.
Conclusion (a) says, more explicitly, that there exist numbers ain > 0, such that
and, for each i, only finitely many ain are ф 0.
proof Let H be the convex hull of the set of all xn; let К be the weak closure of
H. Then xeK. By Theorem 3.12, x is also in the original closure of H. Since
the original topology of X is assumed to be metrizable, it follows that there is a
sequence {jj in H that converges originally to x. ////
To get a feeling for what is involved here, consider the following example.
Let К be a compact Hausdorff space (the unit interval on the real line is a
sufficiently interesting one), and assume that /and fn(n— 1, 2, 3,...) are continuous
complex functions on К such that fn(x) ->/(jt) for every x e K, as n -» oo, and such
that \fn(x)\ < 1 for all n and all x e K. Theorem 3.13 asserts that there are convex
combinations of the/, that converge uniformly to/*
To see this, let C(K) be the Banach space of all complex continuous functions
on K, normed by the supremum. Then strong convergence is the same as uniform
convergence on K. If \x is any complex Borel measure on K9 Lebesgue's dominated

66 GENERAL THEORY
convergence theorem implies that \fn dfi -> \f d\i. Hence /„ -»/ weakly, by the Riesz
representation theorem which identifies the dual of C(K) with the space of all regular
complex Borel measures on K. Now Theorem 3.13 can be applied.
After this short detour we now return to our main line of development.
3.14 The weak*-topology of a dual space Let X again be a topological vector
space whose dual is X*. For the definitions that follow, it is irrelevant whether X*
separates points on X or not. The important observation to make is that every x e X
induces a linear functional fx on X*, defined by
and that {fx: x e X} separates points on X*.
The linearity of each fx is obvious; if fxA =fxAf for all x e X, then Ax = A'x
for all x, and so Л = Л' by the very definition of what it means for two functions to be
equal.
We are now in the situation described by Theorem 3.10, with X* in place of X
and with X in place of X'.
The X-topology of X* is called the weak*-topology of X* (pronunciation: weak
star topology).
Theorem 3.10 implies that this is a locally convex vector topology on X* and
that every linear functional on X* that is weak*-continuous has the form A -> Ax
for some x e X.
The weak*-topologies have a very important compactness property to which we
now turn our attention. Various pathological features of the weak- anfl weak*-
topologies are described in Exercises 9 and 10.
Compact Convex Sets
3.15 The Banach-Alaoglu theorem If V is a neighborhood ofO in a topological
vector space X and if
K = {AeX*: \Ax\ < 1 for every x e V}
then К is weak*-compact.
Note: К is sometimes called the polar of V. It is clear that К is convex and
balanced, because this is true of the unit disc in С (and of the interval [—1, 1] in K).
There is some redundancy in the definition of K, since every linear functional on X
that is bounded on Vis continuous, hence is in X*.

CONVEXITY 67
proof Since neighborhoods of 0 are absorbing, there corresponds to each
xe X & number y(x) < oo such that x e y(x) V. Hence
A) \Ax\<y(x) (xeX,AeK).
Let Dx be the set of all scalars a such that | a | < y(x). Let т be the product
topology on P, the cartesian product of all Dx, one for each x e X. Since each
Dx is compact so is P, by Tychonoff's theorem. The elements of P are the
functions/on X (linear or not) that satisfy
B) ' „ \f(x)\<y(x) (xeX).
Thus К <= X* n P. It follows that К inherits two topologies: one from
X* (its weak*-topology, to which the conclusion of the theorem refers) and the
other, т, from P. We will see that
(a) these two topologies coincide on K, and
(b) К is a closed subset of P.
Since P is compact, (b) implies that К is т-compact, and then (a) implies that
К is weak*-compact.
Fix some Ло e K. Choose xt e X, for 1 < / < n; choose 5 > 0. Put
C) W1={Ae X*: \Axt -Аох£\ <S for 1 < i < n}
and
D) W2 = {feP: \f(Xi) - Aox(\ <S for 1 < / < n].
Let n, xt, and S range over all admissible values. The resulting sets Wx then
form a local base for the weak*-topology of X* at Ло and the sets W2 form a
local base for the product topology т of P at Ло. Since KczP n X*, we have
Wx n К = W2 n K.
This proves (a).
Next, suppose /0 is in the т-closure of К Choose x e X, у e X, scalars
a and /?, and e > 0. The set of all/eP such that \f — fo\ < г at x, at 7, and at
ocx + Py is a т-neighborhood of/0. Therefore A' contains such an / Since this
/is linear, we have ^
/0(ax + /ЗД - af0(x) - ШУ) = (/0 -
so that
|/o(oa + /i»
Since £ was arbitrary, we see that /0 is linear. Finally, if x e V and e > 0, the
same argument shows that there is an feK such that \f(x)—fo{x)\ <s.

68 GENERAL THEORY
Since |/(jc)| < 1, by the definition of K, it follows that |/0(jc)| < 1. We conclude
that/0 e K. This proves (b) and hence the theorem. ////
When X is separable (i.e., when there is a countable dense set in X) then the
conclusion of the Banach-Alaoglu theorem can be strengthened by combining it with
the following fact:
3.16 Theorem If X is a separable topological vector space, if К <= X* and if К is
weak*-compact, then К is metrizable, in the weak*-topology.
Warning: It does not follow that X* itself is metrizable in its weak*-topology. In
fact, this is false whenever Xis an infinite-dimensional Banach space. See Exercise 15.
proof Let {xn} be a countable dense set in X. Put /„(A) = Axn, for Л е I*.
Each /„ is weak*-continuous, by the definition of the weak*-topology. If
/„(Л) =/п(Л') for all n, then Axn = A'xn for all n, which implies that Л = Л',
since both are continuous on X and coincide on a dense set.
Thus {/„} is a countable family of continuous functions that separates
points on X*. The metrizability of К now follows from (c) of Section 3.8. ////
3.17 Theorem If V is a neighborhood ofO in a separable topological vector space
X, and if{An} is a sequence in X* such that
\Anx\ <1 (xeK,/i = 1,2,3,...),
then there is a subsequence {Art.} and there is a A e X* such that
Ax = lim Ащ x (x e X).
i-+ao
In other words, the polar of V is sequentially compact in the weak*-topology.
proof Combine Theorems 3.15 and 3.16. ////
The next application of the Banach-Alaoglu theorem involves the Hahn-
Banach theorem and a category argument.
3.18 Theorem In a locally convex space X, every weakly bounded set is originally
bounded, and vice versa.
Part (d) of Exercise 5 shows that the local convexity of X cannot be omitted
from the hypotheses.

CONVEXITY 69
proof Since every weak neighborhood of 0 in X is an original neighborhood
of 0, it is obvious from the definition of " bounded " that every originally bounded
subset of Xis weakly bounded. The converse is the nontrivial part of the theorem.
Suppose E с X is weakly bounded and U is an original neighborhood of
0 in X.
. Since X is locally convex, there is a convex, balanced, original neighbor-
neighborhood Fof 0 in Xsuch that Fcf/. Let К c= X* be the polar of V:
A) * K={Ae X*: \Ax\ < 1 for all x e V}.
We claim that
B) V= {x e X: \Ax\ < 1 for all Л е К}.
It is clear that V is a subset of the right side of B) and hence so is F, since the
right side of B) is closed. Suppose x0 e X but x0 ф V. Theorem 3.7 (with V
in place of B) then shows that Лх0 > 1 for some AeK. This proves B).
Since E is weakly bounded, there corresponds to each Ael*a number
y(A) < oo such that
'C) |Л*|<у(Л) (хеЕ).
Since A^is convex and weak*-compact (Theorem 3.15) and since the functions
Л -> Ax are weak*-continuous, we can apply Theorem 2.9 (with X* in place of
X and the scalar field in place of Y) to conclude from C) that there is a constant
у < oo such that
D) |Ax|<7 (xeE,AeK).
Now B) and D) show that j~xxe Fez U for all xeE. Since V is balanced,
E) EcztVcztU (t>y).
Thus E is originally bounded. ////
Corollary If X is a normed space, if E a X, and if
F) sup|Ax| <oo (AeP)
xeE
then there exists у < oo such that
G) WI<T (xeE).
proof Normed spaces are locally convex; F) says that E is weakly bounded,
and G) says that E is originally bounded. ////
The following analogue of part (b) of the separation theorem 3.4 will be used in
the proof of the Krein-Milman theorem.

70 GENERAL THEORY
3.19 Theorem Suppose X is a topological vector space on which X* separates
points. Suppose A and В are disjoint, nonempty, compact, convex sets in X. Then
there exists Ael* such that
A) sup Re Ax < inf Re Ay.
xeA ye В
Note that part of the hypothesis is weaker than in (b) of Theorem 3.4 (since local
convexity of X implies that X* separates points on X); to make up for this, it is now
assumed that both A and В are compact.
proof Let Xw be X with its weak topology. The sets A and В are evidently
compact in Xw. They are also closed in Xw (because Xw is a Hausdorff space).
Since Xw is locally convex, (b) of Theorem 3.4 can be applied to Xw in place of
X; it gives us а Л e (Xw)* that satisfies A). But we saw in Section 3.11 (as a
consequence of Theorem 3.10) that (Xw)* = X*. ////
3.20 Extreme points Let К be a subset of a vector space X. A nonempty set
S <= К is called an extreme set of К if no point of S is an internal point of a line interval
whose end points are in К but not in S. Analytically, the condition can be expressed
as follows: If x e К, у e K, 0 < t < 1, and
tx + A - t)y e S,
then x e S and у е S.
The extreme points of К are the extreme sets that consist of just one point.
Recall that the convex hull of a set E cz X is the smallest convex set in X that
contains E, and that the closed convex hull of E is the closure of its convex hull.
At present it is not known whether it is true in every topological vector space X
that every compact convex set has an extreme point. In a large class of spaces, the
supply of extreme points is actually very abundant. This is shown by Theorems
3.21 and 3.22.
3.21 The Krein-Milman theorem Suppose X is a topological vector space on
which X* separates points. If К is a compact convex set in X, then К is the closed
convex hull of the set of its extreme points.
proof Let 0> be the collection of all compact extreme sets of K. Since К е &*,
& Ф 0- We shall use the following two properties of 0>\
(a) The intersection S of any nonempty subcollection of 0* is a member of £РУ
unless S = 0.
(b) IfSe&iAeX*,^ is the maximum of Re A on S, and
then S

CONVEXITY 71
The'proof of (a) is immediate. To prove (b), suppose tx -f A — t)y =
ге5л, xgAT, jG/f, 0 < / < 1. Since z e S and 5е^, we have xg5 and
>> e 5*. Hence Re Ax < /*, Re Ay < ji. Since Re Az = /* and Л is linear, we
conclude: Re Ax = [x = Re Лу. Hence хе5л and jg^.
Choose some Se^. Let 0>' be the collection of all members of ^ that are
subsets of S. Since S e 0>\ 0>' is not empty. Partially order 0>' by set inclusion,
let Qbea maximal totally ordered subcollection of &', and let M be the inter-
intersection of all members of Q. Since Q is a collection of compact sets with the
finite intersection property, M ф 0. By (a), M e 0*'. The maximality of Q.
implies that no proper subset of M belongs to &. It now follows from (b) that
every Л e X* is constant on M. Since X* separates points on X, Mhas only one
point. Therefore M is an extreme point of K.
We have now proved that every compact extreme set of К contains an
extreme point of K. (So far, the convexity of К has not been used.)
If H is the convex hull of the set of extreme points of K, it follows, for
every S e 0>, that H n S is not empty.
Since К is compact and convex, we have H с К. Hence H is compact.
Assume (to get a contradiction) that some x0 e К is not in H. By Theorem
3.19 there is а Л e X* such that ReAx < ReAx0 for every x eH. If KA is
defined as in (b), then KA e 0*. Since H does not intersect KA we have our
contradiction. ////
Remark The convexity of К was used only to show that H is compact.
If X were assumed to be locally convex, the compactness of H would not be
needed, since one could use (b) of Theorem 3.4 in place of Theorem 3.19.
The above argument then proves that KcH. The following version of the
Krein-Milman theorem is thus obtained:
3.22 Theorem If X is a locally convex space and if E is the set of extreme points
of a compact set К in X, then К lies in the closed convex hull of E.
Equivalently, E and К have the same closed convex hull.
The preceding remark raises a question: What can one say about the convex
hull H of a compact set КЧ Even jn a Hilbert space, H need not be closed, and there
are situations in which H is not compact (Exercises 20, 22). In Frechet spaces, the
latter pathology does not occur (Theorem 3.25). The proof of this will depend on
the fact that a subset of a complete metric space is compact if and only if it is closed
and totally bounded. (Appendix A4.)
Let us recall that a subset E of a metric space X\s said to be totally bounded \{ E
is contained in the union of finitely many open balls of radius г, for every г > 0.
The same concept can be defined in any topological vector space X, metrizable or not.

72 GENERAL THEORY
3.23 Definition A set E in a topological vector space X is said to be totally
bounded if to every neighborhood V of 0 in X corresponds a finite setfcl such that
EF V.
If X happens to be a metrizable topological vector space, then these two notions
of total boundedness coincide, provided that we restrict ourselves to invariant metrics
that are compatible with the topology of X. (The proof of this is as in Theorem 1.26.)
3.24 Theorem If X is a locally convex space and H is the convex hull of a totally
bounded set E с X, then H is totally bounded.
proof Let U be a neighborhood of 0 in X. There is a convex neighborhood V
of 0 in X such that V + V с С/, and there is a finite set Ex с X such that
E с Ex + V. Let #! be the convex hull of Ex.
If eu ..., em are the points of Ex and if S is the simplex in Rm consisting
of all f = (f!,..., tm) that satisfy tt > 0 and £/(. = 1, then
maps the compact set S continuously onto H^. Hence Нг is compact.
If x e H, then x = a^ + • • • + an xn, where jct- e £, a; > 0, ]Га; = 1.
There are points yt e Ex such that xt — yi e V\ this follows from the choice
of Ex. Decompose x into the sum
x = x -b x",
where jc' = Х/*^ an(^ x" — Z0^** ~~ ^«)- ^e convexity of V implies that
x" 6 V. It is clear that x' e H1. Hence
H с Hx + V.
Since Hx is compact, there is a finite set F such that HY с JF + F. Thus
HcF+ V+ VczF+ U.
Since f/ was arbitrary, it follows that H is totally bounded. ////
3.25 Theorem Suppose H is the convex hull of a compact set К in a topological
vector space X.
(a) If X is a Frechet space, then H is compact.
(b) IfX= Rn, then H is compact.
proof (a) By Theorem 3.24, H is totally bounded. Since Frechet spaces are
complete metric spaces, the closure H of H is compact.

CONVEXITY 73
(b) Let S be the simplex in Rn + 1 consisting of all t = (tu ..., tn + l)
with tt > 0 and £f£ = 1. By the lemma that follows, x e H if and only if
i= 1
for some t e S and xt- e ^ A < / < n -f 1). In other words, Я is the image of
SxKx- xK
(K occurs n + 1 times.) under the continuous mapping
(/, xl9 ...,xn + 1)-> £/.*..
i = 1
Hence H is compact. ////
Lemma If x lies in the convex hull of a set E cz Rn, then x lies in the convex
hull of some subset of E that contains at most n -f 1 points.
proof It is enough to show that if r > n and x = Yji xt is a convex combination
- of some r -f 1 vectors xt e E, then x is actually a convex combination of some
r of these vectors.
Assume, without loss of generality, that tt > 0 for 1 < / < r + 1. The
r vectors xt- — xr+l A < / < r) are linearly dependent, since r > n. It follows
that there are real numbers at, not all 0, such that
r+l r+1
Yj ai xt — 0 and Л a,- = 0.
Choose m so that |tft//t| < \ajtm\, for 1 < i < r + 1, and define
Then c, > 0, YFi = 1^- = l > x = Iеt xi > an^ cm = 0. ////
Vector-valued Integration *:
Sometimes it is desirable to be able to integrate functions/that are defined on some
measure space Q (with a real or complex measure //) and whose values lie in some
topological vector space X. The first problem is to associate with these data a vector
in X that deserves to be called

74 GENERAL THEORY
i.e., which has at least some of the properties that integrals usually have. For instance,
the equation
ought to hold for every ЛеР, because it does hold for sums, and because integrals
are (or ought to be) limits of sums in some sense or other. In fact, our definition will
be based on this single requirement.
Many other approaches to vector-valued integration have been studied in great
detail; in some of these, the integrals are defined more directly as limits of sums (see
Exercise 23).
3.26 Definition Suppose \i is a measure on a measure space Q, A" is a topological
vector space on which Z* separates points, and/is a function from Q into Xsuch that
the scalar functions Л/аге integrable with respect to /л, for every Ле!*; note that
Л/is defined by
A) (Л/)(<?) = Л(Д<7)) {qeQ).
If there exists a vector у е X such that
B) Ay
for every A e X*, then we define
C) \
Remarks It is clear that there is at most one such y, because X* separates
points on X. Thus there is no uniqueness problem.
Existence will be proved only in the rather special case (sufficient for many
applications) in which Q is compact and/is continuous. In that case,/@ is compact,
and the only other requirement that will be imposed is that the closed convex hull of
f(Q) should be compact. By Theorem 3.25, this additional requirement is automatically
satisfied when X is a Frechet space.
Recall that a Bore I measure on a compact (or locally compact) Hausdorff
space Q is a measure defined on the cr-algebra of all Borel sets in Q; this is the smallest
cr-algebra that contains all open subsets of Q. A probability measure is a positive
measure of total mass 1.
3.27 Theorem Suppose
(a) X is a topologica{ vector space on which X* separates points, and
(b) \x is a Borel probability measure on a compact Hausdorff space Q.

CONVEXITY 75
Iff'- Q -*X is continuous, and if the convex hull H off(Q) has compact closure
H in X, then the integral
0) ' y
exists, in the sense of Definition 3.26.
Moreover, у е Н.
Remark If- v is any positive Borel measure on Q, then some scalar multiple
of v is a probability measure. The theorem therefore holds (except for its last
sentence) with v in place of /x. It can then be extended to real-valued Borel
measures (by the Jordan decomposition theorem) and (if the scalar field of
X is 0) to complex ones.
Exercise 24 gives another generalization.
proof Regard lasa real vector space. We have to prove that there exists
у е Н such that
B) Ay = f (Л/) dfi
JQ
for every A e X*.
Let L = {Au ..., Л„} be a finite subset of X*. Let EL be the set of all
у е Н that satisfy B) for every AeL. Each EL is closed (by the continuity
of Л) and is therefore compact, since H is compact. If no EL is empty, the col-
collection of all EL has the finite intersection property. The intersection of all EL
is therefore not empty, and any у in it satisfies B) for every Л 6 X*. It is there-
therefore enough to prove EL ф 0.
Regard L = (At, ..., Ля) as a mapping from X into R", and put K =
Uf(Q)l Define
C) mi\
We claim that the point m = (mu ..., mn) lies in the convex hull of K.
If t = (tu ..., tn)eRn is not in this hull, then [by Theorem 3.25 and
(b) of Theorem 3.4 and the known form of the linear functionals on Rn] there
are real numbers съ ..., cn such that
() E*i E(,
1 = 1 i=l
if и = («j, ..., un) e К Hence
E) I>A-/(<7)< 2>,'f fe 6 0.

76 GENERAL THEORY
Since ji is a probability measure, integration of the left side of E) gives
This shows that m lies in the convex hull of K. Since К z=L(f(Q)) and L
is linear, it follows that m = Ly for some у in the convex hull H off(Q). For
this у we have
F) Aty = mt = f (Л,/) d/л (\<i< n).
JQ
Hence у е EL. This completes the proof. ////
3.28 Theorem Suppose
(a) X is a topological vector space on which X* separates points,
(b) Q is a compact subset of X, and
(c) the closed convex hull H of Q is compact.
Then у e H if and only if there is a regular Borel probability measure \i on Q
such that
О) У-
Remarks The integral is to be understood as in Definition 3.26, with/(x) = x.
Recall that a positive Borel measure on Q is said to be regular if
B) ii{E) = sup {fi(K) :KczE} = mf {/z(G): E с G}
for every Borel set E cz Q, where К ranges over the compact subsets of E and G
ranges over the open supersets of E.
The integral A) represents every у еН sls sl "weighted average" of g,
or as the "center of mass" of a certain unit"mass distributed over Q.
We stress once more that (c) follows from (b) if X is a Frechet space.
proof Regard X again as a real vector space. Let C(Q) be the Banach space
of all real continuous functions on Q, with the supremum norm. The Riesz
representation theorem identifies the dual space C@* with the space of all
real Borel measures on Q that are differences of regular positive ones. With this
identification in mind, we define a mapping
C)
by
D) -

CONVEXITY 77
Let P be the set of all regular Borel probability measures on Q. The
theorem asserts that ф(Р) = H.
For each x e Q, the unit mass Sx concentrated at x belongs to P. Since
0(<5jc) — *•> we see that Q с ф(Р). Since ф is linear and P is convex, it follows
that H cz ф(Р), where H is the convex hull of Q. By Theorem 3.27, ф(Р) с Н.
Therefore all that remains to be done is to show that ф(Р) is closed in X.
This is a consequence of the following two facts:
(/) P is weak*-compact in C@*.
(if) The mapping ф -defined by D) is continuous if C@* is given its weak*-
topology and if X is given its weak topology.
Once we have (/) and (//), it follows that ф(Р) is weakly compact, hence
weakly closed, and since weakly closed sets are strongly closed, we have the
desired conclusion.
To prove (/), note that
E) {\)
-and that this larger set is weak*-compact, by the Banach-Alaoglu theorem.
It is therefore enough to show that P is weak*-closed.
If heC(Q) and h > 0, put
F)
Since ii->\hdii is continuous, by the definition of the weak*-topology, each
Eh is weak*-closed. So is the set
G)
Since P is the intersection of E and the sets Eh, P is weak*-closed.
To prove (ii) it is enough to prove that ф is continuous at the origin,
since ф is linear. Every weak neighborhood of 0 in X contains a set of the form
(8) W={yeX:\Kiy\<ri for 1 </<«},
where At e X* and r{ > 0. The restrictions of the Л£ to Q lie in C(Q). Hence
(9) V = LeC(Q)*: f Л,&\i < rt for 1 < i < n)
is a weak*-neighborhood of 0 in C@*. But
A0) JAtdn = A,(j x diiix)j = Л, Ш,

78 GENERAL THEORY
by Definition 3.26. It follows from (8), (9), and A0) that ф(У) с W. Hence
ф is continuous. ////
The following simple inequality sharpens the last assertion in the statement of
Theorem 3.27.
3.29 Theorem Suppose Q is a compact Hausdorff space, X is a Banach space,
f: Q -> X is continuous, and ц is a positive Borel measure on Q. Then
/II dfi.
proof. Put у = \fd\i. By the corollary to Theorem 3.3, there is aAel*
such that Ay = \\y\\ and \Ax\ < \\x\\ for all x e X. In particular,
| < 11/0011
for all seQ. By Theorem 3.27, it follows that
Holomorphic Functions
In the study of Banach algebras, as well as in some other contexts, it is useful to
enlarge the concept of holomorphic function from complex-valued ones to vector-
valued ones. (Of course, one can also generalize the domains, by going from <Z to <Zn
and even beyond. But this is another story.) There are at least two very natural
definitions of "holomorphic" available in this general setting, a "weak" one and a
"strong" one. They turn out to define the same class of functions if the values are
assumed to lie in a Frechet space.
3.30 Definition Let Q be an open set in С and let X be a complex topological
vector space.
(a) A function/: п -► .Ж is said to be weakly holomorphic in Q if Л/ is holomorphic
in the ordinary sense for every Л е X*.
(b) A function /: Q, -> X is said to be strongly holomorphic in Q. if
w-*z W — Z
exists (in the topology of X) for every zefi.
Note that the above quotient is the product of the scalar (w — z)'1 and the
vector/(w) — f(z) in X.

CONVEXITY 79
The continuity of the functionals Л that occur in (a) makes it obvious that every
strongly holomorphic function is weakly holomorphic. The converse is true when X
is a Frechet space, but it is far from obvious. (Recall that weakly convergent se-
sequences may very well fail to converge originally.) The Cauchy theorem will play an
important role in this proof, as will Theorem 3.18.
The index of a point z eG with respect to a closed path Г that does not pass
through z will be denoted by Indr (z). We recall that
3.31 Theorem Let п be open in 0, let X be a complex Frechet space, and assume
that
f:Q-+X
is weakly holomorphic. The following conclusions hold:
(a) fis strongly continuous in Q.
(b) The fcauchy theorem and the Cauchy formula hold: If Г is a closed path in Q
*such that Indr (w) = Ofor every w ф Q, then
(i)
and
B) /(z)= .
2ni ■
ifzeQ and Indr (z) = 1. IfTx and T2 are closed paths in Q such that
IndFl (w) = Indr2 (w)
for every w ф Q, then
C) f M)dC= Г /@ JC
Jrt Jr2
(c) f is strongly holomorphic in Q.
The integrals in (b) are to he understood in the sense of Theorem 3.27. Either
one can regard a% as a complex measure on the range of Г (a compact subset of 0), or
one can parametrize Г and integrate with respect to Lebesgue measure on a compact
interval in R.
proof (a) Assume 0 e Q.. We shall prove that / is strongly continuous at 0.
Define
D) Ar = {ze(Z: \z\ < r].

80 GENERAL THEORY
\ )
Then A2r cz Q for some r > 0. Let Г be the positively oriented boundary of
A2r.
Fix Л e X*. Since Л/is holomorphic,
... 1 г (Д/
- 55 Jr к -
if 0 < \z\ < 2r. Let М(Л) be the maximum of |Л/| on A2r. If 0 < |z| < r,
it follows that
F) I
The set of all quotients
G)
is therefore weakly bounded in X. By Theorem 3.18, this set is also strongly
bounded. Thus if V is any (strong) neighborhood of 0 in X, there exists t < со
such that
(8) /(z)-/@)ez/F @<|z| < r).
Consequently, /(z) -»/@) strongly, as z -> 0.
This was the crux of the matter. The rest is now almost automatic.
(b), By (a) and Theorem 3.27, the integrals in A) to C) exist. These
three formulas are correct (by the theory of ordinary holomorphic functions)
if /is replaced in them by Л/ where Л is any member of X*. The formulas are
therefore correct as stated, by Definition 3.26.
(c) Assume, as in the proof of (a), that A2r <= £2, and choose Г as in (a).
Define
(9) j, M
2ni
The Cauchy formula B) shows, aft er a small computation, that
,if 0 < \z\ <2r, where
A1) g(z) = ~f [2reieBrew - z)Tlf{2rew) d6.
In >
Let V be a^ convex balanced neighborhood of 0 in X. Put K =
{/@: K| = 2r}. Then К is compact, so that К a tV for some t <co. If s =

CONVEXITY 81
tr~2 and [z\ < r, it follows that the integrand A1) lies in ^Ffor every 6. Thus
g(z) e s У if | z\ <r. The left side of A0) therefore converges strongly to j, as
The following extension of Liouville's theorem concerning bounded entire
functions does not even depend on Theorem 3.31. It can be used in the study of
spectra in Banach algebras. (See Exercise 4, Chapter 10.)
3.32 Theorem Suppose X is a complex topological vector space on which X*
separates points. Suppose/: @ -> Xis weakly holomorphic andf(<£) is a weakly bounded
subset of X. Then f is constant.
proof For every Л e X*, Л/is a bounded (complex-valued) entire function.
If z e 0, it follows from Liouville's theorem that
Since X* separates points on X, this implies/(z) =/@), for every z e С ////
Part (d) of Exercise 5 describes a weakly bounded set which is not originally
bounded, in an jp-space Xon which X* separates points. Compare with Theorem 3.18.
Exercises
1 Call a set H <= Rn a hyperplane if there exist real numbers au ..., an, с (with at ф 0 for
at least one /) such that H consists of all points x = (xb ..., xn) that satisfy 2 etXt = с
Suppose E is a convex set in R", with nonempty interior, and у is a boundary
point of E. Prove that there is a hyperplane H such that yeH and E lies entirely on
one side of H. (State the conclusion more precisely.) Suggestion: Suppose 0 is an interior
point of E, let M be the one-dimensional subspace that contains y, and apply Theo-
Theorem 3.2.
2 Suppose L2 =L2([—1, 1]), with respect to Lebesgue measure. For each scalar a, let
Ex be the set of all continuous functions/on [—1, 1] such that /@) = a. Show that
each Ea is convex and that each is dense in L2. Thus Ea and Ep are disjoint convex sets
(if а Ф ft) which cannot be separated by any continuous linear functional Л on L2.
Hint: WhatisA(£"a)? *:
3 Suppose Xis a real vector space (without topology). Call a point x0 e A <= JTan internal
point of A if A — Xo is an absorbing set.
(a) Suppose A and В are disjoint convex sets in X, and A has an internal point. Prove
that there is a nonconstant linear functional Л on Xsuch that &(A) n A(B) contains
at most one point. (The proof is similar to that of Theorem 3.4.)
(b) Show (with X = R2, for example) that it may not be possible to have A(A) and &(B)
disjoint, under the hypotheses of (a).

82 GENERAL THEORY
4 Let /°° be the space of all real bounded functions x on the positive integers. Let т be
the translation operator defined on Z00 by the equation
(tx)(/2)=x(«+1) in = 1,2,3,...).
Prove that there exists a linear functional Л on /°° (called a Banach limit) such that
(а) Лтх = Лх, and
ib) lim inf x(n) < Лх < lim sup x(n)
n-*oo n-*ao
for every x e /°°.
Suggestion: Define
M = {л: e /*>: lim Л„ х = Лх exists}
n-*oo
/?(x) = lim sup Л„ x
n-+ со
and apply Theorem 3.2.
5 For 0 <p < oo, let Л be the space of all functions x (real or complex, as the case may
be) on the positive integers, such that
2 \x(n)\p<co.
n=l
For 1 <p < oo, define 11*11, = B |х(и)|р}1/р, and define ||*||e =supB \x(ri)\.
(a) Assume 1 <p < oo. Prove that ||x||P and ||x||co make Л and Z00 into Banach spaces.
If p~l + q~y = 1, prove that (/p)* = Л, in the following sense: There is a one-to-one
correspondence K+-+y between (/p)* and /9, given by
{Ь) Assume 1 <p < oo and prove that Л contains sequences that converge weakly
but not strongly.
(c) On the other hand, prove that every weakly convergent sequence in Л converges
strongly, in spite of the fact that the weak topology of Z1 is different from its strong
topology (which is induced by the norm).
(d) If 0 <p < 1, prove that /p, metrized by
is a locally bounded F-space which is not locally convex but that (/p)* nevertheless
separates points on /p. (Thus there are many convex open sets in Л but not enough
to form a base for its topology.) Show that (Л)* = Z00, in the, same sense as in (я),.
Show also that the set of all x with Щх(п)\ < 1 is weakly bounded but not originally
bounded.
(e) For 0 <p < 1, let тр be the weak*-topology induced on Z00 by /p; see (a) and (d).
lfO<p<r<\, show that rp and rr are different topologies (is one weaker than the
other?) but that they induce the same topology on each norm-bounded subset of
/°°. Hint: The norm-closed unit ball of /°° is weak*-compact.

CONVEXITY 83
Put /„(/) ='eint (—it <t<77); let Lp =Lp(—tt, tt), with respect to Lebesgue measure.
If 1 <p < oo, prove that/, ->0 weakly in Lp, but not strongly.
L°°([0, 1]) has its norm topology (||/||a> is the essential supremum of |/|) and its weak*-
topology as the dual of L1. Show that C, the space of all continuous functions on [0, 1],
is dense in V° in one of these topologies but not in the other. (Compare with the
corollaries to Theorem 3.12.) Show the same with "closed" in place of "dense."
Let С be the Banach space of all complex continuous functions on [0, 1], with the
supremum norm. Let В be the closed unit ball of С Show that there exist continuous
linear functionals Л on С for which A(B) is an open subset of the complex plane; in
particular, |Л| attains no maximum on B.
Let E с L2(-tt, 77) be the set of all functions
fm. n(t) = eimt + mein\
where m, n are integers and 0 < m < n. Let Ei be the set of all g eL2 such that some
sequence in E converges weakly to g. {E± is called the weak sequential closure of E.)
(a) Find all geE^
(b) Find all g in the weak closure Ew of E.
(c) Show that Oe Ew but 0 is not in Eu although 0 lies in the weak sequential closure
This example shows that a weak sequential closure need not be weakly sequentially
closed. The passage from a set to its weak sequential closure is therefore not a closure
operation, in the sense in which that term is usually used in topology. (See also
Exercise 28.)
10 Represent Л as the space of all real functions x on S = {(m, n): m > 1, n > 1}, such that
11x11! = 2 \x(m,n)\ <oo.
Let Co be the space of all real functions у on S such that y(m, n) ->0 as m + n -> 00,
with norm \\y\U = sup \y(m, n)\.
Let M be the subspace of Л consisting of all x e Z1 that satisfy the equations
mx(m, 1) = 2 x(m> n) (w = 1, 2, 3,...).
n = 2
(a) Prove that Л = (c0)*. (See also Exercise 24, Chapter 4.)
(b) Prove that M is a norm-closed subspace ofY1.
(c) Prove that M is weak*-dense in Л [relative to the weak*-topology given by (a)].
(d) Let В be the norm-closed tfnit ball of Л. In spite of (c), prove that the weak*-
closure of M n В contains no ball. Suggestion: If 8 > 0 and m > 2/8, then
if x e M n B, although x(m, 1) = 8 for some x e 35. Thus S/? is not in the weak*-
closure of M n B. Extend this to balls with other centers.
11 Let X be an infinite-dimensional Frechet space. Prove that X*, with its weak*-topology,
is of the first category in itself.

84 GENERAL THEORY
12 Show that the norm-closed unit ball of c0 is not weakly compact; recall that (c0)* = Z1
(Exercise 10).
13 Put/v(O = JV-1 2?-i etnt. Prove that fN -»0 weakly in L2(-tt, tt).
By Theorem 3.13, some sequence of convex combinations of the/v converges to
0 in the L2-norm. Find such a sequence. Show that gN = iV^C/i + Ь /*) will not do.
14 (a) Suppose Q is a locally compact Hausdorff space. For each compact К с п define a
seminorm pK on С(п), the space of all complex continuous functions on П, by
pK(f) = sup {\f(x)\:xeK}.
Give C(O) the topology induced by this collection of seminorms. Prove that to
every Л e C(Q)* correspond a compact К с Q and a complex Borel measure p on К
such that
Л/= Г f dfi (/еОД).
F) Suppose Q. is an open set in <?. Find a countable collection Г of measures with
compact support in Q such that H(C1) (the space of all holomorphic functions in Q)
consists of exactly those /e C(Q) which satisfy J / ф. = 0 for every /x e Г.
75 Let X be a topological vector space on which X* separates points. Prove that the weak*-
topology of X* is metrizable if and only if X has a finite or countable Hamel basis.
(See Exercise 1, Chapter 2 for the definition.)
16 Prove that the closed unit ball of L1 (relative to Lebesgue measure on the unit interval)
has no extreme points but that every point on the "surface" of the unit ball in Lp
A <p < со) is an extreme point of the ball.
17 Determine the extreme points of the closed unit ball of C, the space of all continuous
functions on the unit interval, with the supremum norm. (The answer depends on the
choice of the scalar field.)
7# Let К be the smallest convex set in R2 that contains the points A, 0, 1), A, 0, —1), and
(cos в, sin в, 0), for 0 < 6 < 2тг. Show that К is compact but that the set of all extreme
points of К is not compact. Does such an example exist in R21
19 Suppose К is a compact convex set in R". Prove that every x e К is a convex combina-
combination of at most n + 1 extreme points of K. Suggestion: Use induction on n. Draw a
line from some extreme point of К through x to where it leaves K. Use Exercise 1.
20 Suppose a topological vector space X contains a countable set Е = {е1,е2,ез,..-}
with the following properties:
(a) en->0 as л->оо.
(b) Every xelisa finite linear combination of members of E, x = 2 7n(x)en.
(c) No en is in the closed subspace of X generated by the other et.
For example, X could be the space of all complex polynomials
f(z) = a0 + aiz ^ h anz\
with norm
and with en(z) = n'1^'1 (л = 1, 2, 3, ...).

CONVEXITY 85
Prove that each yn in (b) is in X*. Put K = Eu {0}. Then К is compact.
Prove that the convex hull H of К is closed but not compact and that the extreme
points of H are exactly the points of K.
21 If 0 <p < 1, every feLp (except /= 0) is the arithmetic mean of two functions whose
distance from 0 is less than that of/. (See Section 1.47.) Use this to construct an explicit
example of a countable compact set К in Lp (with 0 as its only limit point) which has
no extreme point.
22 If 0 <p < 1, show that Л contains a compact set К whose convex hull is unbounded.
This happens in spite of the fact that (Л)* separates points on Л; see Exercise 5.
Suggestion: Define xn e *f £ by
xn(n) =np~\ xn(m) = 0 if m Ф n.
Let К consist of 0, xu x2, x3, •.. . If
show that {yN} is unbounded in /p.
23 Suppose /x is a Borel probability measure on a compact Hausdorff space Q, A" is a
Frechet space, and /: Q -> X is continuous. A partition of Q is, by definition, a finite
collection of disjoint Borel subsets of Q whose union is Q. Prove that to every neighbor-
neighborhood V of 0 in X there corresponds a partition {£*,} such that the difference
lies in V for every choice of sieEi. (This exhibits the integral as a strong limit of
" Riemann sums.") Suggestion: Take Kconvex and balanced. If Л e X* and if |Лх| < 1
for every x e V, then \Az\ < 1, provided that the sets Et are chosen so that f(s) —fit) e V
whenever s and / lie in the same Et.
24 In addition to the hypotheses of Theorem 3.27, assume that T is a continuous linear
mapping of X into a topological vector space Y on which Y* separates points, and
prove that
f /ф = f
Jq Jq
Hint: ATe X* for every Л е У*.
25 Let E be the set of all extreme points of a compact convex set К in a topological vector
space Ion which X* separates-points. Prove that to every у е К corresponds a regular
Borel probability measure \x on Q —E such that
'=( xd^x).
Jo
26 Suppose Cl is a region in 0, X is a Frechet space, and/: И -> JSf is holomorphic.
(a) State and prove a theorem concerning the power series representation of/ that is,
concerning the formula /(z) = 2 (z ~ аУс» > where с„ е X.
(b) Generalize Morera's theorem to AWalued holomorphic functions.

86 GENERAL THEORY
(с) For a sequence of complex holomorphic functions in П, uniform convergence on
compact subsets of П implies that the limit is holomorphic. Does this generalize to
X-valued holomorphic functions?
27 Suppose {a*} is a bounded set of distinct complex numbers, /(z) = 2* cnzn is an entire
function with every cn ф 0, and
Prove that the vector space generated by the functions gt is dense in the Frechet space
#(<?) defined in Section 1.45.
Suggestion: Assume /x is a measure with compact support such that \gt d/x = 0
for all i. Put
Prove that ф(м) = 0 for all w. Deduce that J z" dfi(z) - 0 for и = 1, 2, 3, .... Use
Exercise 14.
Describe the closed subspace of Я(<£) generated by the functions gt if some of
the с„ are 0.
28 Suppose X is a Frechet space (or, more generally, a metrizable locally convex space).
Prove the following statements.
(a) X* is the union of countably many weak*-compact sets Е„.
(b) If X is separable, each En is metrizable. The weak*-topology of X* is therefore
separable, and some countable subset of X* separates points on X. (Compare with
Exercise 15.)
(c) If К is a weakly compact subset of X and if x0 e К is a weak limit point of some
countable set E <= K, then there is a sequence {х„} in E which converges weakly
to jc0. Hint: Let Y be the smallest closed subspace of X that contains E. Apply
(b) to Y to conclude that the weak topology of К п Yis metrizable.
Remark: The point of (c) is the existence of convergent subsequences rather
than subnets. Note that there exist compact Hausdorff spaces in which no sequence
of distinct points converges.
29 Let C{K) be the Banach space of all continuous complex functions on the compact
Hausdorff space K, with the supremum norm. For p e K, define Ap e C(K)* by
Лр/=/(р). Show that p~>Ap is a homeomorphism of К into C(K)*, equipped with
its weak*-topology. Part (c) of Exercise 28 can therefore not be extended to weak*-
compact sets.

DUALITY IN BANACH SPACES
The Normed Dual of a Normed Space
Introduction If X and Y are topological vector spaces, 3t(X, Y) will denote the
collection of all bounded linear mappings (or operators) of X into Y, For simplicity,
&(X, X) will be abbreviated to ЩХ). Each ЩХ, Y) is itself a vector space, with
respect to the usual definitions of addition and scalar multiplication of functions.
(This depends only on the vector space structure of Y, not on that of X.) In general,
there are many ways in which &(X, Y) can be made into a topological vector space.
In the present chapter, we shall deal only with normed spaces X and Y. In that
case, $(X, Y) can itself be normed in a very natural way. When Fis specialized to be
the scalar field, so that &(X, Y) is the dual space X* of X, the above-mentioned
norm on &(X, Y) defines a topology on X* which turns out to be stronger than its
weak*-topology. The relations between a Banach space X and its normed dual X*
form the main topic of this chapter.
4.1 Theorem Suppose X and Y are normed spaces. Associate to each Л e f (I, Y)
the number
A) ||A||=sup{||Ax||:xeX, ||*|| <; 1}.

88 GENERAL THEORY
\
This definition of \\A\\ makes &(X, Y) into a normed space. If Y is a Banach space, so is
C,Y).
proof Since subsets of normed spaces are bounded if and only if they lie in
some multiple of the unit ball, ||A|| < oo for every A e 3t(X, Y). If a is a scalar,
then (aA)(x) = a • Ajc, so that
The triangle inequality in Y shows that
IKA, + A2)x\\ = \\A,x + A2x\\ < \\AlX\\ + ||A2*||
for every xe X with \\x\\ < 1. Hence
C) IIAi
If Л Ф 0, then Ax Ф 0 for some xe X; hence ||A|| > 0. Thus Я(Х, Y) is a
normed space.
Assume now that Y is complete -and that {Л„} is a Cauchy sequence in
&(X9Y). Since
D) \\Кх-К*\\<\\К-К\\Ы
and since it is assumed that ||An — Am|| ->0 as n and m tend to oo, {Anx} is a
Cauchy sequence in Yfor every xe X. Hence
E) Ax = lim An x
n-* oo
exists. It is clear that A: Af-> 7is linear. If e > 0, the right side of D) does not
exceed e||x||, provided that m and n are sufficiently large. It follows that
F) ||Лх-Лтх||<вМ
for all large m. Hence ||Ajc|| < (||AJ| + e)||jc||, so that Л е ®{Х9 У), and
|| A - AJ| < e. Thus Am -> Л in the norm of @(X, Y). This establishes the com-
completeness of @(X, Y). ///I
4.2 Duality It will be convenient to designate elements of the dual space Z* of X
by x* and to write
A) <x, x*}
in place of x*(x). This notation is well adapted to the symmetry (or duality) that exists
between the action of X* on X on the one hand and the action of X on X* on the
other. The following theorem states some basic properties of this duality.

DUALITY IN BANACH SPACES 89
4.3 Theorem Suppose В is the closed unit ball of a normed space X. Define
||**||=sup{|<x,**>|:xe*}
for every x* e X*.
(a) This norm makes X* into a Banach space.
(b) Let B* be the closed unit ball ofX*. For every x e X,
\\x\\=suV{\(x,x*):x*eB*}.
Consequently, x* -> <*, x*} is a bounded linear functional on X*, of norm \\x\\.
(c) В* is weak*-compact.
proof Since J*(X, Y) = X*, when Y is the scalar field, (a) is a corollary of
Theorem 4.1.
Fix xe X. The corollary to Theorem 3.3 shows that there exists у* е В*
such that
0) <x,j*> = |WI-
On the other hand,
B) |<x,x*>|<:|WI 11**11 < IWl
for every x* e B*. Part (b) follows from A) and B).
Since the open unit ball Uof Zis dense in B, the definition of \\x*\\ shows
that x* g B* if and only if | (x, x*) | < 1 for every xe U. Part (c) now follows
directly from Theorem 3.15. ////
Remark The weak*-topology of X* is, by definition, the weakest one that
makes all functional
x* -> <x, x*>
continuous. Part (b) shows therefore that the norm topology of X* is stronger
than its weak*-topology; in fact, it is strictly stronger, unless dim X < oo, since
the proposition stated at the end of Section 3.11 holds for the weak*-topology as
well.
Unless the contrary is explicitly stated, X* will from now on denote the
normed dual of X (whenever Xis normed), and all topological concepts relating
to X* will refer to its norm topology. This implies in no way that the weak*-
topology will not play an important role.
We now give an alternative description of the operator norm defined in
Theorem 4.1.

90 GENERAL THEORY
4.4 Theorem If X and Y are normed spaces and if Ae @{X, Y), then
||A||=sup{|<Ax,y*>|:||x||<l, \\y*\\ <1}.
proof Apply (b) of Theorem 4.3 with Fin place of X. This gives
for every xe X. To complete the proof, recall that
4.5 The second dual of a Banach space The normed dual X* of a Banach
space X is itself a Banach space and hence has a normed dual of its own, denoted by
X**. Statement (b) of Theorem 4.3 shows that every x e X defines a unique фх е X**,
by the equation
A) <*,**> = <х*,фх> (х*еХ*),
and that
B) ИФ*И = 11*11" (хеХ).
It follows from A) that ф: X-+ X** is linear; by B), ф is an isometry. Since X is now
assumed to be complete, ф(Х) is closed in X**.
Thus ф is an isometric isomorphism of X onto a closed subspace of X**.
Frequently, Xis identified with ф(Х); then X is regarded as a subspace of X**.
The members of ф{Х) are exactly those linear functional on X* that are con-
continuous relative to its weak*-topology. (See Section 3.14.) Since the norm topology
of X* is stronger, it may happen that ф(Х) is a proper subspace of X**. But there are
many important spaces X (for example, all ZZ-spaces with 1 < p < oo) for which
ф(Х) = X**; these are called reflexive. Some of their properties are given in Exercise 1.
It should be stressed that, in order for X to be reflexive, the existence of some
isometric isomorphism ф of Xonto X** is not enough; it is crucial that the identity A)
be satisfied by ф.
4.6 Annihilators Suppose X is a Banach space, M is a subspace of X, and N is a
subspace of X*; neither M nor N is assumed to be closed. Their annihilators M x and
±N are defined as follows:
M1 - {x* e X*: <x, x*> = 0 for all x e M},
±N^{xeX: (x, x*> = 0 for all x* e N}.
Thus Mx consists of all bounded linear functional on X that vanish on M, and
±N is the subset of X on which every member of N vanishes. It is clear that Mx and ±N

DUALITY IN BANACH SPACES 91
are vector spaces. Since Mx is the intersection of the null spaces of the functional фх>
where x ranges over M (see Section 4.5), M1 is a weak*-closed subspace of X*. The
proof that ±N is a norm-closed subspace of X is even more direct. The following
theorem describes the duality between these two types of annihilators.
4.7 Theorem Under the preceding hypotheses,
(a) 1(M±) is the norm-closure of M in X, and
(b) (^-NI is the'weak*-closure of N in X*.
As regards (a), recall that the norm-closure of M equals its weak closure, by
Theorem 3.12.
proof If x e M, then <x, x*> = 0 for every x* e Mx, so that x e ±(M1). Since
X(MX) is norm-closed, it contains the norm-closure M of M. On the other hand,
if x ф M the Hahn-Banach theorem yields an x* e M1 such that <x, x*> ф 0.
Thus x ф X(MX), and (a) is proved.
Similarly, if x* e N, then <x, x*> = 0 for every x e xjV, so that x* e (ХЛГ)Х.
This weak*-closed subspace of X* contains the weak*-closure N of N. If
'x* ф JV, the Hahn-Banach theorem (applied to the locally convex space X* with
its weak*-topology) implies the existence of an x e ±N such that <x, x*> ^0;
thus x* ф (хЛ0х, which proves (b). - - ////
Observe, as a corollary, that every norm-closed subspace of X is the annihilator
of its annihilator and that the same is true of every weak*-closed subspace of X*.
4.8 Duals of subspaces and of quotient spaces If M is a closed subspace of a
Banach space X, then X/M is also a Banach space, with respect to the quotient norm.
This was defined in the proof of (d) of Theorem 1.41. The duals of M and of X/M can
be described with the aid of the annihilator M1 of M. Somewhat imprecisely, the
result is that
M* - X*/ML and (X/M)* = M1.
This is imprecise because the equalities should be replaced by isometric isomorphisms.
The following theorem describes these explicitly.
4.9 Theorem Let M be a closed subspace of a Banach space X.
(a) The Hahn-Banach theorem extends each m* e M* to a functional x* e X*. Define
от* = x* + M1.
Then о is an L metric isomorphism of M* onto X*/M±.

92 GENERAL THEORY
(b) Let n: X-> X/M be the quotient map. Put 7 = XjM. For each y* e 7*, define
ту* = y*n.
Then т is an isometric isomorphism of 7* onto M1.
proof (a) If x* and x* are extensions of m*, then jc* — jc* is in M1; hence
x* + M1 = x* + M1. Thus <t is well defined. A trivial verification shows that a
is linear. Since the restriction of every jc* g X* to M is a member of M*, the
range of g is all of X*jML.
Fix ra* e M*. If x* g X* extends m*9 it is obvious that \\m*\\ < \\x*\\. The
greatest lower bound of the numbers \\x*\\ so obtained is ||x* + Mx||, by the
definition of the quotient norm. Hence
But Theorem 3.3 furnishes an extension jc* of m* with \\x*\\ — ||^*||. It follows
that ||<x/w*|| = ||/w*||. This completes (a).
(b) If xe X and y* g Y*, then 7cx e 7; hence x->y*nx is a continuous
linear functional on X which vanishes for xe M. Thus ту* е М1. The linearity
of т is obvious. Fix x* e M1. Let N be the null space of x*. Since M с N, there
is a linear functional Л on Fsuch that An = x*. The null space of Л is n(N), a
closed subspace of Y, by the definition of the quotient topology in Y = X/M. By
Theorem 1.18, Л is continuous, that is, Ле 7*. Hence тЛ = Ля = x*. The
range of т is therefore all of M1.
Fix y* g 7*. If у g 7, ||y|| = 1, and r > 1, the definition of the quotient
norm in X/M shows that there is an x0 e X, with ||jco|| < r, such that nx0 = y.
Hence
\<У,У*У\ = 1/*я*0| = |тг%| < ||tj,*|| ||*0|| < г||т^*||,
which implies that
№•11 < 11^*11-
On the other hand, \\nx\\ < \\x\\ for every xe X. Hence
|ty*x| =\y*nx\ < \\y*\\ \\nx\\ < \\y*\\ \\x\\,
which implies that
11^*11 < lb*||.
This completes the proof. ////
Adjoin ts
We shall now associate with each Te ЩХ, 7) its adjoint, an operator Г* е &(Y*9 X*)9
and will see how certain properties of T are reflected in the behavior of T*. If X and 7
are finite-dimensional, every Те &(X, Y) can be represented by a matrix [T]; in that

DUALITY IN BANACH SPACES 93
"case, [T*] is the transpose of [71], provided that the various vector space bases are
properly chosen. No particular attention will be paid to the finite-dimensional case in
what follows, but historically linear algebra did provide the background and much of
the motivation that went into the construction of what is now known as operator
theory.
Many of the nontrivial properties of adjoints depend on the completeness of X
and Y (the open mapping theorem will play an important role). For this reason, it
will be assumed-throughout that J^and У are Banach spaces, except in Theorem 4.10,
which furnishes the definition of T*.
4.10 Theorem Suppose X and Y are normed spaces. To each Te&(X, Y) corre-
corresponds a unique T* e ^(Y*, X*) that satisfies
0) • <Tx,y*} = <x, T*y*)
for all x e X and all y* e Y*. Moreover, T* satisfies
B) ||T*|| = || Щ.
, proof If y* e Y* and ТеЩХ, Y), define
C) T*y* =y*oT,
Being the composition of two continuous linear mappings, T*y* e X*. Also,
<x, T*y*) = (T*y*)(x) = y*(Tx) = <Tx, y*},
which is A). The fact that A) holds for every xe X obviously determines
T*y* uniquely.
Ifjf e Y* and y$eY*, then
= <Tx, уХУ + (Tx, у*2У
= <*, T*yty + <x, T*y*2y
= <x, T*yt + Т*у*2У
for every x e X, so that ^
D) T*(y?+yJ)=T*yf+ T*yJ.
Similarly, Г*(о>>*) = ocT*y*. Thus T*:Y*-> Z* is linear. Finally, (b) of Theo-
Theorem 4.3 leads to

94 GENERAL THEORY
4.11 Notation If Г maps Xinto У, the null space and the range of Г will be deno-
denoted by jV(T) and &(T), respectively:
={j/e Y:Tx = y for some x e X).
The next theorem concerns annihilators; see Section 4.6 for the notation.
4.12 Theorem Suppose X and Y are Banach spaces, andTe^(X, У). Then
JV(T*) = ЩТI and JT(T) = L0t(T*).
proof In each of the following two columns, each statement is obviously equiv-
equivalent to the one that immediately follows and/or precedes it.
У* G Jf(T*
T*y* = 0.
<x, T*y*>
<Tx,y*} =
У* G ^(T)J
Corollaries
:).
=
= 0
0 for all x.
for all x.
X G JT(T).
Гх-0.
<7x,y*> =
<x, T*y*>
X G ±^(T*)
0 for all y*.
= 0 for all y*
////
(a) Ж(Г*) is weak*-closed in У*.
(b) 0t(T) is dense in Y if and only if T* is one-to-one.
(c) T is one-to-one if and only if 0l(T*) is weak*-dense in X*.
Recall that M1 is weak*-closed in У* for every subspace M of Y. In
particular, this is true of &(T)L. Thus (a) follows from the theorem.
As to (b), M(T) is dense in Y if and only if 0t(J)L = {0}; in that case,
Likewise, 1^(T*) = {0} if and only if &(T*) is annihilated by no x e X
other than x = 0; this says that 0t{T*) is weak*-dense in X*.
Note that the Hahn-Banach theorem 3.5 was tacitly used in the proofs of
(b) and (c).
There is a very useful analogue of (b) that allows us to decide, in terms of T*,
whether 0t(T) = Y, that is, whether Г maps X onto Y. This is given in Theorem 4.15
and will be obtained by first looking for conditions on T* which imply that T has
closed range (Theorem 4.14).
4.13 Lemma Suppose U and V are the open unit balls in the Banach spaces X and Y,
respectively. Suppose ТеЩХ, У), and о 0.

DUALITY IN BANACH SPACES 95
(a) If the closure of T(U) contains cV, then T(U) id cV.
(b) Ifc\\y*\\ < ||TV|| for every y* e Y*, then T(U) => cV.
proof (a) Take с = 1, without loss of generality. Then T(U) => V. To every
yeY and every £>0 corresponds therefore an x e X with ||x|| < ||>>|| and
\\y - 7*|| < г.
Pick уле V. Pick sn > 0 so that
Assume n > 1 and ^rt is picked. There exists xn such that ||xj| < \\yn\\ and
\\УП-ТХП\\<8П. Put
By induction, this process defines two sequences {xn} and {yn}. Note that
1|х„+1||<1к+1|| = №„-Гх„||<г„.
Hence
swi< и*, n + 1>„<|ь>1ц + £8„<1.
/i= 1 n= 1 /i= 1
It follows that x = ^xrt is in U (see Exercise 23) and that
N N
Tx = lim X 7X = lim X (л ~ Л+ i) = 3;i
N-*oo /i=l N-*co /i=l
since ^+1 -> 0 as TV -> oo. Thus ^ = Гх e T(l/), which proves (a).
Note that the preceding argument is just a specialized version of part of
the proof of the open mapping theorem 2.11.
(b) Let E be the closure of T(U), pick y0 e У,уоф E. Since E is closed,
convex, and balanced, Theorem 3.7 yields а у* е У* such that
\<У,У*>\<К\<Уо,У*>\
for all у e E. UxeU then 7* e E, so that
It follows that
Ф*\\<\\Т*у*\\<1,
and therefore
i <1<Уо,;>*>|£1Ы111^*11 ^"'Iboll.
or ||JoII > c. Thus cFc£, and (b) follows from (a). ////

96 GENERAL THEORY
4.14 Theorem If X and Y are Banach spaces and if Те @{Х, Y), then each of the
following three conditions implies the other two:
(a) &(T) is closed in Y.
(b) &(T*) is weak*-closed in X*.
(c) 0£{T*) is norm-closed in X*.
Remark Theorem 3.12 implies that (a) holds if and only if M{T) is weakly
closed. However, norm-closed subspaces of X* are not always weak*-closed
(Exercise 7, Chapter 3).
proof It is obvious that (b) implies (c). We will prove that (a) implies (b) and
that (c) implies (a).
Suppose (a) holds. By Theorem 4.12 and (b) of Theorem 4.7, JV(T)L is the
weak*-closure of ^(T*). To prove (b) it is therefore enough to show that
JV(T)L a m{J*\
Pick x* e Jf(J)L. Define a linear functional Л on M(T) by
Л7* = <*,**> (xeX).
Note that Л is well defined, for if Tx = 7У, then x - x' e ^T(T); hence
(x - x\ x*) = 0.
The open mapping theorem applies to
T\X->0liJ)
since M(T) is assumed to be a closed subspace of the complete space Y and is
therefore complete. It follows that there exists К < oo such that to each у e @t(T)
corresponds anxel with Tx = y, \\x\\ < K\\y\\, and
| ЛИ = |ЛГх| = \<х,х*У\£К\\у\\\\х*\\.
Thus Л is continuous. By the Hahn-Banach theorem, some у* е У* extends Л.
Hence.
<7x, у*} = ЛГх = <jc, x*) (x e X).
This implies x* = T*y*. Since x* was an arbitrary element of ^Г(ТI, we have
shown that JV(T)L с 0t(T*). Thus (b) follows from (a).
Suppose next that (c) holds. Let Z be the closure of &(T) in Y. Define
Se@(X,Z) by setting Sx = 7x. Since ^(*S) is dense in Z, Corollary F) to
Theorem 4.12 implies that
S*:Z*-+X*
is one-to-one.

DUALITY IN BANACH SPACES 97
If z* eZ*, the Hahn-Banach theorem furnishes an extension y* of z*; for
every x e X,
<x, TV> = <7x, y*) - <6X z*> = <*, S*z*).
Hence S*z* = T*^*. It follows that 5* and T* have identical ranges. Since (c)
is assumed to hold, &(S*) is closed, hence complete.
Apply the open mapping theorem to
Since S* is one-to-one, the conclusion is that there is a constant с > 0 which
satisfies
ф*|| < ||S*z*||
for every z* e Z*. Hence *S: X-* Z is an open mapping, by (b) of Lemma 4.13.
In particular, S(X) = Z. But M(T) = #(S), by the definition of *S. Thus
ЩТ) = Z, a closed subspace of Г.
This completes the proof that (c) implies (a). ////
The following consequence is useful in applications.
4.15 Theorem Suppose Xand Yare Banach spaces, and Те ЗВ(Х, Y). Then
(a) <%(T)=Y
if and only if
(b) ||T*}>*|| > c\\y*\\ for some constant с > 0 and for every у* е У*.
proof Statement (a) holds if and only if&(T) is dense and closed. By Theorem
4.14 and Corollary (b) of Theorem 4.12, (a) is therefore equivalent to
(с) Т* is one-to-one and &(T*) is norm-closed in X*.
lf(c) holds, the open mapping theorem, applied to Г*: У* -> ^(T*), gives
(b). Conversely, (b) obviously implies that T* is one-to-one; also, the inverse
image (under T*) of any Cauchy sequence is a Cauchy sequence, so that ^(Г*) is
complete and hence closed. ////
Compact Operators
4.16 Definition Suppose J^and Fare Banach spaces, and Uis the open unit ball
in X. An operator Те &(Х, Y) is said to be compact if the closure of T(U) is compact
in Y.
Since Fis a complete metric space, the subsets of У whose closure is compact are
precisely the totally bounded ones. Thus Те &(X, Y) is compact if and only if T(U)

98 GENERAL THEORY
is totally bounded. Also, Tis compact if and only if every bounded sequence {xn} in X
contains a subsequence {х„.} such that {Txn.} converges to a point of Y.
Many of the operators that arise in the study of integral equations are compact.
This accounts for their importance from the standpoint of applications. They are in
some respects as similar to linear operators on finite-dimensional spaces as one has
any right to expect from operators on infinite-dimensional spaces. As we shall see,
these similarities show up particularly strongly in their spectral properties.
4.17 Definitions (a) Suppose I is a Banach space. Then &(X) [which is an
abbreviation for &(X, X)] is not merely a Banach space (see Theorem 4.1) but also an
algebra: If Se &(X) and ТеЩХ), one defines STe ®(X) by
(ST)(x) = S(T(x)) (xeX).
The inequality
iisii < iis
is trivial to verify.
In particular, powers of Те M(X) can be defined: T° = /, the identity mapping on
X, given by Ix = x, and Tn = TT"~\ for n = 1,2, 3, ....
(b) An operator Те ЩХ) is said to be invertible if there exists S e &(X) such that
ST = T= TS.
In this case, we write S = T. By the open mapping theorem, this happens if and only
if Jf(T) = {0} and 0t(T) = X.
(c) The spectrum a(T) of an operator Те 08 (X) is the set of all scalars X such that
T - XI is not invertible. Thus X e cr(T) if and only if at least one of the following two
statements is true:
@ The range of T - XI is not all of X.
(if) T — XI is not one-to-one.
If (ii) holds, X is said to be an eigenvalue of T; the corresponding eigenspace is
J^(T — XI); each x e jV(T — XI) (except x = 0) is an eigenvector of T\ it satisfies the
equation
Tx = Xx.
Here are some very easy facts which will illustrate these concepts.
4.18 Theorem Let X and Y be Banach spaces.
(a) If Те &(X9 Y) and dim M(T) < oo, then T is compact.
(b) If Те &(Х, У), Т is compact, and Ш(Т) is closed, then dim Ш(Т) < oo.
(c) The compact operators form a closed subspace of &(X, 7), in its norm-topology.

DUALITY IN BANACH SPACES 99
(d) If Те &(X), T is compact, and Л Ф 0, then dim jV{T -ll)<oo.
(e)' If dim X= oo, Te&(X), and Г is compact, then 0 6 a(T).
(/) IfSe ЩХ), T e &(X), and T is compact, so are ST and TS.
proof Statement (a) is obvious. If M(T) is closed, then ЩТ) is complete
(since У is complete), so that T is an open mapping of X onto 0t{J)\ if Tis com-
compact, it follows that $(T) is locally compact; thus (b) is a consequence of
Theorem 1.22. '
Put У =yT(T - XI) in (d). The restriction of Г to У is a compact operator
whose range is У. Thus~(//) follows from (b), and so does (e), for if 0 is not in
G(T), then &(T) = X. The proof of (/) is trivial.
If S and T are compact operators from X into У, so is S + T, because the
sum of any two compact subsets of У is compact. It follows that the compact
operators form a subspace X of &(X9 Y). To complete the proof of (c), we now
show that I is closed. Let Те ЩХ, У) be in the closure of I, choose r > 0,
and let U be the open unit ball in X. There exists Se£ with \\S - T\\ < r.
Since S(U) is totally bounded, there are points xl9..., xn in U such that S(U)
is covered by the balls of radius r with centers at the points Sxt. Since
||Sx - Tx\\ < r for every x e U, it follows that T(l/) is covered by the balls of
radius 3r with centers at the points Txt. Thus T(U) is totally bounded, which
proves that T eX. ////
The main objective of the rest of this chapter is to analyze the spectrum of a
compact Те &(Х). Theorem 4.25 contains the principal results. Adjoints will play an
important role in this investigation.
4.19 Theorem Suppose X and Y are Banach spaces and Te^(X, У). Then T is
compact if and only if T* is compact.
proof Suppose T is compact. Let {y*} be a sequence in the unit ball of У*.
Define
Since \fn(y) —fn(y')\ < \\y — /II, {/„} isequicontinuous. SinceT(U)has compact
closure in У (as before, U is the unit ball of X), Ascoli's theorem implies that
{fn} has a subsequence {/,.} that converges uniformly on T(U). Since
\\T*yt - 7V* || = sup | (Tx, yt, - j;*> I
the supremum being taken over x e U, the completeness of X* implies that
{T*y*t} converges. Hence T* is compact.

100 GENERAL THEORY
i
The second half can be proved by the same method, but it may be more
instructive to deduce it from the first half.
Let ф: X -> X** and ф: Y-» Y** be the isometric embeddings given by the
formulas
<x, **> = <**, фх} and <y, y*> = <y*9
as in Section 4.5. Then
<y*, фТх} = <7x, у*> = <*, TV) = <T*y*, </>*> = <
for all x e X and у* е Y*, so that
^Г= Г**ф.
If jc e U, then фх lies in the unit ball I/** of X**. Thus
iAT(l/) с T**(l/**).
Now assume that T* is compact. The first half of the theorem shows that
r**. x** -> Y** is compact. Hence Г**A7**) is totally bounded, and so is its
subset \j/T(U). Since ф is an isometry, T(U) is also totally bounded. Hence Tis
compact. ////
4.20 Definition Suppose M is a closed subspace of a topological vector space X. If
there exists a closed subspace N of X such that
X=M + N and MnN = {0},
then M is said to be complemented in X. In this case, Xis said to be the direct sum of M
and TV, and the notation
is sometimes used.
We shall see examples of uncomplemented subspaces in Chapter 5. At present
we need only the following simple facts.
4.21 Lemma Let M be a closed subspace of a topological vector space X.
(a) If X is locally convex and dim M < oo, then M is complemented in X.
(b) If dim (X/M) < oo, then M is complemented in X.
The dimension of X/M is also called the codimension of M in X.
proof (a) Let {ex, ..., en} be a basis for M. Every x e M has then a unique
representation
x = ccl(x)e1 + •- + an(x)en.

DUALITY IN BANACH SPACES 101
Each oil is & continuous linear functional on M (Theorem 1.21) which extends to a
member of X*, by the Hahn-Banach theorem. Let N be the intersection of the
null spaces of these extensions. Then X — M®N.
(b) Let n: X-+X/M be the quotient map, let {ex,..., e„} be a basis for
X/M, pick xt e X so that nxi = et A < i < n), and let N be the vector space
spanned by {jq, ..., xn}. Then X = M ® N. ////
4.22 Lemma If M is a subspace of a normed space X, ifM is not dense in X, and if
r > 1, then there exists x e X such that
\\x\\ < r but \\x - y\\ > 1 for ally e M.
proof There exists xxe X whose distance from M is 1, that is,
Choose y1 e M such that \\xx — yx \\ < r, and put x = xx — yx. ////
4.23 Theorem If X is a Banach space, Те ЩХ), T is compact, and к Ф 0, then
T — XI has closed range.
proof By (d) of Theorem 4.18, dim JT{T - XI) < oo. By (a) of Lemma 4.21, X
is the direct sum of J^(T — XI) and a closed subspace M. Define an operator
Se ЩМ, Х)Ъу
A) Sx = Tx - Xx.
Then S is one-to-one on M. Also, 0t{S) = &(T - XI). To show that 0t{S) is
closed, it suffices to show the existence of an r > 0 such that
B) • r||jc|| < ||Sjc|| for all xeM.
For if B) holds, and if {Sxn} is a Cauchy sequence, so is {*„}; the completeness of,
M(S) is a consequence.
If B) fails for every r > 0, there exists {xn} in M such that ||*J| = 1, Sxn -> 0,
and (after passage to a subsequence) Txn -> x0 for some x0 e X. (This is where
compactness of T is used.) It follows that Xxn -» x0 . Thus x0 e M, and
Sx0 = lim (XSxn) = 0.
Since S is one-to-one, x0 = 0. But \\xn\\ = 1 for all л, and x0 = lim Ях„, and so
ll*o II = Ul > 0- This contradiction proves B) for some r > 0. ////
4.24 Theorem Suppose Xis a Banach space, Те &(X), Tis compact, r > 0, andE
is a set of eigenvalues X of T such that \X\ > r. Then

102 GENERAL THEORY
(a) for each XeE, M(T - XI) Ф X, and
(b) E is a finite set.
proof We shall first show that if either (a) or (b) is false then there exist closed
subspaces Mn of X and scalars Xne E such that
A) M, с М2 с М3 с • • •, М„ # Мв + 1,
B) T(Mn)aMn for«>l,
and
C) (Т-Хп1){Мп)^Мп_, for я > 2.
The proof will be completed by showing that this contradicts the com-
compactness of T.
Suppose (a) is false. Then ЩТ - Xo I) = X for some Ao e E. Put S =
T — 1OI, and define Mn to be the null space of Sn. (See Section 4.17.) Since Ao is
an eigenvalue of T, there exists xi e Ml9 xx # 0. Since ^(S) = X, there is a se-
sequence {xj in X such that 5xn + 1 = xn, л = 1, 2, 3, Then
D) S"*^ !=*!#<) but SB+4 + i=^1=0.
Hence Mn is a proper closed subspace of Mn + l. It follows that A) to C) hold,
with ln = l0. [Note that B) holds because ST = TS.]
Suppose (b) is false. Then E contains a sequence {Я„} of distinct eigenvalues
of T. Choose corresponding eigenvectors en, and let Mn be the (finite-dimen-
(finite-dimensional, hence closed) subspace of X spanned by {ex,..., en}. Since the ln are
distinct, {el9..., en} is a linearly independent set, so that Mn_! is a proper sub-
subspace of Mn. This gives A). If x e Mn, then
x = cclel + •■• + anen,
which shows that Гл: е Mn and
(Г- Яи/)лг = a^ - Xn)e, + • • • + *n_,(Xn_, - Xn)en_, eMn^.
Thus B) and C) hold.
Once we have closed subspaces Mn satisfying A) to C), Lemma 4.22 gives
us vectors yn e Mn, for n = 2, 3, 4, ..., such that
E) 1Ы1<2 and ||л-*И>1 if xeMn-t.
If 2 < m < и, define
F) z = 7>,,,-(r-;.„,?)>"„.
By B) and C), zeMn_l. Hence E) shows that
Ул - TyJ = \\Х„у„ - z\\ = \Xn\ \\у„ - X;'z\\ > K| > r.

DUALITY IN BANACH SPACES 103
The sequence {Tyn} has therefore no convergent subsequences, although {yn} is
bounded. This is impossible if Г is compact. ////
4.25 Theorem Suppose X is a Banach space, Те &(X), and T is compact,
{a) If ХФ0, then the four numbers
a = dim Jf(T-XI)
a* = dim Jf(T* - XI)
P* = dim X*I®(T* - XI)
are equal and finite.
(b) If X Ф 0 and Xe o(T) then X is an eigenvalue of T and of T*.
(c) o(T) is compact, at most countable, and has at most one limit point, namely, 0.
Note: The dimension of a vector space is here understood to be either a non-
negative integer or the symbol oo. The letter / is used for the identity operators on
both X and X*; thus
(T - XI)* = T* - U* = T* - XI,
since the adjoint of the identity on X is the identity on X*.
The spectrum cf(T) of T was defined in Section 4.17. Theorem 4.24 contains a
special case of (a): fi = 0 implies a = 0. This will be used in the proof of the inequal-
inequality D) below.
It should be noted that a(T) is compact even if T is not (Theorem 10.13). The
compactness of T is needed for the other assertions in (c).
proof Put S = T - XI, to simplify the writing.
We begin with an elementary observation about quotient spaces. Suppose
Mo is a closed subspace of a locally convex space Y, and A: is a positive integer
such that к < dim YfM0. Then there are vectors yt, ..., yk in Y such that the
vector space Mt generated by Mo and yu ..., yc contains M,_! as a proper sub-
space. By Theorem 1.42, each. Mt is closed. By Theorem 3.5, there are con-
continuous linear functionals Al9..., Ak on У such that Л,^ = 1 but Aty = 0 for all
у е Mi__i. These functionals are linearly independent. The following conclusion
is therefore reached: ■ if £ denotes the space of all continuous linear functionals
on Y that annihilate Mo, then
A) dim Y/Mo < dim 2.

104 GENERAL THEORY , }
Apply this with Y = X, MQ = ЩБ). By Theorem 4.23, &(S) is closed.
Also, S - ^(SI = ^(S*), by Theorem 4.12, so that A) becomes
B) p<a*.
Next, take Y = X* with its weak*-topology; take Mo = ^E*). By
Theorem 4.14, ^(S*) is weak*-closed. Since X now consists of all weak*-
continuous linear functional on X* that annihilate ^E*), L is isomorphic to
±^E*) = ^E) (Theorem 4.12), and A) becomes
C) P* < oc. S
Our next objective is to prove that
D) a < p.
Once we have D), the inequality
E) a* < (I*
is also true, since T* is a compact operator (Theorem 4.19). Since a < со by
(d) of Theorem 4.18, (a) is an obvious consequence of the inequalities B) to E).
Assume that D) is false. Then a > p. Since a < oo, Lemma 4.21 shows that
X contains closed subspaces E and f such that dim F = ft and
F) X = jr(S) ®E = ®(S) 0 F.
Every x e X has a unique representation x = xx + x2, with x\ e jV(S), x2 e -E.
Define тг: X-> ^KE) by setting жх = jcx. It is easy to see (by the closed graph
theorem, for instance) that тг is continuous.
Since we assume that dim jV(S) > dim F, there is a linear mapping ф of
jV{S) onto i7 such that фх0 = 0 for some jc0 t^ 0. Define
G) Фх = Tx + фтгх (x e Z).
Then Ф e ^(JT). Since dim &(ф) < со, фтг is a compact operator; hence so is Ф
(Theorem 4.18).
Observe that
(8) Ф-Я/
Since л:0 e ^K(iS), ях0 = x0, and so ^тгх0 = 0. It follows that X is an eigenvalue
of Ф (with eigenvector jc0). Hence
(9) #(Ф - U) Ф X,
by Theorem 4.24.
Since nx — 0 for every xe E, (8) shows that
A0) (Ф - XI){E) = 5(^) = S(X) =

DUALITY IN BANACH SPACES 105
If x e ^V(S), then nx = x, and (8) gives
(Ф - U)(JT(S)) = ^(^E)) = F.
It follows from A0) and A1) that
A2) m(® - XI)
The contradiction between (9) and A2) shows that D) is true. This completes the
proof of (a).
Part (b) follows from (a), for if X is not an eigenvalue of T, then a(T) = 0,
and (a) implies that p(T) = 0, that is, that &(T-M)=X. Thus T - XI is
invertible, so that X ф о(Т).
It now follows from (b) of Theorem 4.24 that 0 is the only possible limit
point of <j(T), that o(T) is at most countable, and that o(T) u {0} is compact.
If dimJf<oo, then <j(T) is finite; if dim X = сю, then Oea(T), by (e) of
Theorem 4.18. Thus cf(T) is compact. This gives (c) and completes the proof of
the theorem. ////
Exercises
Throughout this set of exercises, X and Y denote Banach spaces, unless the contrary is
explicitly stated.
1 Let ф be the embedding of X into X** described in Section 4.5. Let r be the weak
topology of X, and let a be the weak*-topology of X**—the one induced by X*.
(a) Prove that <f> is a homeomorphism of (X, т) onto a dense subspace of (X**, a).
(b) If В is the closed unit ball of X, prove that ф(В) is o--dense in the closed unit ball
of X**. (Use the Hahn-Banach separation theorem.)
(c) Use (a), (b), and the Banach-Alaoglu theorem to prove that X is reflexive if and
only if В is weakly compact.
(d) Deduce from (c) that every norm-closed subspace of a reflexive space X is reflexive.
(e) If X is reflexive and Y is a closed subspace of X, prove that X/ Y is reflexive.
(/) Prove that X is reflexive if and only if X* is reflexive.
Suggestion: One half follows from (c); for the other half, apply (d) to the subspace
ф(Х) of X**.
2 Which of the spaces c0, t1, Л,у°° are reflexive? Prove that every finite-dimensional
normed space is reflexive. Prove that C, the supremum-normed space of all complex
continuous functions, on the unit interval, is not reflexive.
3 Prove that a subset E of &(X, Y) is equicontinuous if and only if there exists M < со
such that ||Л|| < M for every AeE.
4 Recall that X* = &(Xy d\ if <£ is the scalar field. Hence Л* е 0Ц&, X*) for every
Л g X*. Identify the range of Л*.
5 Prove that Te ЩХ, Y) is an isometry of X onto Y if and only if T* is an isometry of
Y* onto X*.

106 GENERAL THEORY
6 Let cr and r be the weak*-topologies of X* and Y*, respectively, and prove that £ is a
continuous linear mapping of (Y*,r) into (X*, a) if and only if S = T* for some
Те &(Х, Y).
7 Let L1 be the usual space of integrable functions on the closed unit interval /, relative
to Lebesgue measure. Suppose ТеЩЬ1, Y), so that T* e &(Y*,L00). Suppose
&(T*) contains every continuous function on /. What can you deduce about 77
8 Prove that (ST)* = T*S*. Supply the hypotheses under which this makes sense.
9 Suppose S e &{X\ T e &(X).
(a) Show, by an example, that ST= I does not imply TS = /.
(b) However, assume T is compact, show that
S(I - T) = / if and only if (/ - T)S = I,
and show that either of these equalities implies that I — (I — T)'1 is compact.
10 Assume Te&(X) is compact, and assume either that dim J=oo or that the scalar
field is <?. Prove that a(T) is not empty. However, <j(T) may be empty if dim X < со
and the scalar field is R.
11 Suppose dim X < со and show that the equality a = /8 of Theorem 4.25 reduces to the
statement that the row rank of a square matrix is equal to its column rank.
12 Suppose Те &(Х, Y) and M(T) is closed in Y. Prove that
dim Jf(T) = dim X*I&(T*\
dim JT{T*) = dim Yffi{T).
This generalizes the assertions a = /?* and a* = /8 of Theorem 4.25.
13 (a) Suppose Те &(X, Y\ Tn e &(X, Y) for n =1, 2, 3, ..., each Tn has finite-dimen-
finite-dimensional range, and lim \\T — Tn\\ = 0. Prove that Tis compact.
(b) Assume Y is a Hilbert space, and prove the converse of (a): Every compact
Те &(Х, Y) can be approximated in the operator norm by operators with finite-
dimensional ranges. Hint: In a Hilbert space there are linear projections of norm 1
onto any closed subspace. (See Theorems 5.16, 12.4.)
14 Define a shift operator S and a multiplication operator M on £2 by
if /2=0,
(Mx)(n) = (n + l)-1^/*) if n > 0.
Put T = MS. Show that Г is a compact operator which has no eigenvalue and whose
spectrum consists of exactly one point. Compute \\Tn ||, for n = 1, 2, 3,..., and compute
lim^oo НПГ.
15 Suppose /л is a finite (or cr-finite) positive measure on a measure space О, /л х ju, is the
corresponding product measure on Q x fi, and КеЬ2(/л х /л). Define
(Tf)(s) = f

DUALITY IN BANACH SPACES 107
(a) Prove that Те @(L2({i)) and that
\\T\\2 < jj \K(s,t)\2 dfi(s) dfi(t).
(b) Suppose ai, bt are members of L2({jl), for 1 <i <n, put Ki(s, t) = 2 cii{s)bi{t\ and
define 7\ in terms of Ki as Г was denned in terms of K. Prove that dim M(Ti) < n.
(c) Deduce that Г is a compact operator on Ь2((л). Hint: Use Exercise 13.
(d) Suppose A e &', А Ф 0. Prove: Either the equation
Tf-Xf=g
has a unique solution feL2(jj) for every geL\\£) or there are infinitely many
solutions for some# and none for others. (This is known as the Fredholm alternative.)
0) Describe the adjoint of T
16 Define
and define Те &(L2@, 1)) by
GШ = f K(s, /)/(/) Л @ < j < 1).
(a) Show that the eigenvalues of Гаге (птг)-2, п = 1, 2, 3, ..., that the corresponding
eigenfunctions are sin nirx, and that each eigenspace is one-dimensional. Hint: If
Л Ф 0, the equation 77= A/implies that/is infinitely differentiable, that A/" H-/= 0,
and that /@) =/A) = 0. The case A = 0 can be treated separately.
(b) Show that the above eigenfunctions form an orthogonal basis for L2@, 1).
(c) Suppose g(t) = ^cn sin rnrt. Discuss the equation Tf— Xf=g.
(d) Show that T is also a compact operator on C, the space of all continuous functions
on [0, 1]. Hint: If {fi} is uniformly bounded, then {Tft} is equicontinuous.
17 If L2 =L2@, oo) relative to Lebesgue measure, and if
= ~ \f(t)dt
s j0
prove that Те &(L2) and that Г is not compact. (The fact that ||Г|| < 2 is a special case
of Hardy's inequality. See p. 72 of [23].)
18 Prove the following statements.
(a) If {xn} is a weakly convergent sequence in X, then {||лг„||} is bounded.
(b) If Те &(X, Y) and xn -»* weakly, then Txn -> Tx weakly.
(c) If Те @{X, Y\ \fxn-+x weakly, and if Г is compact, then \\Txn - Tx\\ ->0.
(^) Conversely, if X is reflexive, if Te&(X, Y\ and if ||Гхп - Гх||->0 whenever
лг„ -> x weakly, then T is compact. Hint: Use (c) of Exercise 1, and part (c) of Exercise
28 in Chapter 3.
(e) If Xis reflexive and Ге #(ЛГ, ^0» then ^is compact. Hence 01{T) ф t1. Hint: Use
(c) of Exercise 5 of Chapter 3.
(/) If У is reflexive and Te$?(c0, Y\ then Г is compact.

108 GENERAL THEORY
19 Suppose Y is a closed subspace of Xy and x% e X*. Put
jc8>|:xey, IMI < 1},
In other words, fi is the norm of the restriction of xt to У, and 8 is the distance from
xt to the annihilator of Y. Prove that ^ = 8. Prove also that S = ||jc* — xo II for at
least one x* e Y1.
20 Extend Sections 4.6 to 4.9 to locally convex spaces. (The word " isometric" must of
course be deleted from the statement of Theorem 4.9.)
21 Let В and B* be the closed unit balls in X and X*, respectively. The following is a
converse of the Banach-Alaoglu theorem: IfE is a convex set in X* such that E n (rB*)
is weak*-compact for every r > 0, then E is weak*-closed. (Corollary: A subspace of X*
is weak*-closed if and only if its intersection with B* is weak*-compact.)
Complete the following outline of the proof.
(i) E is norm-closed.
(ii) Associated to each fc X its polar
The intersection of all sets P(F), as F ranges over the collection of all finite subsets
of r~iB, is exactly rB*.
(ш) The theorem is a consequence of the following proposition: If in addition to the
stated hypotheses, E n B* = 0, then there exists x e X such that Re <jc, x*> > 1 for
every jc* e E.
(iv) Proof of the proposition: Put Fo = {0}. Assume finite sets Fo, ..., Fk-i have been
chosen so that iFt с В and so that
A) P(F0) П-- n
Note that A) is true for к = 1. Put
^о) n • • • n P(Ffc-0 nEn(k+ 1)B*.
If P(F)n Q ф 0 for every finite set F^k'^, the weak*-compactness of Q,
together with (ii), implies that (kB*) n Q Ф 0, which contradicts A). Hence there
is a finite set Fk с &- 4i? such that A) holds with A: + 1 in place of k. The construction
can thus proceed. It yields
B) Enf] P(Fk) = 0.
Arrange the members of (J Fk in a sequence {*„}. Then ||xn II -> 0. Define Г: X* -> c0
by
Then 7Х£) is a convex subset of c0. By B),
||!T**||= sup |<xn

DUALITY IN BANACH SPACES 109
for every x* e E. Hence there is a scalar sequence {<*„}, with 2 I a« I < °°> sucn tnat
Re | «„<*,,, **>>1
for every x* e E. To complete the proof, put x = ^anxn.
22 Suppose Те ЩХ), Г is compact, А Ф 0, and S = T— XL
(a) If jV(Sn) =±Jf(Sn + l) for some nonnegative integer n, prove that jV(Sn) =
for A: = 1,2,3,....
(Z>) Prove that {a) must happen for some n. (Hint: Consider the proof of Theorem 4.24.)
(c) Let n be the smallest nonnegative integer for which (a) holds. Prove that dim jV(S")
is finite, that
x =jr(sn) © m(sn\
and that the restriction of S to M(Sn) is a one-to-one mapping of 0l{Sn) onto
Suppose {xn} is a sequence in a Banach space X, and
2 ||jcJ|
n = l
Prove that the series 2 *n converges to some x e X. Explicitly, prove that
lim ||*-(*1 + --- + *и)||=0.
я-» oo
Prove also that ||jc || < M. (These facts were used in the proof of Lemma 4.13.)
24 Let с be the space of all complex sequences
for which JCoo = lim xn exists (in C). Put ||jc|| = sup \xn\. Let c0 be the subspace of с that
consists of all x with Xoo =0.
(a) Describe explicitly two isometric isomorphisms и and v, such that и maps c* onto ix
and v maps c* onto Л. ■
(b) Define S: co->cby Sf=f. Describe the operator vS*u ~x that maps Л to Л.
(c) Define T: c->c0 by setting
Prove that Г is one-to-one-and that Tc = c0. Find ЦГЦ and ИГ!!- Describe the
operator uT*v -1 that maps Л to Л.

SOME APPLICATIONS
A Continuity Theorem
One of the very early theorems in functional analysis (Hellinger and Toeplitz, 1910)
states that ifT is a linear operator on a Hilbert space H which is symmetric in the sense
that
GX У) = (x, Ту)
for all xe H and у eH, then T is continuous. Here (x, y) denotes the usual Hilbert
space inner product. (See Section 12.1.)
If {xn} is a sequence in H such that ||jcj ->0, the symmetry of T implies that
Txn -> 0 weakly. (This depends on knowing that all continuous linear functionals од
H are given by inner products.) The Hellinger-Toeplitz theorem is therefore a con-
consequence of the following one.
5.1 Theorem Suppose X and Y are F-spaces?, 7* separates points on Y,T: X-+ Y
is linear, and ATxn ->0for every Л e 7* whenever xn -> 0. Then T is continuous.

SOME APPLICATIONS 111
proof Suppose xn -» x and Txn -+y. If Л e Y*, then
AT(xn-x)->0
so that
Ay = lim ATxn = ATx.
Consequently, у = Tx, and the closed graph theorem can be applied. ////
In the context of Banach spaces, Theorem 5.1 can be stated as follows: If
T: X-+ Y is linear, if \\xn\\ ->0 implies that Txn-+O weakly, then \\xn\\ ->0 actually
implies that \\Txn\\ -> 0.
To see that completeness is important here, let X be the vector space of all
complex polynomials /such that/(O) =/A) = 0, put
% 11/11 = (f,J
and define T: X-> X by (Tf)(x) = if\x). Then (Tf g) = (/ Tg), but T is not contin-
continuous.
Closed Subspaces of Lp-spaces
The proof of the following theorem of Grothendieck also involves the closed graph
theorem.
5.2 Theorem Suppose 0 < p < oo, and
{a) fi is a probability measure on a measure space Q.
(b) S is a closed sub space of LF{\i).
(c) ScL^Qi).
Then S is finite-dimensional.
proof Let / be the identity map that takes S into L00, where S is given the
//-topology, so that S is complete. If {/,} is a sequence in S such that/, -»/in S
and/ -> g in L00, it is obvious that/= # a.e. Hence/ satisfies the hypotheses of
the closed graph theorem, and we conclude that there is a constant К < oo such
that
О) ' 11/11» < K\\f\\p
for all feS. As usual, ||/||p means (J |/|рфI/р, and Ц/Ц^ is the essential
supremum of |/|. If p < 2 then ||/||p < ||/||2 . If 2 < p < oo, integration of
the inequality

112 GENERAL THEORY
]
leads to Ц/Ц^ < Kpl2\\f\\2. In either case, we have a constant M < oo such that
B) \\f\\o><M\\f\\2 C/e5).
In the rest of the proof we shall deal with individual functions, not with
equivalence classes modulo null sets.
Let {фи ..., фп} be an orthonormal set in S, regarded as a subspace of L2.
Let Q be a countable dense subset of the euclidean unit ball В of C". If с =
(q,..., О e Д define/, = £ ^. Then ||/J2 < 1, and so Ш„ < M. Since
Q is countable, there is a set Q' с П, with ^(£У) = 1, such that | fc(x) \ < M for
every cg<2 and f°r еиегу xe€l'. If x is fixed, c-+\fc(x)\ is a continuous
function on B. Hence |/c(x) | < M whenever с е В and x e Ur. It follows that
Z I Ф£х)\2 ^ ^2 f°r every x e Q.'. Integration of this inequality gives n < M2.
We conclude that dim S < M2. This proves the theorem. ////
It is crucial in this theorem thatL00 occurs in the hypothesis (c). To illustrate this
we will now construct an infinite-dimensional closed subspace of Ll which lies in L4.
For our probability measure we take Lebesgue measure on the unit circle, divided by
2tc.
5.3 Theorem Let E be an infinite set of integers such that no integer has more than
one representation as a sum of two members ofE. Let PE be the vector space of all finite
sums f of the form
A) f(eie)= f c{ri)eM
{)e
—oo
in which c(n) = 0 whenever n is not in E. Let SE be the Ll-closure ofPE. Then SE is a
closed subspace ofL4.
An example of such a set is furnished by 2k, к = 1, 2, 3, ... . Much slower growth
can also be achieved.
proof If/is as in A), then
f\eie) = X с(пJе2Ш + £ c(n)c(m)eiin+m)e.
n пФт
Our combinatorial hypothesis about E implies that
so that
B)

SOME APPLICATIONS 113
Holder's inequality, with 3 and § as conjugate exponents, gives
О)
It follows from B) and C) that
D)" ,,J/ll4<21/4||/||2 and \\f\\2<2ll2\\f\\i
for every fe PE. Every L1-Cauchy sequence in PE is therefore also a Cauchy
sequence inL4. Hence SE с Z,4. The obvious inequality \\f\\t < ||/||4 then shows
that SE is closed in L4. ////
An interesting result can be obtained by applying a duality argument to the sec-
second inequality D). Recall that the Fourier coefficients g(n) of every g eU° satisfy
Yj \д(п)\2 < °°- The next theorem shows that nothing more can be said about the
restriction of § to E.
5.4 Theorem If E is as in Theorem 5.3 and if
£ \a(n)\2= A2 < oo
— oo
then there exists g e U° such that g(n) = a(n)for every ne E.
proof If fe PE, the preceding proof shows that
\Zf(n)a(n)\ <А&\?(п)\2}1/2 =A\\f\\2 <2'l2A\\f\\v
Hence/-> Y Kn)aip) is a linear functional on PE which is continuous relative to
the Z^-norm. By the Hahn-Banach theorem, this functional has a continuous
linear extension to L1. Hence there exists g eL°° (with \\g\\n < 2l/2A) such that
t Kn)a{n) = ~ f f(e-ie)g(ew) dO (fe PE).
-oo Z7tJ-n
With f(eie) = еш (n e E\ this shows that g(n) = a{n). ////
к*
The Range of a Vector-valued Measure
We now give a rather striking application of the theorems of Krein-Milman and
Banach-Alaoglu.
Let 5Ш be a c-algebra. A real-valued measure X on 5Ш is said to be nonatomic if
every set £ e 5Ш with | Я | (E) > 0 contains a set Л e50lwith0< |Л|С4)< Щ(Е). Here
| A| denotes the total variation measure of k\ the terminology is as in [23].

114 GENERAL THEORY
5.5 Theorem Suppose fil9..., jin are real-valuednonatomic measures on a o-algebra
Ш. Define
, (Ееm).
Then ji is a function with domain Ш whose range is a compact convex subset of Rn.
proof Associate to each bounded measurable real function g the vector
inR". Put о = \fit\ + ••• + \jin\. If gt= g2a.e. [a],thenAgl=Ag2. Hence
Л may be regarded as a linear mapping of L°°(cr) into R".
Each Hi is absolutely continuous with respect to a. The Radon-Nikodym
theorem [23] shows therefore that there are functions hieL1{a) such that dfxt =
ht da A < i < n). Hence Л is a weak*-continuous linear mapping of L°°(cr) into
Rn; recall that L»(g) = L\a)*. Put
K = {geL™(a):0<g<l}.
It is obvious that К is convex. Since g e К if and only if
0 < jfg da < jfda
for every nonnegative/eL1^), AT is weak*-closed. And since A" lies in the closed
unit ball of LQO(cr), the Banach-Alaoglu theorem shows that ^Cis weak*-compact.
Hence A(K) is a compact convex set in R".
We shall prove that /*(9Л) - A(K).
If Xe is tne characteristic function of a set E e Ш, then Xe e К and V>(E) =
A^f. Thus 1л(Ш) cz A(K). To obtain the opposite inclusion, pick a point p e A(K)
and define
Kp = {geK:Ag=p}.
We have to show that Kp contains some /E, for then p = ц(Е).
Note that iTp is convex; since A is continuous, Kp is weak*-compact. By
the Krein-Milman theorem, Kp has an extreme point.
Suppose g0 e Kp and g0 is not a characteristic function in L°°(cr). Then
there is a set E e Ш and an r > 0 such that a(E) > 0 and r < go< I — r on E.
Put 7 = /£ • £°°((т). Since c^Zs) > 0 and a is nonatomic, dim Y > n. Hence there
exists g e Y, not the zero element of L°°(cr), such that A# = 0, and such that
— r<g<r. It follows that g0 + g and g0 — g are in Kp. Thus g0 is not an
extreme point of Kp.
Every extreme point of Kp is therefore a characteristic function. This
completes the'proof. ////

SOME APPLICATIONS 115
A Generalized Stone-Weierstrass Theorem
The theorems of Krein-Milman, Hahn-Banach, and Banach-Alaoglu will now be
applied to an approximation problem.
5.6 Definitions Let C(S) be the familiar sup-normed Banach space of all contin-
continuous complex functions on the compact HausdorfT space S. A subspace A of C(S) is
an algebra if fg e A- whenever fe A and g e A. A set E с S is said to be A-antisym-
metric if every/e A which is real on £is constant on E; in other words, the algebra AE
which consists of the restrictions /1E of the functions fe A to E contains no non-
constant real functions.
For example, if S is a compact set in С and if A consists of all/e C(S) that are
holomorphic in the interior of S, then every component of the interior of Sis A-antisym-
metric.
Suppose А с C(S\ pe S,q e S, and write p ~ q provided that there is an ^-anti-
^-antisymmetric set E which contains both p and q. It is easily verified that this defines an
equivalence relation in S and that each equivalence class is a closed set. These equiv-
equivalence classes are the maximal >4-antisymmetric sets.
5.7 \ Bishop's theorem Let A be a closed subalgebra of C(S) which contains the
constant functions. Suppose g e C(S) and g\Ee AEfor every maximal A-antisymmetric
set E. Then g e A.
Stated differently, the hypothesis on g is that to every maximal >4-antisymmetric
set E corresponds a function/e A which coincides with g on E; the conclusion is that
one/exists which does this for every E, namely,/ = g.
A special case of Bishop's theorem is the Stone-Weierstrass theorem:
If A is a closed subalgebra of C(S) which contains the constants, which separates
points on S, and which is self-adjoint {that is, fe A whenever fe A), then A — C(S). x
For in this case the real-valued members of A separate points on S. Since no
^-antisymmetric set contains therefore more than one point, every g e C(S) satisfies
the hypothesis of Bishop's theorem.
proof The annihilator A1 of A consists of all regular complex Borel measures
ц on S such that \fd\i = 0 for every/e A. Define
where ||/i|| = |^|E). Then ^Tis convex, balanced, and weak*-compact, by (c) of
Theorem 4.3. If К = {0}, then A1 = {0}; hence A = C(S), and there is nothing
to prove.

116 GENERAL THEORY
i •
i )
Assume К Ф {0}, and let \i be an extreme point of К Clearly, \\x\ = 1. Let
ЕЫ the support of ц; this means that Eis compact, that \ji\(E) = \\ji\\, and that E
is the smallest set with these two properties. Consider an/e A such that 0 <
f(x) < 1 for every x e E, and define
Since Л is an algebra, cr e A1 and т e AL. Since 0 </< 1 on E, \\a\\ > 0 and
||т|| > 0. Also,
Ml + IMI = f fd\n\ + f A -f)d\ii\ = \fi\{E) = 1.
This shows that \x is a convex combination of the measures c^ = <7/||cr|| and
Ti — T/IIхII- Both of these are in K. Since \x is extreme in /£, ц = dj^. In other
words, /ф = ||cr|| d[i9 so that/(;c) = ||cr|| for every x e E. Since A contains the
constants, it follows that every/e A which is real on E is constant on E.
So far we have proved that the support of \i is A-antisymmetric if \x is an
extreme point of K.
If g satisfies the hypothesis of the theorem, it follows that j g d\x — 0 for
every \x that is extreme in K, hence for every \x in the convex hull of these extreme
points. Since \x -»j g dp, is a weak*-continuous function on K, the Krein-
Milman theorem implies that J g d\i = 0 for every ji e K, hence for every fie A1.
Thus every continuous linear functional on C(S) that annihilates A also
annihilates g. Hence g e A, by the Hahn-Banach separation theorem. ////
Here is an example that illustrates Bishop's theorem:
5.8 Theorem Suppose
(a) К is a compact subset ofRnx€ and
(b) ift = (tl,...,tn)eR",theset
Kt={zeC:(t,z)eK}
does not separate C. If g e C(K), define gt on Kt by gt(z) = g(t, z).
Assume that g e C(K), that each gt is holomorphic in the interior of Kt and that
e > 0. Then there is a polynomial P in the variables tu ...,tn, z such that
\P(t, z) - g(t9 z)\ < 8
for every (t, z) e K.
proof Let A be the closure irsC(K) of the set of all polynomials P(t, z). Since
the real polynomials on Rn separate points, every Л-antisymmetric set lies in

SOME APPLICATIONS 117
some Kt.- By Theorem 5.7 it is therefore enough to show that to every t e Rn
corresponds an/e A such that/, = gt.
Fix / e Rn. By Mergelyan's theorem [23] there are polynomials Pt(z) such
that
and \Pt\ < 2~l if i > 1. There is a polynomial g on Rn that peaks at t, in the
sense that Q(t) = 1 but | g(»| < 1 if s # Г and £TS # 0. Consider a fixed i > 1.
The functions фт defined on AT by
form a monotonically decreasing sequence of continuous functions whose limit
is < 2 "' at every point of K. Since К is compact, it follows that there is a positive
integer mt such that фт.(я, z) <2~l aX every point of K. The series
I(s,z)=fjQmis)Pi(z)
.converges uniformly on K. Hence/e A, and obviously/ = gt. ////
Two Interpolation Theorems
The proof of the first of these theorems involves the adjoint of an operator. The second
furnishes another application of the Krein-Milman theorem.
The first one (due to Bishop) again concerns C(S). Our notation is as in Theo-
Theorem 5.7.
5.9 Theorem Suppose Y is a closed subspace ofC(S), К is a compact subset of S,
and | ji | (K) = Ofor every це Y1. If g e C(K) and \g\ < 1, it follows that there exists
fe Ysuch thatf\K = g and \f\ < 1 on S.
Thus every continuous function on A" extends to a member of Y. In other words,
the restriction map/-»/|K maps Y onto C(K).
This theorem generalizes the following special case.
Let A be the disc algebra, i.e., the set of all continuous functions on the closure
of the unit disc U in € which are holomorphic in U. Take S = T, the unit circle. Let
Y consist of the restrictions to T of the members of A. By the maximum modulus
theorem, Y is a closed subspace of C(T). If К с Т is compact and has Lebesgue
measure 0, the theorem of F. and M. Riesz [23] states precisely that К satisfies the
hypothesis of Theorem 5.9. Consequently, to every g e C(K) corresponds? an fe A such
thatf= g on K.

118 GENERAL THEORY
)
proof Let p: F-> C(K) be the restriction map defined by pf=f\K. We have
to prove that p maps the open unit ball of Y onto the open unit ball of C(K).
Consider the adjoint p*: M(K)-* Y*, where M(K) = C(X)* is the Banach
space of all regular complex Borel measures on K, with the total variation norm
\\fi\\ = \n\(K). For each цеМ(К), p*\i is a bounded linear functional on Y;
by the Hahn-Banach theorem, р*/л extends to a linear functional on C(S), of the
same norm. In other words, there exists a e M(S), with \\a\\ = ||p*/j||, such that
for every /e Y. Regard \i as a member of M(S), with support in K. Then
(Т-/16Г1, and our hypothesis about К implies that a(E) = /г(£) for every
Borel set E a K. Hence \\fi\\ < \\a\\. We conclude that |И < ||p*/x||. By F) of
Lemma 4.13, this inequality proves the theorem. ////
Note: Since ||p*|| = ||p|| < 1, we also have ||cr|| < \\fi\\ in the preceding proof. It
follows that и — \i. Hence р*ц has a unique norm-preserving extension to C(S).
Our second interpolation theorem concerns finite Blaschkeproducts, i.e., functions
В of the form
where |c| = 1 and |ak| < 1 for 1 < k < N. It is easy to see that the finite Blaschke
products are precisely those members of the disc algebra whose absolute value is 1 at
every point of the unit circle.
The data of the Pick-Nevanlinna interpolation problem are two finite sets of
complex numbers, {z0, ..., zj and {w0,..., wn}, all of absolute value less than 1, with
zt Ф Zj if i Ф j. The problem is to find a holomorphic function/in the open unit disc £/,
such that | f(z) \ < 1 for all z e U, and such that
f(zt) = wt @ < i < n).
The data may very well admit no solution. For example, if {z0, zx} = {0, i} and
{w0, wj = {0, |}, the Schwarz lemma shows this. But if the problem has solutions,
then among them there must be some very nice ones. The next theorem shows this.^
5.10 Theorem Let {z0, ..., zn}, {w0, ..., wn} be Pick-Nevanlinna data. Let E be
theset of all holomorphic functions fin U such that \f\ < 1 andf(zt) = WiforO <i <n.
If E is not empty, then E contains a finite Blaschke product.

SOME APPLICATIONS 119
proof Without loss of generality, assume z0 = w0 = 0. We will show that there
is a holomorphic function F in U which satisfies
A) Re F(z) > 0 for z e U, F@) = 1,
/04 J7( 'r \ R far 1 <f i <Г и
\~) •* \ i) — г i — lUl 1 _^; t _i ft,
and which has,the form
where ck > 0, £ ck = 1, and \ak\ = 1. Once such an F is found, put В =
(F- l)/(F+ 1). This is a finite Blaschke product that satisfies B(zt) = wt for
0 < i < n.
Let К be the set of all holomorphic functions F in U that satisfy A).
Associate to each \x e M(T) = C(T)* the function
D) ЗД= Г Ji^Me") B 6 17).
If P is the set of all BotqI probability measures on T, then /г<->/^ is a one-to-one
correspondence between P and AT. (Theorems 11.12 and 11.19 of [23].) Define
А:М(Т)->СпЪу
E) . л/i = (ад,..., ад).
Since £ is assumed to be nonempty, there exists ix0 e P such that
F) Л/10 =/? = (ft,..., A).
Since P is convex and weak*-compact, and since Л is linear and weak ^continu-
^continuous, A(P) is a convex compact set in <£" = .R2". Since f$ e A(P), p is a convex
combination of N <2n + 1 extreme points of Л(Р). (Exercise 19, Chapter 3.)
If у is an extreme point of Л(Р), then A-1(y) is an extreme set of K, and every
extreme point of Л" 1(y) (their existence follows from the Krein-Milman theorem)
is an extreme point of P. It follows that there are extreme points ц19 ..., juN of P
and positive numbers ck with £ ck = 1, such that
G)
Being an extreme point of P, each \ik that occurs in G) has a single point
ak e Г for its support; hence
ifFis now defined by C), it follows from G) and (8) that F satisfies A) and B).

120 GENERAL THEORY
A Fixed Point Theorem
Fixed point theorems play an important role in many parts of analysis and topology.
The one that we shall now prove is due to Kakutani; it will be used to prove the exis-
existence of a Haar measure on any compact group. The proof of Kakutani's theorem
involves only the most basic properties of locally convex spaces.
5.11 Theorem Suppose
(a) К is a nonempty compact convex set in a locally convex space X,
(b) G is an equicontinuouS group of linear mappings of X onto X, and
(c) A{K) с К for every AeG.
Then G has a common fixed point in K; that is, there exists p e К such that Ap = p
for every AeG.
Part (b) of the hypothesis should perhaps be made more explicit. Equicontinuity
is defined in Section 2.3. To say that G is a group means that every Л е G is a one-to-
one mapping of X onto X whose inverse A also belongs to G and that Л1Л2 е G
whenever At e G (i = 1, 2). Here (A^)* = Л1(Л2х), of course. Hypothesis (b) is
satisfied, for instance, when G is a group of linear isometries on a normed space X.
proof Let Q be the collection of all nonempty compact convex sets H с К such
that A(H) с Н for every AeG. Partially order Q. by set inclusion. Note that
Q ф 0, since К e Q. By Hausdorff's maximality theorem, Q contains a maximal
totally ordered subcollection Qo . The intersection Ho of all members of Qo is a
minimal member of Q. The theorem will be proved by showing that Ho contains
only one point. To do this, we shall consider a set H e Q which contains at least
two points, and we shall prove that some Ht e Q is a proper subset of H.
Before doing this, we prove that X has a local base consisting of balanced
convex sets U that satisfy A(U) cz U for every AeG.
Let V be a convex neighborhood of 0 in X. Since G is equicontinuous,
there is a balanced neighborhood Vt of 0 such that AfT^ с V for every AeG.
Let U be the convex hull of the union of all sets AiVJ, as A ranges over G. Then
U is convex and balanced, and U с V, since V is convex. Every ueU has the
form
u = clAlv1 + "- + cnAnvn,
where c{ > 0, £ ct = 1, Л, e G, vt e Vv If A ie G, then
Аи = qM^j + • * • + cnAAnvn
lies also in U, because AAf e G. Hence A(U) cz U.

SOME APPLICATIONS 121
Now suppose Ней, and Я contains at least two points. Then Я - Я ф
{0}, and some set U as above fails to cover H — H. Since Я — Я is compact,
H — Я с ,?£/ for some -s1 > 0. Let г be the greatest lower bound of these numbers
s. Then t > 1. Put W =tU. Then l^is a convex balanced open set such that
A) Л@О с JF for every Л е G,
B) .... H-Hcz{\+r)W ifr>0,
C) , A - г)Ж does not cover H- H if 0 < r < 1.
Properties A) and B) are obvious. Since W\s convex,
A -r)W cz(l -r)W
this last set does not cover H — H; hence C) holds.
Since Я is compact, H contains points xl9 ..., xn such that
D) H<=\J(x
•Put r = 1/Dи), and define
E) Я.=ЯпД(
It is clear that Ях is compact and convex.
Suppose x e Ht and у e H. Since Л~ Х(Я) с Я, у = Лд^ for some j^ e Я.
By E), х е уг + A - г)Ж Hence (I) implies that
Ах е Ayl + A - r)A(W) с у + A - г)Ж
It follows that A(Hi) с Я! for every Ae G.
By C), there are points хеН, у е Я, such that * - у does not lie in
A — r)W. Any such x is not in Hl. Thus Hx Ф H.
To complete the proof, we have to show that Ях ф 0. We do this by
showing that Ht contains the point
F) .* -l-(x +•••+*)
П
Since Я is convex, x0 e H. Fix у e H. By D), there exists/such that ^
G) yexj-
If i ф], 1 < i < w, property B) implies that
(8) }exj + (l +

122 GENERAL THEORY
Add the relations G) and (8), divide by n, and use the convexity of W to obtain
У " X° E ~n [\ + (" " 1)A +
since r = l/D«). Thus x0 e у + A - r)PT, for every у е Н. Hence jc0 e Hu and
the proof is complete. ////
Haar Measure on Compact Groups
5.12 Definitions A topological group is a group G in which a topology is defined
that makes the group operations continuous. The most concise way to express this
requirement is to postulate the continuity of the mapping ф: G x G -> G defined by
ф(х,у) =ху~1.
For each ae G, the mappings x -* ax and jc -> xa are homeomorphisms of G
onto G; so is л: -> jc. The topology of G is therefore completely determined by any
local base at the identity element e.
If we require (as we shall from now on) that every point ofG is a closed set, then
the analogues of Theorems 1.10 to 1.12 hold (with exactly the same proofs, except for
changes in notation); in particular, the Hausdorff separation axiom holds.
If/is any function with domain G, its left translates Ls/and its right translates
i?s/are defined, for every se G,hy
{LJ)(x) =f(sx), (Rsf)(x) =/(») (jc e G).
A complex function /on G is said to be uniformly continuous if to every 8 > 0
corresponds a neighborhood V of e in G such that
-/«I <*
whenever s e G, t e G, and .y? e V.
A topological group G whose topology is compact is called a compact group; in
this case, C(G) is, as usual, the Banach space of all complex continuous functions on G,
with the supremum norm.
5.13 Theorem Let Gbe a compact group, suppose fe C(G), and define HL(f) to be
the convex hull of the set of all left translates off Then
(a) fis uniformly continuous, and
(b) HL{f) is a totally bounded subset of C(G).
In other words, the closure of HL(f) in C(G) is compact. (Appendix A4.)

SOME APPLICATIONS 123
proof Fix e > 0. Since/is continuous there corresponds to each a e G a neigh-
neighborhood Wa of e such that \f{t) — f(a)\ < e for all tmaWa. The continuity of
the group operations gives neighborhoods Va of e that satisfy VaV~x с Wa.
Since G is compact, there is a finite set A cz G such that
G=[)aVa.
as A
Put
V- П V-
aeA
Assume x~ly e V. Choose ae Aso that у е aVa. Then |/(.y) —f{fl)\ < £•
Also, \f(x) —f(a)\ < e, because
Hence |/(jc) -/(y)| < 2s. This proves (л).
Since (sx)~l(sy) = x~*y for every se G, it follows that
\(LJ)(x) - (Lj)(y)\ = |/(^) -/(^)| < 2e
whenever x~VeF. Every g e HL(f) is a finite sum of the form £ csLs/, with
4>0, X^1- Hence
Ш-0(У)\ <2e
if x.); e V and ^ e HL{f). This proves that #L(/) is an equicontinuous subset
of C(G). Now (b) follows from Ascoli's theorem. (Appendix A5.) ////
5.14 Theorem On every compact group G exists a unique regular Borel probability
measure m which is left-invariant, in the sense that
A) (fdm= \{LJ)dm [seGJeC(G)l
This m is also right-invariant:
B) \ fdm=\ (RJ) dm [s e GJ e C(G)]
and it satisfies the relation
C) f /(*) Жи(х) = T f(x-x) dm(x) [/e C(G)].
JG JG
This m is called the Haar measure of G.
proof The operators Ls satisfy LsLt =Lts9 because
(Ь,Ь,Л(х) = (L,fKsx) =f(tsx) = (Ltsf)(x).

124 GENERAL THEORY
Since each Ls is an isometry of C(G) onto itself, {Ls: s e G} is an equicontinuous
group of linear operators on C(G). If /e C(G), let Kf be the closure of HL(f).
By Theorem 5.13, Kf is compact. It is obvious that Ls(Kf) = Kf for every tf e G.
The fixed point theorem 5.11 now implies that Kf contains a function ф such that
Ь5ф = ф for every s e G. In particular, ф(,у) = ф(е), so that ф is constant. By
the definition of Kf, this constant can be uniformly approximated by functions
in HL(f).
So far we have proved that to each/e C{G) corresponds at least one con-
constant с which can be uniformly approximated on G by convex combinations of
left translates of/ Likewise, there is a constant с which bears the same relation
to the right translates of/ We claim that cr = c.
To prove this, pick e > 0. There exist finite sets {at} and {bj} in G, and there
exist numbers af > 0, /?7- > 0, with £ <*i = 1 = X Pj > sucn
D)
and
E)
< e
(xeG).
Put jc = bj in D); multiply D) by /?7-, and add with respect to/ The result is
F) |c-5>i/^/(«i6»)<i
Put x = яj in E), multiply E) by at, and add with respect to i, to obtain
G)
i.j
< e.
Now F) and G) imply that с = с'.
It follows that to each /e C{G) corresponds a unique number, which we
shall write Mf, which can be uniformly approximated by convex combinations
of left translates of/; the same Mf is also the unique number that can be uni-
uniformly approximated by convex combinations of right translates of / The
following properties of M are obvious:
(8)
(9)
A0)
A1)
Mf>0
Ml = 1.
^(a/) = uMf if a is a scalar.
M(LJ) =Mf = M(Rsf) for every s e G.
We now prove that
A2)
M{f+g)=Mf+Mg.

SOME APPLICATIONS 125
Pick£>0. Then
A3)
< e
(xeG)
for some finite set {at} <= G and for some numbers a,- > 0 with £ a,- = 1. Define
A4) h(x) = Zotig(aix).
- - - i
Then h e KgJ9 hence Kh a Kg, and since each of these sets contains a unique con-
constant function, we have Mh = M#. Hence there is a finite set {bj} с G, and there
are numbers fij > 0 with £ /?y = 1, such that
A5)
by A4), this gives
A6)
< S
(xeG);
(x e G).
Replace x by bjX in A3), multiply A3) by fij, and add with respect toy, to obtain
\A7) Mf- ^OLiPjKatbjx) I < e (xeG).
h j I
Thus
A8)
Since £ oLipj = 1, A8) implies A2).
<2e (x e G).
The Riesz representation theorem, combined with (8), (9), A0), and A2),
yields a unique regular Borel probability measure m that satisfies
A9)
Mf=\fdm (feC(G));
properties A) and B) now follow from A1).
To prove C), denote the right side of C) by M'f, and observe that M' also
satisfies properties (8) to A2), hence that M' = M. ////
Uncomplemented Subspaces
Complemented subspaces of a topological vector space were defined in Section 4.20;
Lemma 4.21 furnished some examples. It is also very easy to see that every closed
subspace of a Hilbert space is complemented (Theorem 12.4). We will now show that
some very familiar closed subspaces of certain other Banach spaces are, in fact, not
complemented. These examples will be derived from a rather general theorem about

126 GENERAL THEORY
compact groups of operators that have an invariant subspace; its proof uses vector-
valued integration with respect to Haar measure.
We begin by looking at some relations that exist between complemented sub-
spaces on the one hand and projections on the other.
5.15 Projections Let X be a vector space. A linear mapping P: X-*X is called
a projection in X if
i.e., if P(Px) = Px for every x e X.
Suppose P is a projection in X, with null space Л(Р) and range 0t(P). The fol-
following facts are almost obvious.
(a) &(P) = Л {I ~P) = {xe X: Px = x}.
(b) JT(P) = &(I - P).
(c) 0t{P) n Л(Р) = {0} and X = 0t(P) + Л{Р\
(d) If A and В are subspaces of X such that A n В = {0} and X = Л + В, then there
is a unique projection P in X with ^ = ^£(P) and В = Л(Р).
Since (/ - P)P = 0, Ш(Р) с ^Г(/ - P). If jc e Л {I - P), then x-Px = 0, and so
jc = Px e 0t(P). This gives (a); F) follows by applying (a) to / - P. If* 6 ^(P) n ^K"(P),
then jc = Px = 0; if x e X, then jc = Px + (jc - Px), and x - Px e Л{Р). This proves
(c). If A and В satisfy (d), every x e X has a unique decomposition jc = xr + x", with
jc' e Л, х" е Б. Define Px = xr. Trivial verifications then prove (d).
5.16 Theorem
(a) If P is a continuous projection in a topological vector space X, then
Х=М(Р)@Л(Р).
(b) Conversely, if X is an F-space and if X = A@B, then the projection P with range
A and null space В is continuous.
Recall that we use the notation X = A © В only when A and В are closed sub-
spaces of X such that A n В = {0} and A + В = X.
proof Statement (a) is contained in (c) of Section 5.15, except for the assertion
that ffl(P) is closed. To see the latter, note that M{P) = ЛA - P) and that / - P
is continuous.
Next, suppose P is the projection with range A and null space B, as in (b).
To prove that P is continuous we verify that P satisfies the hypotheses of the
closed graph theorem: Suppose xn->x and Pxn-*y. Since PxneA and A is

SOME APPLICATIONS 127
closed, we. have у e A, hence у = Py. Since xn — Pxn e В and В is closed, we
have x — у eB, hence Py = Px. It follows that у = Px. HenceP is continuous.
Corollary A closed subspace of an F-space X is complemented in X if and only
if it is the range of some continuous projection in X.
5.17 Groups of linear operators Suppose that a topological vector space X and
a topological group G are related in the following manner: To every seG corre-
corresponds a continuous linear operator Ts: X-+X such that
Te = I, Tst = TsTt (seG,teG);
also, the mapping 0, x) -+TSx of G x X into X is continuous.
Under these conditions, G is said to act as a group of continuous linear operators
on X.
5.18 Theorem Suppose
(a) X is a Frechet space,
(b) Y is a complemented subspace of X,
(c) G is a compact group which acts a$ a group of continuous linear operators on X, and
{d) Ts(Y) с Yfor every seG.
Then there is a continuous projection Q of X onto Y which commutes with every Ts.
proof For simplicity, write sx in place of Tsx. By (b) and Theorem 5.16, there
is a continuous projection P of X onto Y. The desired projection Q is to satisfy
s~lQs = Q for all seG. The idea of the proof is to obtain Q by averaging the
operators s~lPs with respect to the Haar measure m of G: define
A) Qx = f s~ lPsx dm(s) (x e X).
JG
To show that this integral exists, in accordance with Definition 3.26, put
B) fx(s) = s~lPsx (seG).
By Theorem 3.27, it suffices to show that/x: G -*X is continuous. Fix s0 e G; let
U be a neighborhood of fx(s0) in X. Put у = Ps0 x, so that
C) »,-'у=Ш.
Since (s, z) -> sz is assumed to be continuous, s0 has a neighborhood Vx and у
has a neighborhood W such that
D) s'\W)cU

128 GENERAL THEORY
Also, .s0 has a neighborhood V2 such that
E) PsxeW ifseV2.
The continuity of P was used here. If s e Vx n V2, it follows from B), D), and
E) that/x(^) e U. Thus/X is continuous.
Since G is compact, each/x has compact range in X. The Banach-Steinhaus
theorem 2.6 implies therefore that {s~1Ps: s e G} is an equicontinuous collection
of linear operators on X. To every convex neighborhood Ul of 0 in X corre-
corresponds therefore a neighborhood U2 of 0 such that s~iPs(U2) с U1. It now
follows from A) and the convexity of и± that Q(U2) с U^ (See Theorem 3.27.)
Hence Q is continuous. The linearity of Q is obvious.
If jc e X, then Psx e Y, hence s~xPsx e Y by (d), for every s e G. Since Y
is closed, Qx e Y.
If x e Y, then ^x e F, Psx = sx, and so sr~1Psx = x, for every s e G. Hence
Qx = x.
These two statements prove that б is a projection of X onto Y. To com-
complete the proof, we have to show that
F) 2^0 =s0Q for every ^0 e G.
Note that ^-1Р^0 = ■Уо(^о)^(^о)« It: now follows from A) and B) that
Qs0 x= \ s~iPss0 x dm(s)
= f sofx($so) dm{s)
= \ sQfx(s) dm(s)
= s0 f fx(s) dm(s) = s0 Qx.
JG
The third equality is due to the translation-invariance of w; for the fourth
(moving s0 across the integral sign), see Exercise 24 of Chapter 3. ////
5.19 Examples In our first example, we take X — I}y Y=H1. Here L1 is the
space of all integrable functions on the unit circle, and H1 consists of those fe L1 that
satisfy /(«) = 0 for all n < 0. Recall that f(n) denotes the nth Fourier coefficient of/:-
A) }(n) = 1 f_J(O)e-M dB (n = 0, ±1, ±2,...).
Note that we write fF) in place off(ew\ for simplicity.

SOME APPLICATIONS 129
For G we take the unit circle, i.e., the multiplicative group of all complex numbers
of absolute value 1, and we associate to each eis e G the translation operators ts
defined by
B) (
It is a simple matter to verify that G then acts on L1 as described in Section 5.17 and
that
C) (\jnn) = einsf(n).
Hence т^Я1) = Я1 for every real s. (See Exercise 12.)
If Я1 were complemented in L1, Theorem 5.18 would imply that there is a con-
continuous projection Q of L1 onto Я1 such that
D) *sQ = Qb for alb.
Let us see what such a Q would have to be.
Put en(9) = еш. Then xsen = einsen) and
E) Q^sen = e^Qen,
since Q is linear. It follows from D) and E) that
F) (Qen)(s + в) = ein\Qen){6).
Put cn = (QO@). with 0 = 0, F) becomes
G) 6*» = ^ (л=0, ±1, ±2,...).
So far we have just used D). Since Qen e H1 for all n, cn — 0 when w < 0. Since
6/=/for every/G Я1, с„ = 1 when и > 0. Thus Q (if it exists at all) is the "natural"
projection of L1 onto Я1, the one that replaces /(я) by 0 when n < 0. In terms of
Fourier series,
0
To get our contradiction, consider the functions
(9)
These are the well-known Poisson*kernels. Explicit summation of the series (9) shows
that/r 2:0. Hence
A0) Il/J, = f Г \№\ de = ±f U0) d9 = 1
for all r. But

130 GENERAL THEORY
and Fatou's lemma implies that || Qfr\\t ->oo as r -> 1, since J 11 — eie |  d6 = oo. By
A0), this contradicts the continuity of Q.
Hence Hl is not complemented in L1.
The same analysis can be applied to A and C, where С is the space of all continu-
continuous functions on the unit circle, and A consists of those/e С that have f(n) = 0 for all
n < 0. If A were complemented in C, the operator Q described by (8) would be a
continuous projection from С onto A. Application of Q to real-valued fe С shows
that there is a constant M < oo that satisfies
A2) sup |/@)| <M-sup|Re/@)|
в в
for every/e A. To see that no such M can exist, consider conformal mappings of the
closed unit disc onto tall thin ellipses.
Hence A is not complemented in С
However, the projection (8) is continuous as an operator in Lp, if 1 < p < oo.
Hence Hp is then a complemented subspace of LP. This is a theorem of M. Riesz
(Th. 17.26 of [23]).
We conclude with an analogue of (b) of Theorem 5.16; it will be used in the proof
of Theorem 11.31.
5.20 Theorem Suppose X is a Banach space, A and В are closed suhspaces of X,
and X — A + B. Then there exists a constant у < oo such that every x e X has a
representation x — a + b, where aeA, b e B, and \\a\\ + \\b\\ < y\\x\\.
This differs from (b) of Theorem 5.16 inasmuch as it is not assumed that
A n В = {0}.
proof Let Y be the vector space of all ordered pairs (a, b), with aeA, b e B,
and componentwise addition and scalar multiplication, normed by
Since A and В are complete, Y is a Banach space. The mapping Л: Y-+ X
defined by
A(a, b) = a + b
is continuous, since ||a + b\\ < ||(д, 6)||, and maps Yonto X. By the open mapping
theorem, there exists у < oo such that each xelis A(a, b) for some {a, b) with
Exercises
1 Define measures /xb /x2 on the unit circle by
dfit = cos в d9, dfi2 = sin 9 dd
and find the range of the measure fi = (fili /x2).

SOME APPLICATIONS 131
2 Construct two functions/and g on [0, 1] with the following property: If
diii =/(*) dx, dfj,2 = g(x) dx, fx = (/xb /x2),
then the range of/x is the square with vertices at A, 0), @, 1), (—1, 0), @, —1).
3 Suppose that the hypotheses of Theorem 5.9 are satisfied, that ф e C(S), ф>0,д£ С (К),
and |g | < ф | к • Prove that there exists /e Y such that/| K = g and | /1 < ф on S. Hint:
Apply Theorem 5.9 to the space of all functions//</>, with/e Y.
4 Supply the details of the proof that every extreme point of P has its support at a single
point. (This refers to the end of the proof of Theorem 5.10.)
5 Prove the analogues of Theorems 1.10 to 1.12 that are alluded to in Section 5.12. (Do
not assume that G is commutative.)
6 Suppose G is a topological group and H is the largest connected subset of G that contains
the identity element e. Prove that Я is a normal subgroup of G, that is, a subgroup that
satisfies x'^Hx — H for every x e G. Hint: If A and В are connected subsets of G, so
are AB and A'1.
7 Prove that every open subgroup of a topological group is closed. (The converse is
obviously false.)
8 Suppose m is the Haar measure of a compact group G, and К is a nonempty open set in
G. Prove that m(V)>0.
9 Put ejfi) = eine. Let L2 refer to the Haar measure of the unit circle. Let A be the smallest
closed subspace of L2 that contains en for n = 0, 1, 2,..., let В be the smallest closed
subspace of L2 that contains e.n -f- nen for n = 1, 2, 3, Prove the following:
(a) AnB = {0}.
(b)IfX = A + B then X is dense in L2, but X Ф L2.
(c) Although X = A@B, the projection in X with range A and null space В is not
continuous. (The topology of X is, of course, the one that X inherits from L2. Com-
Compare with Theorem 5.16.)
10 Suppose X is a Banach space, P e ^(X), Q e ^(X), and P and б are projections.
(a) Show that the adjoint P* of P is a projection in X*.
(b) Show that \\P - Q\\ > 1 if FG = G^ and P Ф Q.
11 Suppose P and Q are projections in a vector space X.
(a) Prove that P + Q is a projection if and only if PQ = QP = 0. In that case,
./ПР + G) = Л{Р) п Jf(Q\
0t{P) n M{Q) = {0}.
If PQ = QP, prove that P£> is* a projection and that
(c) What do the matrices
show about part (b) ?
(J о) - (i D

132 GENERAL THEORY
12 Prove that the translation operators rs used in Example 5.19 satisfy the continuity
property described in Section 5.17. Explicitly, prove that
if r->5 and#->/in L1.
13 Use the following example to show that the compactness of G cannot be omitted from
the hypotheses of Theorem 5.18. Take X = Ll on the real line R, relative to Lebesgue
measure; /e Y if and only if JR/= 0; G = R with the usual topology; G acts on L1 by
translation: (rsf)(x) =/(s -f x). The joint continuity property is satisfied (see Exercise
12), rs F= Y for every s, and Y is complemented in X. Yet there is no projection of
X onto У (continuous or not) that commutes with every rs.
14 Suppose S and T are continuous linear operators in a topological vector space, and
T = TST.
Prove that T has closed range. (See Theorem 5.16 for the case S = /.)
75 Suppose Л is a closed subspace of CCS), where S is a compact Hausdorff space; suppose
fi is an extreme point of the unit ball of A1; and suppose/e C(S) is a rea/ function such
that
for every g e A. Prove that/is then constant on the support of fi. (Compare with Theo-
Theorem 5.7.) Show, by an example, that the conclusion is false if the word " real" is omitted
from the hypotheses.

PART TWO
Distributions and
Fourier Transforms

о
TEST FUNCTIONS AND DISTRIBUTIONS
Introduction
6.1 The theory of distributions frees differential calculus from certain difficulties that
arise because nondifferentiable functions exist. This is done by extending it to a class
of objects (called distributions or generalized functions) which is much larger than the
class of differentiable functions to which calculus applies in its original form.
Here are some features that any such extension ought to have in order to be
useful; our setting is some open subset of Rn:
(a) Every continuous function should be a distribution.
(b) Every distribution should have partial derivatives which are again distributions.
For differentiable functions, the new notion of derivative should coincide with
the old one. (Every distribution should therefore be infinitely differentiable.)
(c) The usual formal rules of calculus should hold.
(d) There should be a supply of convergence theorems that is adequate for handling
the usual limit processes.
To motivate the definitions to come, let us temporarily restrict our attention to
the case n — 1. The integrals that follow are taken with respect to Lebesgue measure,
and they extend over the whole line R, unless the contrary is indicated.

136 DISTRIBUTIONS AND FOURIER TRANSFORMS
A complex function / is said to be locally integrable if / is measurable and
Ik I /I < °° f°r every compact К с R. The idea is to reinterpret / as being something
that assigns the number J/ф to every suitably chosen " test function " ф, rather than as
being something that assigns the number/(x) to each xeR. (This point of view is
particularly appropriate for functions that arise in physics, since measured quantities
are almost always averages. In fact, distributions were used by physicists long before
their mathematical theory was constructed.) Of course, a well-chosen class of test
functions must be specified.
We let Q) — $)(R) be the vector space of all ф е С°°(^) whose support is compact.
Then J/ф exists for every locally integrable / and for every ф e Q). Moreover, £d is
sufficiently large to assure that/is determined (a.e.) by the integrals J/ф. (To see this,
note that the uniform closure of $) contains every continuous function with compact
support.) If/happens to be continuously differentiable, then
A)
If/ e С°°(Д), then
B) ]/{к)ф = (- \)к\/ф{к) (фе@,к=1,2,3,.. .)•
The compactness of the support of ф was used in these integrations by parts.
Observe that the integrals on the right sides of (I) and B) make sense whether f is
differentiable or not and that they define linear functionals on Q).
We can therefore assign a "fcth derivative " to every/that is locally integrable:
/(fc) is the linear functional on 0 that sends ф to (-l)fcJ/£(fc). Note that/itself
corresponds to the functional ф -* \/ф.
The distributions will be those linear functionals on Q) that are continuous with
respect to a certain topology. (See Definition 6.7.) The preceding discussion suggests
that we associate to each distribution Л its "derivative" Л' by the formula
C) Л'(ф)=-Л(ф') (фе@).
It turns out that this definition (when extended to n variables) has all the desirable
properties that were listed earlier. One of the most important features of the resulting
theory is that it makes it possible to apply Fourier transform techniques to many
problems in partial differential equations where this cannot be done by more classical
methods.
Test Function Spaces
6.2 The space <2)(Q) Consider a nonempty open set Q, с Rn. For each compact
К cQ, the Frechet space Q)K was described in Section 1.46. The union of the spaces
<2iK, as К ranges over all compact subsets of Q,, is the test function space ^(П). It is

TEST FUNCTIONS AND DISTRIBUTIONS 137
clear-that @{п) is a vector space, with respect to the usual definitions of addition and
scalar multiplication of complex functions. Explicitly, ф e <3(Q) if and only if ф е
and the support of ф is a compact subset of Cl.
Let us introduce the norms
A) || ф \\N = max {| П«ф(х) \:xeu,\a\<N},
for ф e ^(Q) and N - 0, 1, 2,...; see Section 1.46 for the notations Da and | a|.
The restrictions of these norms to any fixed Q)K c= <$(Q) induce the same topology
on <3K as do the seminorms pN of Section 1.46. To see this, note that to each К corre-
corresponds an integer No such that К a KN for all N> No. For these N, \\ф\\ц= Рм(Ф)
if ф e Q)K. Since
B) il^lU^ ll0IU+i and Рц(Ф)^Рц+1(ф),
the topologies induced by either -sequence of seminorms are unchanged if we let N
start at Л^о rather than at 1. These two topologies of ^ coincide therefore; a local
base is formed by the sets
C) Fjv = [ф е @к: \\ф\\я <—) (N = 1, 2, 3,...).
The same norms A) can be used to define a locally convex metrizable topology
on ^(O); see Theorem 1.37 and (b) of Section 1.38. However, this topology has the
disadvantage of not being complete. For example, take n = 1, п = R, pick ф е :
with support in [0, 1], ф > 0 in @, 1), and define
2 m
Then {фт} is a Cauchy sequence in the suggested topology of @(R), but lim фт does not
have compact support, hence is not in @(R).
We shall now define another locally convex topology т on @(Q) in which Cauchy
sequences do converge. The fact that this т is not metrizable is only a minor incon-
inconvenience, as we shall see.
6.3 Definitions Let Obea nonempty open set in Rn.
(a) For every compact К с п, xK denotes the Frechet space topology of Q)K, as
described in Sections 1.46 and 6.2.
(b) P is the collection of all convex balanced sets W а @(п) such that Q)K n Wetk
for every compact К с п.
(c) т is the collection of all unions of sets of the form ф + W, with ф е @(п) and
We p.
Throughout this chapter, К will always denote a compact subset of п.

138 DISTRIBUTIONS AND FOURIER TRANSFORMS
6.4 Theorem
(a) т is a topology in @(Q), and ft is a local base for т.
(b) т makes @(Q) into a locally convex topological vector space.
proof Suppose Vxex, V2 e т, ф e V1 n V2. To prove {a), it is clearly enough
to show that
A) (j)+Wc=:V1r\V2
for some We p.
The definition of т shows that there exist ф{ e @(п) and Wt e p such that
B) феф.+ W^V, (/=1,2).
Choose К so that Q)K contains фъ ф2, and ф. Since Q)K n Wt is open in @K,
we have
C) ф_ф.£A_5.)Ж.
for some ^f > 0. The convexity of Wt implies therefore that
D) Ф-Ф1 + Si Wx с A - бМ + Й, Wt = ^,
so that
E) ф + Й, Wx c^.-l^c F, A = 1, 2).
Hence A) holds with W = E^!) n E2 PF2), and (a) is proved.
Suppose next that фх and ф2 are distinct elements of ^(Д), and put
F) W= {фе@(п): \\ф\\0 < \\ф± - ф2||0},
where ||ф||0 is as in A) in Section 6.2. Then We p and фг is not in ф2 + W. It
follows that the singleton {0J is a closed set, relative to т.
Addition is т-continuous, since the convexity of every We p implies that
G) OAi + \W) + (ф2 + i W) = Oh + W + ^
for any ^ e ^(Q), ф2 е Qj(u).
To deal with scalar multiplication, pick a scalar a0 and а ф0 е ^(fl). Then
(8) аф -аофо = а(ф - ф0) + (а - а0) Фо •
, there exists S > 0 such that <5ф0 e \ W. Choose с so that 2c( | a01 + 5) = 1.
Since W is convex and balanced, it follows that
(9) aф-aoфoeW
whenever \a — ао|%< д and ф — фое cW.
This completes the proof. ////

TEST FUNCTIONS AND DISTRIBUTIONS 139
Note: From now on, the symbol <3(Q) will denote the topological vector space
), t) that has just been described. All topological concepts related to @(п) will
refer to this topology т.
6.5 Theorem
(a) A convex balanced subset V of @(Q) is open if and only if V e ft.
(b) The topology тк of any Q)K с @(Q) coincides with the subspace topology that
@K inherits from'@(u).
(c) If E is a bounded subset of @(£l), then E с <2)K for some К с= Q, and there are
numbers MN < oo such that every ф е Е satisfies the inequalities
W\N<MN (N=0,1,2,...).
(d) @(Q) has the Heine-Borel property.
(e) If{4>i} is a Cauchy sequence in ^(O), then {ф J c= @Kfof some compact К с= Q, and
lim ||<£.-фу||„ = 0 (^=0,1,2,...)-
(/) If фх -> 0 in the topology of @(п), then there is a compact Кап which contains
the support of every фс, and Ваф1->0 uniformly, as /-> oo, for every multi-
index a.
(g) In @(Q), every Cauchy sequence converges. t . ,. .
Remark In view of (b), the necessary conditions expressed by (c), (e), and
(/) are also sufficient. For example, if Ecz@K and \\ф\\м < MN < oo for
every ф e E, then E is a bounded subset of SfK (Section 1.46), and now (b) implies
that E is also bounded in
proof Suppose first that Vet. Pick ф e ®K n V. By Theorem 6.4, ф + W с V
for some We p. Hence
ф + @jj; nW)cz@KnV.
Since Q)K n Ж is open in <2)K, we have proved that
A) 9Kr\VexK if Vex and^cQ.
Statement (a) is an immediate consequence of A), since it is obvious that
P а т.
One-half of (b) is proved by A). For the other half, suppose Eetk. We
have to show that E = @Kn К for some Vet. The definition of тк implies
that to every ф e E correspond N and д > 0 such that
B) {

140 DISTRIBUTIONS AND FOURIER TRANSFORMS
Put }Уф = {фе 0(Q): \\ij/\\N< S}. Then W+ e p, and
C) @кп(ф+ЦГф) = ф + (@к п Цгф) <= Д.
If Fis the union of these sets ф + 1¥ф, one for each ф е Е, then Fhas the desired
property.
For (c), consider a set E a 3)(Q) which lies in no QJK. Then there are func-
functions фт e E and there are distinct points xm e Q, without limit point in Q, such
that фт(хт) фО(т = 1, 2, 3,...). Let W be the set of all ф е $(п) that satisfy
D) \Ф(хт)\<т-1\фт(хт)\ (m= 1,2,3,...).
Since each К contains only finitely many xm, it is easy to see that <2)K n W exK.
Thus We p. Since фт ф mW, no multiple of W contains E. This shows that E
is not bounded.
It follows that every bounded subset E of <3(Q) lies in some @K. By (b), E
is then a bounded subset of <&K. Consequently (see Section 1.46)
E) sup {U\\N: феЕ}<оо (N = 0, 1, 2,...).
This completes the proof of (c).
Statement (d) follows from (c), since Q)K has the Heine-Borel property.
Since Cauchy sequences are bounded (Section 1.29), (c) implies that every
Cauchy sequence {</>;} in <3(п) lies in some @K. By (b), {фс} is then also a Cauchy
sequence relative to tk . This proves (e).
Statement (/) is just a restatement of (e).
Finally, (g) follows from (b), (e), and the completeness of @K. (Recall that
@K is a Frechet space.) ////
6.6 Theorem Suppose Л is a linear mapping of @(Q) into a locally convex space
Y. Then each of the following four properties implies the others:
{a) A is continuous.
(b) A is bounded.
(c) 1/ф{ -> 0 in 0(Q) then Аф{ -> 0 in Y.
(d) The restrictions of A to every @к с @(П) are continuous.
proof The implication (a) -> (b) is contained in Theorem 1.32.
Assume Л is bounded and ф(->0 in @(п). By Theorem 6.5, ф(->0 in
some SfK, and the restriction of Л to this $)K is bounded. Theorem 1.32, applied
to Л: @к -> Г, shows that Аф1 -> 0 in Y. Thus (b) implies (c).
Assume (c) holds, {ф(} cz <%K, and ф1 -> 0 in Q)K. By (b) of Theorem 6.5,
фг -> 0 in ^(Q). Hence (c) implies that Лф^ -> 0 in Y, as / -> oo. Since @K is
metrizable, (J) follows.

TEST FUNCTIONS AND DISTRIBUTIONS 141
To prove that (d) implies (a), let U be a convex balanced neighborhood of
0 in F, and put V = Л " *( U). Then V is convex and balanced. By (a) of Theorem
6.5, V is open in $J{Q) if and only if <3K n V is open in $)K, for every
This proves the equivalence of (a) and (d). ////
Corollary Every differential operator D* is a continuous mapping of
into
proof Since \\Da(j)\\N < \\<l>\\N+\a\ for # = 0, 1, 2,..., Da is continuous on each
@k. ~ Illl
6.7 Definition A linear functional on @(п) which is continuous (with respect to
the topology т described in Definition 6.3) is called a distribution in п.
The space of all distributions in п is denoted by <3'{п).
Note that Theorem 6.6 applies to linear functional^ on @(п). It leads to the
following useful characterization of distributions.
6.8 Theorem If A is a linear functional on @(Q), the following two conditions
are equivalent:
(a) Ae&(Cl).
(b) To every compact К czQ, corresponds a nonnegative integer N and a constant
С < oo such that the inequality
holds for every ф е $)K.
proof This is precisely the equivalence of (a) and (d) in Theorem 6.6> combined
with the description of the topology of Q)K by means of the seminorms ||ф||^
given in Section 6.2. ////
Note: If Л is such that one N will do for all К (but not necessarily with the same
C), then the smallest such N is called the order of Л. If no N will do for all K, then Л
is said to have infinite order.
6.9 Remark Each хеп determines a linear functional Sx on @(п), by the formula
Зх(ф) = ф(х).
Theorem 6.8 shows that 3X is a distribution, of order 0.
If x = 0, the origin of Rn, the functional S = 30 is frequently called the Dirac
measure on R".
Since BK, for К cz Q, is the intersection of the null spaces of these Sx, as x ranges
over the complement of K, it follows that each Q)K is a closed subspace of @(Q). [This

142 DISTRIBUTIONS AND FOURIER TRANSFORMS
follows also from Theorem 1.27 and part (b) of Theorem 6.5, since each Qjk is com-
complete.] It is obvious that each Q)K has empty interior, relative to @(Q). Since there is a
countable collection of sets ^cfi such that ^(П) = \]BK., 9{п) is of the first
category in itself. Since Cauchy sequences converge in @(Q) (Theorem 6.5), Baire's
theorem implies that ^(Q) is not metrizable.
Calculus with Distributions
6.10 Notations As before, п will denote a nonempty open set in Rn. If a = (a1?
..., an) and P — (pl9..., f$n) are multi-indices (see Section 1.46) then
A) |a| =at + ••• + <*„,
B) Da = D\l • • • D'n9 where #,=/-,
dXj
C) p < a means j^f < a,- for 1 < / < «,
D) a±/»-=(«i±P1,...,«.±W.
If x e Л" and у е R", then
E) х-у = х1у1 + ---+хяуя,
F) ^[«(х-хУ'^Сх?+--+^I/2.
The fact that the absolute value sign has different meanings in A) and in F)
should cause no confusion.
If x e Rn and a is a multi-index, the monomial xa is defined by
G) x" = x? •••*?■.
6.11 Functions and measures as distributions Suppose/is a locally integrable
complex function in п. This means that/is Lebesgue measurable and $K | f(x) | dx < oo
for every compact К cz Q,; dx denotes Lebesgue measure. Define
A) А/ф)=\ <t>(x)f(x)dx
Since
B)
Theorem 6.8 shows that Af e @'(Q).
It is customary to identify the distribution Af with the function/and to say that
such distributions " are " functions.

TEST FUNCTIONS AND DISTRIBUTIONS 143
Similarly, if fi-is a complex Borel measure on Q, or if jjl is a positive measure on Q
with fi(K) < oo for every compact К с O, the equation
C) \(Ф)
defines a distribution Лд in П, which is usually identified with \i.
6.12 Differentiation of distributions If a is a multi-index and Ae$J'{u), the
formula
A) (Г>"А)(ф) = (- 1)ИЛ(Да<£) [</> e
(motivated in Section 6.1) defines a linear functional DaA on ^(O). If
B) \Аф\ <С\\ф\\„
for all фе@к, then
C) 1
Theorem 6.8 shows therefore .that DaA e @'{п).
Note that the formula
D) "• D"DPA = Da+ph
holds for every distribution Л and for all multi-indices a and ft, simply because the
operators Da and Dp commute on C°°(O):
6.13 Distribution derivatives of functions The ath distribution derivative of a
locally integrable function/in Q. is, by definition, the distribution DaAf.
If Daf also exists in the classical sense and is locally integrable, then Z)a/is also a
distribution in the sense of Section 6.11. The obvious consistency problem is whether
the equation
A) 'iTA, = A»,
always holds under these conditions.
More explicitly, the question is whether
B) (-1)M f f(x)(D^)(x) dx = f (i)aj)(x)<Kx) dx
for every ф е

144 DISTRIBUTIONS AND FOURIER TRANSFORMS
If/ has continuous partial derivatives of all orders up to N9 integrations by part
give B) without difficulty, if |a| < N.
In general, A) may be false. The following example illustrates this, in the case
12= 1.
6.14 Example Suppose Q, is a segment in R, and/is a left-continuous function
of bounded variation in Q. If D = d/dx, it is well known that (Df)(x) exists a.e. and
that DfeL1. We claim that
A) DAf = Л,
where fi is the measure defined in Q. by
B) Vila, *)) =f(b) -f(a).
Thus DAf = ADf if and only iffis absolutely continuous.
To prove A), we have to show that
for every ф е Щп), that is, that
C)
But C) is a simple consequence of Fubini's theorem, since each side of C) is equal to the
integral of ф\х) over the set
D) {(x,y):xeQ,yeQ,x<y}
with respect to the product measure of dx and d\x. The fact that ф has compact
support in Q. is used in this computation.
6.15 Multiplication by functions Suppose Л e @\U) and / e C°°(fi). The right
side of the equation
A) (/А)(ф) = А(/ф) [фе®(ОЯ
makes sense because /ф e @(Q) when ф e ^(O). Thus A) defines a linear functional
/Л on @(Q). We shall see that/Л is, in fact, a distribution in Q.
Observe that the notation must be handled with care: If/ e @(п), then Л/is a
number, whereas /Л is a distribution.
- The proof that/Л e &(Q) depends on the Leibniz formula
B)
valid for all/and g in C°°(O) and all multi-indices a, which is obtained by iteration of
the familiar formula
C) (mi?)' = u'v + uv'.

TEST FUNCTIONS AND DISTRIBUTIONS 145
The numbers ca/? 'are positive integers whose exact value is easily computed but is
irrelevant to our present needs.
To each compact К a Q, correspond С and N such that \Лф\ < С\\ф\\м for
all фе<Зк. By B), there is a constant C, depending on /, K, and N, such that
\\/ф\\м<С'\\ф\\м?огфе@к. Hence
D) ' , |(/Л)(ф)| <CC'U\\N {фе®к\
By Theorem 6.8,/Л e S)\u).
Now we want to show that the Leibniz formula B) holds with Л in place of g,
so that
E) D\fh) = £ сОГ-'Л^Л).
The proof is a purely formal calculation. Associate to each ue R" the function hu
defined by
hu(x) = exp (u • x).
Then Dahu = uahu. If B) is applied to hu and /г^ in place of/and g, the identity
F) , (u + v)a=Y< Cbfitf-'vP {ueR\ veRn)
P<a.
is obtained. In particular,
y<a
Hence
otherwise.
Apply B) to D%£Da~Y), and use G), to obtain the identity
(8) Z( —1)|д'са0^(Ф^а ~Pf) = (-1)|а|/^аф-
The point of all this is that (8) gives E). For if ф е Я(п), then

146 DISTRIBUTIONS AND FOURIER TRANSFORMS
)
6.16 Sequences of distributions Since О)\Щ is the space of all continuous linear
functionals on Щ£1), the general considerations made in Section 3.14 provide a
topology for @'(Q)—its weak*-topology induced by @(Q)—which makes @f(Q) into
a locally convex space. If {At} is a sequence of distributions in O, the statement
A) Л; -> Л in 0'(fl)
refers to this weak*-topology and means, explicitly, that
B) \\тА1ф = Аф [ф
In particular, if {ft} is a sequence of locally integrable functions in П, the state-
statements "// -* Л in £^'(Ц)" or "{/;} converges to Л in the distribution sense " mean that
C) lim f ф(х)Мх)с1
i->oo Jfi
х =
for every ф е
The simplicity of the next theorem, concerning termwise differentiation of a
sequence, is rather striking.
6.17 Theorem Suppose At e @'(Q)for i = 1, 2, 3,..., and
A) Лф = limA^
exists (as a complex number) for every ф e @(Q). Then A e &(Q.), and
B) DaA{r-> DaA in @'(п\
for every multi-index a.
proof: Let К be an arbitrary compact subset of Q. Since A) holds for every
фе@к, and since @K is a Frechet space, the Banach-Steinhaus theorem 2.8
implies that the restriction of Л to @K is continuous. It follows from Theorem
6.6 that Л is continuous on ^(Q); in other words, Л e <2)'(Q). Consequently A)
implies that
= (— l)'a| lim Л;(£)аф) = lim
j -♦ oo i -+ oo
6.18 Theorem TfAt^A in @r(U) andg^g in C°°(Q), then дгАг ->^Л in @'(п).
Note: The statement ** gt -> g in С°°(Д)" refers to the Frechet space topology of
C°°(O) described in Section 1.46.
proof Fix ф e 9(U). Define a bilinear functional В on C°°(Q) x $'(п) by

TEST FUNCTIONS AND DISTRIBUTIONS 147
.Then В is separately continuous, and Theorem 2.17 implies that
B(9i > Ai) -» B(9> A) as / -> oo.
Hence
Localization
6.19 Local equality Suppose Л£ e !3(Q) (i = 1, 2) and со is an open subset of Q.
The statement
A) At = A2 in со
means, by definition, that Л^ф =.Л2 Ф for every ф е ®(cd).
For example, if/is a locally integrable function and \i is a measure, then Лг = О
in со if and only if f(x) = 0 for almost every xeco, and Лд = 0 in со if and only if
ц(Е) = 0 for every Borel set E а со.
This definition makes it possible to discuss distributions locally. On the other
hand, it is also possible to describe a distribution globally if its local behavior is
known. This is stated precisely in Theorem 6.21. The proof uses partitions of unity,
which we now construct.
6.20 Theorem If Г is a collection of open sets in Rn whose union is Q, then there
exists a sequence {^J с £^(Д), with \^{ > 0, such that
(a) each фг has its support in some member ofT,
oo
(b) X $i{x) = 1 for every xeQ,
(c) to every compact К c П correspond an integer m and an open set W ^> К such that
A) *i(*) + -"+ *„(*) = !
for all xeW.
Such a collection {ф^ is called a locally finite partition of unity in Q, subordinate
to the open cover Г of Q. Note that it follows from (b) and (c) that every point of О
has a neighborhood which intersects the supports of only finitely many фг. This is the
reason for calling {ф^ locally finite.
proof Let S be a countable dense subset of Q. Let {BVB2,B3, ...} be a
sequence that contains every closed ball Bt whose center pt lies in 3, whose
radius r{ is rational, and which lies in some member of Г. Let Vt be the open
ball with center pt and radius rJ2. It is easy to see that О = (j Vt.

148 DISTRIBUTIONS AND FOURIER TRANSFORMS
The construction described in Section 1.46 shows that there are functions
4>i e 0(Q) such that ф; > 0, ф,- = 1 in Vt, фг = 0 off Bt. Define фг = фи and,
inductively,
B) ^i=O-W-(l-^l+i O'>1)-
Obviously, фь = 0 outside B{. This gives (a). The relation
C) ^1 + ... + ^J=i_(i_01)...(i_^)
is trivial when f = 1. If C) holds for some /, addition of B) and C) yields C)
with i + l in place of L Hence C) holds for every i. Since фг = 1 in Vi9 it
follows that
D) №) + ■••+Ы*)= I ifxeV1u--'-^Vm.
This gives (b). Moreover, if К is compact, then К a Vt и -- - и Vm for some
m9 and (c) follows. ////
6.21 Theorem Suppose Г is an open cover of an open set Q. с Rn9 and suppose that
to each coeT corresponds a distribution Л^ е Q}\co) such that
A) Лш, = Ла» in со' n w"
whenever со' п со" Ф 0.
Then there exists a unique Л е $'(п) such that
B) Л = Лю in со
for every со е Г.
proof Let {фг} be a locally finite partition of unity, subordinate to Г, as in
Theorem 6.20, and associate to each i a set со^еГ such that со{ contains the
support of фг.
If ф e ^(Q), then ф = £ Фг Ф- Only finitely many terms in this sum are
different from 0, since ф has compact support. Define
C) л^ = |
It is clear that Л is a linear functional on
To show that Л is continuous, suppose ф] -> 0 in @(Q). There is a compact
К a Q, which contains the support of every ф]. If m is chosen as in part (c) of
Theorem 6.20, then
D) * Лфj = f Лв|(!^у) U = 1, 2, 3,...).

TEST FUNCTIONS AND DISTRIBUTIONS 149
Since il/i4>j -> 0 in @(coi), asy-> oo, it follows from D) that Лфу -> О. By Theorem
6.6, A e ®\Cl).
To prove B), pick ф е 3>{to). Then
E) ф.фе^со.псо) (i = 1,2,3,...)
so that A) implies Л^.О/^ф) = Л^.ф). Hence
F) ' Лф = £Лт(^ф) -
which proves B).
This gives the existence of Л. The uniqueness is trivial since B) (with cot
in place of со) implies that Л must satisfy C). ////
Supports of Distributions
6.22 Definition Suppose Л e @\Q). If со is an open subset of п and if Лф = 0
for every ф e Q){co), we say that Л vanishes in со. Let W be the union of all open cocQ
in which Л vanishes. The complement of W (relative to Q) is the support of A.
6.23 Theorem If W is as above, then A vanishes in W.
proof W is the union of open sets со in which Л vanishes. Let Г be the collec-
collection of these w's, and let {\l/t} be a locally finite partition of unity in W, sub-
subordinate to Г, as in Theorem 6.20. If ф е Q){W), then ф = %фгф. Only finitely
many terms of this sum are different from 0. Hence
since each фг has its support in some со е Г. ////
The most significant part of the next theorem is (d). Exercise 20 complements it.
6.24 Theorem Suppose A e @f(Q) and SA is the support of A.
(a) If the support of some ф e @(Q) does not intersect SA, then Лф = 0.
(b) IfSA is empty, then A = 0. f •
(c) #фе С00(£2) and ф = 1 in some open set V containing SA, then фА = А.
(d) If SA is a compact subset ofQ, then A has finite order; in fact, there is a constant
С < oo and a nonnegative integer N such that
\4\<с\\ф\и
for every ф е Qj{Q). Furthermore, A extends in a unique way to a continuous
linear functional on C°°(Q).

150 DISTRIBUTIONS AND FOURIER TRANSFORMS
proof Parts (a) and (b) are obvious. If ф is as in (c) and if ф е @(£l), then the
support of ф — фф does not intersect SA. Thus Аф = А(фф) = (фА)(ф), by (a).
If »SA is compact, it follows from Theorem 6.20 that there exists ф е <3(Q)
that satisfies (c). Fix such a ^; call its support K. By Theorem 6.8, there exist
ct and N such that |Лф| < cJ^IU for all фе$к. The Leibniz formula shows
that there is a constant c2 such that \\фф\\м < с2\\ф\\н for every ф е
Hence
for every ф е
Since Лф = Л(^ф) for all ф е $(п), the formula
A) Л/=А(фЛ [/бСю(П)]
defines an extension of Л. This extension is continuous, for if ft -> 0 in C°°(O),
then each derivative of/f tends to 0, uniformly on compact subsets of Q; the
Leibniz formula shows therefore that ф/с-*0 in ^(Q); since Ле^'(A), it
follows that Л/, -> 0.
If/e C°°(£2) and if АГ0 is any compact subset of Q, there exists ф е @(п)
such that ф=/опК0. It follows that 0(fi) is dense in C°°(Q). Each Л е @'(п)
has therefore at most one continuous extension to C°°(Q). ////
Note: In (a) it is assumed that ф vanishes in some open set containing SA, not
merely that ф vanishes on SA.
In view of (b), the next simplest case is the one in which SA consists of a single
point. These distributions will now be completely described.
6.25 Theorem Suppose A e $Jf(Q), peQ, {/?} is the support of A, and A has order
N. Then there are constants ca such that
A) л= £ cjrsp9
|a|<JV
where Sp is the evaluation functional defined by
B) др{ф) = ф(р).
Conversely, every distribution of the form A) has p for its support (unless ca = 0
for all a).
proof It is clear that the support of Da5p is {/?}, for every multi-index a. This
- proves the converse.
To prove the nontrivial half of the theorem, assume that/? = 0 (the origin
of jRn), and consider а ф е @(Q) that satisfies
C) ФаФШ = 0 for all a with | a | < N
Our first objective is to prove that C) implies Аф — 0.

TEST FUNCTIONS AND DISTRIBUTIONS 151
Jf rj > 0, there is a compact ball К cz D, with center at 0, such that
D) \D^\<rjmK, lf\a\ = N.
We claim that
E) . |DVWI<^"Ml^r"|e| (xeK,\<x\<N).
When |aj~= N, this is D). Suppose 1 < / < N, assume E) is proved for
all a with |a| ?= /*, and suppose \fi\ = i — 1. The gradient of Орф is the vector
F) grad
Our inductioiThypothesis implies that
G) |(grad Dty)(x)\ < n • цпн-1\х\н-'х (x e K),
and since (Врф)@) — 0 the mean value theorem now shows that E) holds with
P in place of a. Thus E) is proved.
Choose an auxiliary function ф e Q)(Rn), which is 1 in some neighborhood
of 0 and whose support is in the unit ball В of R". Define
(8) ФМ ф
If r is small enough, the support of фг lies in rB cz K. By Leibniz' formula
(9) D"(W)W =
It now follows from E) that
A0)
as soon as r is small enough; here С depends on n and N.
Since Л has order N, there is a constant Q such that \Аф\ < Сг\\ф\\м for
all ф e Q)K. Since фг = 1 in a neighborhood of the support of Л, it now follows
from A0) and (c) of Theorem 6.24 that
Since ц was arbitrary, we have proved that Аф = О whenever C) holds.
In other words, Л vanishes on the intersection of the null spaces of the
functional Da50 (|a| < N), since
(И) AГдо)Ф = (-l)|a|
The representation A) follows now from Lemma 3.9. ////

152 DISTRIBUTIONS AND FOURIER TRANSFORMS
Distributions as Derivatives
It was pointed out in the introduction to this chapter that one of the aims of the theory
of distributions is to enlarge the concept of function in such a way that partial dif-
differentiations can be carried out unrestrictedly. The distributions do satisfy this
requirement. Conversely—as we shall now see—every distribution is (at least locally)
Da/ for some continuous function / and some multi-index a. If every continuous
function is to have partial derivatives of all orders, no proper subclass of the distribu-
distributions can therefore be adequate. In this sense, the distribution extension of the function
concept is as economical as it possibly can be.
6.26 Theorem Suppose A e @'(п), and К is a compact subset ofU. Then there is
a continuous function f in Q and there is a multi-index a such that
(i) A^ = (-i)
for every ф е $)K.
proof Assume, without loss of generality, that Kcz Q, where Q is the unit
cube in R", consisting of all л: = (xl9..., xn) with 0 < xt < 1 for i — 1,...,«.
The mean value theorem shows that
B)
xeQ
for i = 1,..., л. Put T = Dy D2 - • - Dn. For у e Q, let Q{y) denote the subset
of Q in which xt < yt A < / < «). Then
C)
If N is a nonnegative integer and if B) is applied to successive derivatives of ф>
C) leads to the inequality
D) 1№11л^тах|(Г»(х)|£ Г \(Тм+1ф)(х)\с1х,
xeQ JQ
for every ф е Qjq .
Since A e @'(Q), there exist N and С such that
E)
Hence D) shows that
F) | Л«£ I fS С f I (Т"+1ф)(х) \dx (ф

TEST FUNCTIONS AND DISTRIBUTIONS 153
By C); T is one-to-one on Q)Q, hence on <3K. Consequently, TN+1:
<3K-+Q)K is one-to-one. A functional Av can therefore be defined on the range
YofTN+l by setting
G) А,Т^1ф^Аф (фе@к)9
and F) shows that
(8) ""
The Hahn-Banach theorem therefore extends Ax to a bounded linear functional
on Ll(K). In other words, there is a bounded Borel function g on К such that
(9) Аф = А^+1ф = Г д(х)(Т»+1ф)(х) dx (ф e @K).
K
Define g(x) = 0 outside К and put
A0) /GO =["'••• f g(x) dx.--dxy (ye R").
Then/is continuous, and n integrations by parts show that (9) gives
A1) Аф = (-1)" f Пх)(Т»+2ф)(х) dx (ф е ®K\
This is A), with a = (N + 2,..., N -f 2), except for a possible change in sign.
When Л has compact support, the local result just proved can be turned into a
global one:
6.27 Theorem Suppose К is compact, V and Q are open in Rn, and К a V c= п.
Suppose also that Л е @'(п), that К is the support of A, and that A has order N. Then
there exist finitely many continuous functions fp in О {one for each multi-index f$ with
Pi<N+ 2 for i = 1,...,«) with supports in V, such that
0) Л
These derivatives are, of course, to be understood in the distribution sense: A)
means that
B) АФ = I (-1I f МхХ&фХх) dx [ф е
proof Choose an open set W with compact closure W, such that К с= W and
We V. Apply Theorem 6.26 with Win place of K. Put a = (N + 2,..., N + 2).

154 DISTRIBUTIONS AND FOURIER TRANSFORMS
)
The proof of Theorem 6.26 shows that there is a continuous function/in Q such
that
C) Л0 = (-1)
We may multiply/by a continuous function which is 1 on PFand whose support
lies in V, without disturbing C).
Fix ф е £^(Д), with support in W, such that ф = 1 on some open set
containing K. Then C) implies, for every ф е £^(П), that
5>,
This is B), with
//? = (-l)"-<%/-Da-'ty {p<a). ЦП
Our next theorem describes the global structure of distributions.
6.28 Theorem Suppose A e &(Q). There exist continuous functions ga in Q, one
for each multi-index a, such that
(a) each compact К с Q intersects the supports of only finitely many ga, and
(b) Л =
If A has finite order, then the functions дл can be chosen so that only finitely many
are different from 0.
proof There are compact cubes Qt and open sets Vx (i = 1, 2, 3,...) such that
Qt a Vtcz O, Q is the union of the Qt, and no compact subset of Q intersects
infinitely many V-t. There exist 0fe^(Ff) such that ф{ = 1 on Qt. Use this
sequence {<^J to construct a partition of unity {фг}, as in Theorem 6.20; each
ф i has its support in V{.
Theorem 6.27 applies to each ф-ьА. It shows that there are finitely many
continuous functions fia with supports in Vi9 such that
A) M =
Define
B) - g.=

TEST FUNCTIONS AND DISTRIBUTIONS 155
These 'sums are locally finite, in the sense that each compact Кап
intersects the supports of only finitely many Да. It follows that each ga is
continuous in п and that (a) holds.
Since ф = Yj{I/i Ф> ^ог everY Ф е ^(Д)>we have Л = £ ^-Л, and therefore
A) and B) give (b).
The final assertion follows from Theorem 6.27. ////
Convolutions
Starting from convolutions of two functions, we shall now define the convolution of a
distribution and a test function and then (under certain conditions) the convolution of
two distributions. These are important in the applications of Fourier transforms to
differential equations. A characteristic property of convolutions is that they commute
with translations and with differentiations (Theorems 6.30, 6.33, 6.37). Also, dif-
ferentiatons may be regarded as convolutions with derivatives of the Dirac measure
(Theorem 6.37).
It will be convenient to make a small change in notation and to use the letters
w, v,... for distributions as well as for functions.
6.29 Definitions In the rest of this chapter, we shall write Of and 9)' in place
of @{Rn) and @'(Rn). If и is a function in Rn, and x e R\ xxu and й are the func-
functions defined by
A) (т,и)(у) = и(у-*), й{у) = и{-у) (ye IF).
Note that
B) (?xu)(y) = u(y -x) = u(x - y).
If и and v are complex functions in Rn, their convolution и * v is defined by
C) (u*v)(x)= f u(y)v(x - y) dy,
Rn
provided that the integral exists for all (or at least for almost all) x e R", in the Lebesgue
sense. Because of B),
D) («•»)(*)= f u{y)(txv)(y)dy.
This makes it natural to define
E) (и * ф)(х) = и(тхф) (иеЯ',фе@,хе Rn),
for if и is a locally integrable function, E) agrees with D). Note that и * ф is a function.

156 DISTRIBUTIONS AND FOURIER TRANSFORMS
)
The relation J (тхи) • v = J и • (t-xv), valid for functions и and v, makes it
natural to define the translate тх и of и е $}' by
F) (гхи)(ф) = и(т.хф) {фе®,хе R").
Then, for each x e Rn, тх и е Яд'; we leave the verification of the appropriate continuity
requirement as an exercise.
6.30 Theorem Suppose ue $', ф е Я, ф е@. Then
(a) тх(и * ф) = (tx u) * ф = и * (тх ф)/ог all x e Rn;
(b) и*феСсо and
D\u *ф) = (Dau) * ф = и * (В"ф)
for every multi-index a ;
(c) и * (ф * ф) = (и * ф) * \j/.
proof For any у е Rn,
(tx(u * ф))(у) = (м * ф)(у - a;) = и(ту-хф),
((TXU) * ф)(у) = (ТЯЕ/)(Т7Й = И(Т7.ЯЙ,
{и * (тхф))(у) = и(ту(тхфУ) = и(ту_хф),
which gives (a); the relations
т,т_я = т7_х and (тхфУ = т_хф
were used. In the sequel, purely formal calculations such as the preceding ones
will sometimes be omitted.
If и is applied to both sides of the identity
A) W
one obtains part of (&), namely,
To prove the rest of (b), let e be a unit vector in R", and put
B) |/г = г(т0-тгв) (r>0).
Then (л) gives
C) Пг(и*ф) = и*{г}гф).
As r -> 0, г]гф -* £>еФ in ^> where Z)e denotes the directional derivative in the
direction e. Hence

TEST FUNCTIONS AND DISTRIBUTIONS 157
for each x e R", so that
D) lim (u * (rjr ф))(х) = (u*(De
r->0
By C) and D) we have
E) . Пе(и*ф) = и*(Веф)
and iteration of E) gives (b).
To prove, (c), we begin with the identity
F) (Ф*ФШ=
JRn
Let Ki and K2 be the supports of ф and ф. Put K = Kx + K2. Then
is a continuous mapping of R" into Q)K, which is 0 outside K2. Therefore F)
may be written as a ^-valued integral, namely,
G) (Ф*ФГ= f
•: K
and now Theorem 3.27 shows that
= f $(s)u(xs$) ds = f l|/(-sXu*ф)(s)dst
JK2 JRn
or
(8) (и *(ф
То obtain (8) with jc in place of 0, apply (8) to т-хф in place of ф, and
appeal to (a). This proves (c). ////
6.31 Definition The term approximate identity on Rn will denote a sequence
of functions hj of the form
bj(x)=fh(jx) G=1,2,3,...),
where h e 3)(Rn), /z > 0, and $Rn h(x) dx = \.
6.32 Theorem Suppose Щ} is an approximate identity on Rn, ф е @,andue $)'.
Then
(a) lim ф * hj = ф in @,
j-*co
(b) lim и * hj — и in $)'.
j-*cc

158 DISTRIBUTIONS AND FOURIER TRANSFORMS
Note that (b) implies that every distribution is a limit, in the topology of 9)\ of a
sequence of infinitely differentiable functions.
proof It is a trivial exercise to check that /* hj->f uniformly on compact
sets, if/ is any continuous function on Rn. Applying this to Иаф in place of/,
we see that £>а(ф * hj) -> £)аф uniformly. Also, the supports of all ф * hj lie in
some compact set, since the supports of the hj shrink to {0}. This gives (a).
Next, (a) and statement (c) of Theorem 6.30 give (b), because
и(ф) = (и * ф)@) = lim (и * (hj * ф))ф)
= lim ((и * hj) * 0)(O) = lim (u * hj)($). ////
6.33 Theorem
(a) If и е 9'and
A) Ьф = и*ф (фе@)9
then L is a continuous linear mapping of 13 into C°° which satisfies
B) txL = Ltx (xeRn).
(b) Conversely, ifL is a continuous linear mapping of 9) into C(Rn), and ifL satisfies
B), then there is a unique и e QI such that A) holds.
Note that (b) implies that the range of L actually lies in C00.
proof (a) Since xx(u * ф) = и * (тхф), A) implies B). To prove that L is
continuous, we have to show that the restriction of L to each 9)K is a continuous
mapping into C00. Since these are Frechet spaces, the closed graph theorem can
be applied. Suppose" that ф^ф in Q)K and that и * фi->fm С00; we have to
prove that/= и * ф.
Fix x e Rn. Then тхфг -+тхф in ^, so that
f(x) = lim (м * фг)(х) = lim и(тл&) = и(тхф) = (и * ф)(х).
(b) Define и(ф) = (L<^)@). Since ф -> ф is a continuous operator on ^, and
since evaluation at 0 is a continuous linear functional on С, и is continuous on
$J. Thus и e Qf\ Since L satisfies B),
(Ьф)(х) = (T_X
The uniqueness of w is obvious, for if ue & and и* ф = 0 for every
ф g 0, then
!/(£) = (и*ф)@)=0
for every ф e $); hence w = 0. ////

TEST FUNCTIONS AND DISTRIBUTIONS 159
6.34 Definition - Suppose now that wef and that и has compact support. By
Theorem 6.24, и extends then in a unique fashion to a continuous linear functional on
C°°. One can therefore define the convolution of и and any ф е С00 by the same
formula as before, namely,
(и*ф)(х) = и(тхф) (xeR").
6.35 Theorem Suppose ue$)' has compact support, and ф е C°°. Then
(a) тх(и * ф) = (тхи\* ф = и*(тхф) ifx e Rn,
(b) иъфеС™ and
D\u * ф) = (Dau) * ф = и * (В*ф).
If in addition, ф е Q>, then
(c) и * ф e Q), and
(d) и* (ф *ф) = (u * ф) * ф — (u * i^) * ф.
proof The proofs of (a) and (b) are so similar to those given in Theorem 6.30
that they need not be repeated. To prove (c), let К and H be the supports of
и and ф, respectively. The support of ххф is x — H. Therefore
unless К intersects x — H, that is, unless x e K-\- H. The support of и * ф thus
lies in the compact set К + H.
To prove (d), let W be a bounded open set that contains K, and choose
ф0 e & so that ф0 = ф in Ж + Я. Then (ф * i^)v = (ф0 * i/0v in W, so that
A) («*W
If —s e H, then т3ф = ts$0 in W; hence и * ф — и * ф0 in — H. This gives
B) ((и * ф) * ДО0) = ((« * Ф
Since the support of и * ф lies in jST + 77,
C) ((и*ф
The right sides of A) to C) are ^qual, by Theorem 6.30; hence so are
their left sides. This proves that the three convolutions in (d) are equal at the
origin. The general case follows by translation, as at the end of the proof of
Theorem 6.30. ////
6.36 Definition If и e &', v e $)', and at least one of these two distributions has
compact support, define
A) Ьф = и * О * ф) (фе $).

160 DISTRIBUTIONS AND FOURIER TRANSFORMS
Note that this is well defined. For if v has compact support, then v * ф e Q), ajnd
ЬфеС^; if и has compact /Support, then again ЬфеС™, since г;*феС°°. Also,
txL = Lt x, for all x e Rn. These assertions follow from Theorems 6.30 and 6.35.
The functional ф -»(Ьф)@) is in fact a distribution. To see this, suppose ф{ -»0
in Q). By (a) of Theorem 6.33, v * <^ -> 0 in C00; if, in addition, v has compact support
then v * 4>i -» 0 in ^. lt follows, in either case, that (Ьф^ф) -» О.
The proof of F) of Theorem 6.33 now shows that this distribution, which we
shall denote by и * v, is related to L by the formula
B) Ьф = (w * v) * ф (ф g 0).
In other words, w * v e $)' is characterized by
C) (u * v) * ф = w * (v * ф) (ф g 0).
6.37 Theorem Suppose и e $)', v e Q)\ w e $)'.
(a) If at least one of u, v has compact support, then и * v — v * u.
(b) IfSu and Sv are the supports of и and v, and if at least one of these is compact, then
(c) If at least two of the supports Su, Sv, Sw are compact, then (u * v) * w = и * (v * w),
(d) If Ь is the Dirac measure and a is a multi-index, then
Dau = (Dad) * u.
In particular, и = 3 * и.
(e) If at least one of the sets Su, Sv is compact, then
Da(u *v) = (Dau) *v = u* (Dav)
for every multi-index a.
Note: The associative law (c) depends strongly on the stated hypotheses; see
Exercise 24.
proof (a) Pick ф g О), ф е @. Since convolution of functions is commutative,
(c) of Theorem 6.30 implies that
(u * v) * (ф * ф) = и * (v * (ф * ф))
= и * ((v * ф) * ф) = и * (ф * (v * ф)).
If Sv is compact, apply (c) of Theorem 6.30 once more; if Su is compact, apply
{d) of Theorem 6.35; in either case
A) (и * v) * (ф * ф) = (u * I/O * (i; * ф).

TEST FUNCTIONS AND DISTRIBUTIONS 161
Since ф * ф = ф * ф, the same computation gives
B) (v * u) * (ф * ф) = (v * ф) * (и * ф).
The two right members of A) and B) are convolutions of functions (one
in *?, one in C00); hence they are equal. Thus
C) - ((u * v) * ф) * ф — ((v * и) * ф) * ф.
Two applications of the uniqueness argument used at the end of the proof of
Theorem 6.33 new give и * v = v * u.
(b) If ф е 9), a simple computation gives
D) (i/ ^
By (a) we may assume, without loss of generality, that Sv is compact. The proof
of (c) of Theorem 6.35.shows that the support of v * ф lies in Sv — £^. By D),
(w * г)(ф) = 0 unless 5И intersects S^,- Sv, that is, unless 5^ intersects 5Ы + Sv.
(c) We conclude from (b) that both
(w * v) * w and w * (v * w)
are defined if at most one of the sets Su, Sv, Sw fails to be compact. Jf ф е 13, it
follows directly from Definition 6.36 that
E) (u * (v * w)) * ф = u* ((v * w) * ф) = и * (v * (w * ф)).
If £w is compact, then
F) ((u * y) * w) * ф = (w * v) * (w * ф) = и * (у * (w * ф))
because w * ф е @,Ъу (с) of Theorem 6.35. Comparison of E) and F) gives (c)
whenever Sw is compact.
lfSw is not compact, then Su is compact, and the preceding case, combined
with the commutative law (a), gives
и * (v * w) = и * (w * v) — (w * v) * и
= w * (v * u) = w * (w * y) = (u * f) * H\
(<f) If ф e @, then S * ф = ф, because
E * <£)(*) = 5(t^) = (rJ)@) = ф(-х) = <£(*).
Hence (c) above and (b) of Theorem 6.30 give
(Dau) *ф = и*Пасф = и* D\d *ф) = и* {D*d) * ф.
Finally, (e) follows from (d), (c), and (a):
Щи * v) = G)^) * (M * i?) = ((/)а^) *«)*i; = (/)aw) * г;
and
((ITS) *u)*v = (u* D*S) * v = w * /

162 DISTRIBUTIONS AND FOURIER TRANSFORMS
Exercises
1 Suppose / is a complex continuous function in R", with compact support. Prove that
фР3 ->/uniformly on Rn, for some ф е @ and for some sequence {Pj} of polynomials.
2 Show that the metrizable topology for Q)(Q) that was rejected in Section 6.2 is not com-
complete for any O.
3 If Eh an arbitrary closed subset of R", show that there is an/e Cx(Rn) such that f(x) = 0
for every x e E and f(x) > 0 for every other x e R".
4 Suppose A g $)'{Q) and Аф>0 whenever ф e ^(O) and </• > 0. Prove that A is then a
positive measure in О (which is finite on compact sets).
5 Prove that the numbers caP in the Leibniz formula are
С-'=Д/?,1(Г,'-0,)!
6" («) Suppose cm = exp {—(aw!)!}, w = 0, 1, 2,.... Does the series
converge for every ф е C°°(jR)?
(^) Let О be open in Rn, suppose Л£ e @'(Cl\ and suppose that all At have their supports
in some fixed compact К с О. Prove that the sequence {AJ cannot converge in
9>'{£l) unless the orders of the Aj are bounded. Hint: Use the Banach-Steinhaus
theorem.
(c) Can the assumption about the supports be dropped in (b) ?
7 Let О = @, oo). Define
Prove that A is a distribution of infinite order in O. Prove that A cannot be extended
to a distribution in R; that is, there exists no Ao e 3)'{R) such that Ao = A in @, oo).
8 Characterize all distributions whose supports are finite sets.
9 (a) Prove that a set E<=@(Q) is bounded if and only if
sup{|A^|: фе E) < oo
for every A e @'(п).
(b) Suppose {<Д|} is a sequence in £^(П) such that {A^j} is a bounded sequence of numbers,
- for every A e ^'(^)- Prove that some subsequence of {</»j} converges, in the topology
(c) Suppose {Aj} is a sequence in Я)'{£1) such that {Aj ф} is bounded, for every ф е
Prove that some subsequence of {Aj} converges in $)'($$) and that the convergence is
uniform on every bounded subset of ^(O). Hint: By the Banach-Steinhaus theorem,
the restrictions of the A3 to Q)K are equicontinuous. Apply Ascoli's theorem.

TEST FUNCTIONS AND DISTRIBUTIONS 163
10 Suppose {ft} is a sequence of locally integrable functions in О (an open set in Rn) and
lim f \Mx)\dx = O
for every compact К <= П. Prove that then Dafi ->0 in @'(Q), as / -> oo, for every multi-
index a.
11 Suppose О is open in R2, and {/J is a sequence of harmonic functions in О that con-
converges in the distribution sense to some Л е Q)\D)\ explicitly, the assumption is that
= lim f Л(х)ф(х) dx [ф е
Prove then that {/)} converges uniformly on every compact subset of О and that Л is a
harmonic function. Hint: If/is harmonic, f(x) is the average of/over small circles
centered at x.
12 Recall that 8 (the Dirac measure) is the distribution defined by 8(ф) = ф@), for ф e 3)(R).
For which/e C™{R) is it true that/S' = 0? Answer the same question for/S". Conclude
that a function /e C™(R) may vanish on the support of a distribution Л е ^'(i?) al-
although/Л^О.
75 If ^ e 0(fi) and Л е ^'(^X does either of the statements
imply the other?
14 Suppose К is the closed unit ball in Rn, Л e Q)\Rn) has its support in K, and /e C00^")
vanishes on AT. Prove that /Л = 0. Find other sets К for which this is true. (Compare
with Exercise 12.)
15 Suppose К с: Кс П, ^ is compact, К and О are open in Rn, Л е Q)f{fl) has its support
in K9 and {</»,} c^(Q) satisfies
lim [sup |(/)^i)W| =0
(a)
for every multi-index a. Prove that then
Jim МФд = 0.
i-юо
16 The preceding statement becomes false if К is replaced by К in the hypothesis (a). Show
this by means of the following example, in which О = R. Choose cx > c2 > ' • * > 0,
such that 2 C] < °°; define
Лс£ = 2 @fe/) - <£@)) (^ e @{R))\
and consider functions ф( e 3){R) such that ф{(x) = 0 if л: <ct+1^i(x) = l//if cf <л: <Ci.
Show also that this Л is a distribution of order 1.
However, for certain K, V can be replaced by К in the hypothesis (a) of Exercise

164 DISTRIBUTIONS AND FOURIER TRANSFORMS
15. Show that this is so when К is the closed unit ball of Rn. Find other sets К for which
this is true.
17 If Ae@'(R) has order N, show that Л = DN + 2f, for some continuous function /. If
Л = 8, what are the possibilities for/?
18 Express 8 e Q)'(R2) in the form given by Theorem 6.27, as explicitly as you can.
19 Suppose Л e Q)f(u), ф e ^(O), and (Da(j>)(x) = 0 for every jc in the support of Л and for
every multi-index a. Prove that Л^ == 0. Suggestion: Do it first for distributions with
compact support, by the method used in Theorem 6.25.
20 Prove that every continuous linear functional on С°°(П) is of the form/-»A/, where Л
is a distribution with compact support in O; this is a converse to (d) of Theorem 6.24.
21 Let C™(T) be the space of all infinitely differentiable complex functions on the unit circle
T in <Z. One may regard С°°(Г) as the subspace of C™(R) consisting of those functions
that have period 2тг. Suppose
/(z)= 2*-*"
n = 0
converges in the open unit disc U in (Z. Prove that each of the following three properties
of/implies the other two:
(а) There exist p < со and у < со such that
\an\<yn" (/1 = 1,2,3,...)-
(б) There exist /? < со and у < со such that
(c) linw \п-п/{геьв)ф{е1в) d9 exists (as a complex number) for every ^ e С^
22 For и е ^СЮ, show that
и — rxu
>Du in
as x ->0. (The derivative of м may thus still be regarded as a limit of quotients.)
23 Suppose {/} is a sequence of locally integrable functions in Rny such that
Jim (/ * ф)(х)
exists, for each ф e @(Rn) and each x e jR". Prove that then {Da(fi * ф)} converges uni-
uniformly on compact sets, for each multi-index a.
24 Let H be the Heaviside function on i?, defined by
if*>0,
and let 8 be the Dirac measure.
(а) Show that (Я * ф)(х) = Jloo ^(j) Л, if ^ e
(б) Show that 8' * Я = 8.
(с) Show that 1 * 8' = 0, (Here 1 denotes the locally integrable function whose value is
1 at every point and which is thought of as a distribution.)

TEST FUNCTIONS AND DISTRIBUTIONS 165
(d) It follows that the associative law fails:
1 *(8'*H) =
but
25 Here is another characterization of convolutions analogous to Theorem 6.33. Suppose L
is a continuous linear mapping of Q) into C00 which commutes with every Da, that is,
(a) '
Then there is а и е £^' such that
Suggestion: Fix феЗ)у put
let De be the directional derivative used in the proof of Theorem 6.30, and show that
(Deh)(x) = (ПеЬтхф)(х) - (LtxПеф)(х),
whichis 0 if (a) holds. Thus h(x) = /z@), which implies that rxL =Ltx.
Can the assumption that the range of L is in C00 be weakened?
26 If /e L4( — oo, — 8) u (S, oo)) for every 8 > 0, define its principal value integral to be
f
f /(jc) Лс = lim ( f + f )f(x) dx,
if the limit exists. For ^ e ^(i?), put
Show that
dx.
27 Find all distributions и е $)'(Rn) that satisfy at least one of the following two conditions:
(а) тхи = и for every jcei?",
(б) D*u = 0 for every a with |a| = 1.

7
FOURIER TRANSFORMS
Basic Properties
7.1 Notations (a) The normalized Lebesgue measure on Rn is the measure
mn defined by
dmn(x) = Bnyn/2 dx.
The factor Bя)"л/2 simplifies the appearance of the inversion theorem 7.7 and the
Plancherel theorem 7.9. The usual Lebesgue spaces Lf, or LP(Rn), will be normed by
means of mn:
H/llP = jJ^I/|^mnJ A<;р<оо).
It is also convenient to redefine the convolution of two functions on Rn by
(/* ff)(x) = f /(* - y)g{y) dmn{y)
R"
whenever the integral exists.
(b) For each t e Rn, the character et is the function defined by
et(x) = eif'x = exp {i'(f 1*1 + ••• + /„xn)} (x e Rn).

FOURIER TRANSFORMS 167
Each et_ satisfies the functional equation
et(x + y) = et(x)et{y).
Thus et is a homomorphism of the additive group Rn into the multiplicative group
of the complex numbers of absolute value 1.
(c) The Fourier transform of a function/e l}(Rn) is the function /defined by
, /@= f fe-tdmn (teRn).
Rn
The term "Fourier transform" is often also used for the mapping that takes/to/.
Note that
(d) If a is a multi-index, then
The use of Da in place of Da simplifies some of the formalism. Note that
where, as before, f = r^1 • • • fnn. If P is a polynomial of « variables, with complex
coefficients, say
the diiferential operators P(D) and P{ — D) are defined by
It follows that
P(D)et=P(t)et (teRn).
(e) The translation operators тл are defined, as before, by
7.2 Theorem Suppose f, g e L\Rn), x e R". Then
(c) (/*дГ = /д-
(d) IfX>0 and h(x) = /(*/A), ?Aot й@ = Xnf(Xt).

168 DISTRIBUTIONS AND FOURIER TRANSFORMS
proof It follows from the definitions that
(т,/Г@ = J(*x/)-e-, = jf'x-'e-
and
An application of Fubini's theorem gives (c); (d) is obtained by a linear change
of variables in the definition of/. ////
7.3 Rapidly decreasing functions This name is sometimes given to those
fe СЮ(ЯЯ) for which
A) sup sup(l + \x\2)N\(Daf)(x)\ <oo
for N = 0, 1, 2,... . (Recall that | x |2 = 5] *f.) In other words, the requirement is
that P - Daf is a bounded function on JR", for every polynomial P and for every multi-
index a. Since this is true with A + \x\2)NP(x) in place of P(x), it follows that every
P-^/lies in L1^").
These functions form a vector space, denoted by S?n, in which the countable
collection of norms A) defines a locally convex topology, as described in Theorem
1.37.
It is clear that @(R") с <fn.
7.4 Theorem
(a) £fn is a Frechet space.
(b) If P is a polynomial, g e Уп, and a is a multi-index, then each of the three
mappings
f^Pf, f^gf, f-+Daf
is a continuous linear mapping of'£fn into £fn.
(c) Iffe Уп and P is a polynomial, then
(P(D)fr = Pf and (Pfr = P(- D)f.
(d) The Fourier transform is a continuous linear mapping of ^n into Sfn.
[Part (d) will be strengthened in Theorem 7.7.]
proof (a) Suppose {/]•} is a Cauchy sequence in S?n. For every pair of multi-
indices a and p the functions xpDY^x) converge then (uniformly on Rn) to a
bounded function gaP, as / -> oo. It follows that
and hence that/f -*g00 in £fn. Thus Sfn is complete.

FOURIER TRANSFORMS 169
(b) If/e^'„, it is obvious that Dafe £fn, and the Leibniz formula implies
that Pf and gf are also in £fn. The continuity of the three mappings is now an
easy consequence of the closed graph theorem.
(c) If/e Уп, so is P(D)f9 by ф\ and
(P(D)f) *et = f* P(D)et = /* P(f)e, = P(t)[f* et].
Evaluation of these functions at the origin of Rn gives the first part of
(c), namely,
If / = (tlv...., tn) and I' = (ft + e, r2,..., <„)> e # 0, then
The dominated convergence theorem can be applied, since XifeL1, and yields
1
-7-/@= f x
This is the case P(x) = xl of the second part of (c); the general case follows by
iteration.
(d) Suppose fe^n and g(x)} = (- \)^xj{x). Then ^ e ¥n; now-(c)
implies that g = Z>e/ and P • Daf= P • g = (P(D)gY, which is a bounded
function, since Д/% e L)(Rn). This proves that /e £?n. If/£ -> / in У>п, then
fi->fin L1^"). Therefore/£@-*/@ for a11 ^e^"- That/-*/is a continuous
mapping of £fn into Уп follows now from the closed graph theorem. ////
7.5 Theorem Iffe Ll(R"), thenfe C0{Rn\ and \\f\\m < \\f\\,.
Here C0(Rn) is the supremum-normed Banach space of all complex continuous
functions on R" that vanish at infinity.
proof Since |tf,(X)| = 1, it is clear that
0) |/@l <H/lli (feL\teRn).
Since Q>(Rn) с 5^и, ¥n is dense in L\Rn). To each/e Ll(Rn) correspond func-
functions ft e ¥n such that ||/ - f^ -> 0. Since / 6 ^я с С0(^и) and since A)
implies that /f ->/ uniformly on ,R", the proof is complete. ////
The following lemma will be used in the proof of the inversion theorem. It
depends on the particular normalization that was chosen for mn.

170 DISTRIBUTIONS AND FOURIER TRANSFORMS
7.6 Lemma If фп is defined on Rn by
О) «*) = exp{-i|x|2}
then4>ne$fn, фп = фп, and
B) ФМ= Г kdmn.
JRn
proof It is clear that фп е Уп. Since ф1 satisfies the differential equation
C) / + xy = 0,
a short computation, or an appeal to (c) of Theorem 7.4, shows that фг also
satisfies C). Hence ^/фх is a constant. Since фх(О) — 1 and
&@) = f Ф, dm, = B7Г)-1/2 f" exp{-ijc2} dx = 1,
we conclude that ^ = ф1. Next,
D) ф„(д:) = ^(х1)---ф1(х„) (xeJP)
so that
E) 4@ = <to---to a e no-
it follows that ф„ = ф„ for all л. Since $„@) = J ф„ dmn, by definition, and
since фп = ф„, we obtain B). ////
7.7 The inversion theorem
(a) Ifge<?n,then
A) g(x)= f §exdmn (x e R").
(b) The Fourier transform is a continuous, linear, one-to-one mapping of 9>n onto
«^л» of period A, whose inverse is also continuous.
(c) Iffe Ll (Rn), fe L\R"), and
B) /0(x)= f fexdmn (xeR"),
JRn
thenf(x) = fo(x) for almost every x e R".
proof If/and g are in Ll(R"), Fubini's theorem can be applied to the double
integral
f f f(x)g(y)e-ix-ydmn(x)dmn(y)
JRn JRn
to yield the identity
C) f fgdmn= f fgdmn.
JRn JRn

FOURIER TRANSFORMS 171
To prove part (a), take ge^n, ф e £fn, f(x) = ф(х/Х), where X > 0.
By (d) of Theorem 7.2, C) becomes
f g(t)Xn$(Xt) dmn(t) = \ ф(^\д(у) dmn(y\
JRn JRn \A/
or
D) г "йг(т)^(О dmn(t) = I ф[-\д{у) dmn{y).
As Д-> oo, g(t/X)->g(O) and ф(у/Л) -+ ф@), boundedly, so that the dominated
convergence theorem can be applied to the two integrals in D). The result is
E) fl@) f $<1т„ = ф@)\ gdmn
JRn JRn
If we specialize ф to be the function фп of Lemma 7.6,. E) gives the case x = 0
of the inversion formula A). The general case follows from this, since (a) of
Theorem 7.2 yields
= (*-,0X0)= f (T-xgTdmn= f 0
This completes part (a).
To prove part (b), we introduce the temporary notation Ф# = д. The
inversion formula A) shows that Ф is one-to-one on £fn, since g = 0 obviously
implies g = 0. It also shows that
F) Ф2# = <7
where, we recall, g(x)—g(—x), and hence that Ф4д — g. It follows that Ф
maps £fn onto £fn. The continuity of Ф has already been proved in Theorem
7.4. To prove the continuity of Ф, one can now either refer to the open
mapping theorem or to the fact that Ф = Ф3.
To prove (c), we return to the identity C), with g e S?n. Insert the inversion
formula A) into C) and use Fubini's theorem, to obtain
G) f fo0dmn=( f$dmn {geSTa).
JRn +-JRn
By (b), the functions ff cover all of S?n. Since @(Rn) с ¥п, G) implies that
(8) f (fo-f№dmn = 0
JRn
for every ф e !3(Rn), hence (by a uniform approximation described in Exercise 1
of Chapter 6) for every continuous ф with compact support. It follows that
/o-/ = 0a.e. ■ ////

172 DISTRIBUTIONS AND FOURIER TRANSFORMS
7.8 Theorem Iffe 9>n andge£fn, then
(a) f*ge<7n,and
proof By (c) of Theorem 7.2, (/* дУ = fg, or
(О Ф{/*д) = Ф/-Фд,
in the notation used in the proof of (b) of Theorem 7.7. With / and $ in place
of/and g, A) becomes
B) ф(/ * g) = ф2/- Ф2# = JS = Ш" = Ф2Ш-
Now apply Ф to both sides of B) to obtain (b). Note that fg e £fn; hence
F) implies that/* g e £fn, and this gives (a), since the Fourier transform maps
&nonto9>n. ЦП
7.9 The Plancherel theorem There is a linear isometry 4? of L2(Rn) onto l}{Rn)
which is uniquely determined by the requirement that
4>f = f for every feSTn.
Observe that the equality xF/= /extends from Sfn to L1 n I}, since Sfn is dense
in L2 as well as in L1. This gives consistency: The domain of Ш is L2, /was defined in
Section 7.1 for all/e L1, and Ч1/ = /whenever both definitions are applicable. Thus
*F extends the Fourier transform from L1 n L2 to L2. This extension Ч1 is still called
the Fourier transform (sometimes the Fourier-Plancherel transform), and the notation
/will continue to be used in place of *F/, for any/e L?(Rn).
proof If/and g are in 9?n, the inversion theorem yields
f fidrn,- f 3(x)rfmn(x) f f{t)eixt dmn(t)
JRn JRn JRn
= f ?(t)dmn(t)l g(x)eixldmn(x).
JRn JRn
The last inner integral is the complex conjugate of g(t). We thus get the Parseval
formula
0) f fgdmn=\ f§dmn
Ifg=f9 A) specializes to
B) ' Il/L = Il/L

FOURIER TRANSFORMS 173
Note that 9>n is dense in L2(Rn), for the same reason that £fn is dense in
l}(Rn). Thus B) shows that/-»/is an isometry (relative to the L2-metric) of
the dense subspace £?n ofL2(Rn) onto 9)n. (The mapping is onto by the inversion
theorem.) It follows, by elementary metric space arguments, that /-* / has a
unique continuous extension W: L2(Rn) -»l}(Rn) and that this 4* is a linear
isometry onto l}(Rn). Some details of this are given in Exercise 13. ////
It should be noted that the Parseval formula A) remains true for arbitrary /
and g in L2(Rn).
That the Fourier transform is an L2-isometry is one of the most important
features of the whole subject.
Tempered Distributions
Before we define these, we establish the following relation between $fn and Q>(Rn).
7.10 Theorem
(a) &ф) is dense in &>n.
(b) The identity mapping ofQ){Rn) into 9>n is continuous.
These statements refer, of course, to the usual topologies of Q}(Rn) and S?n,
as defined in Sections 6.3 and 7.3.
proof (a) Choose fe 9>n, ф e 3){Rn) so that ф = 1 on the unit ball of Rn,
and put
A) fM = Лх)Ф(гх) (х eR\r> 0).
Then fr e <2){Rn). If P is a polynomial and a is a multi-index, then
P(x)D«{f- fr)(x) = P(x) X cafi(ir-'f)(x)rMD'[l - ф(гх)].
Our choice of ф shows that Dfi[l - ф(гх)] = О for every multi-index ft when
|jc| < 1/r. Since/e Sfn, we haye P • Da~pfe C0(Rn) for all j8 < a. It follows
that the above sum tends to 0, uniformly on Rn, when r -»0. Thus fr ~> / in
S?n, and (a) is proved.
(b) If ^ is a compact set in Rn, the topology induced on Q)K by $fn is
clearly the same as its usual one (as defined in Section 1.46), since each
A + | x\ 2)N is bounded on K. The identity mapping of $K into 9?n is therefore
continuous (actually, a homeomorphism), and now (b) follows from Theorem
6.6 Illl

174 DISTRIBUTIONS AND FOURIER TRANSFORMS
7.11 Definition If i: 3>(Rn)-> Sfn is the identity mapping, if L is a continuous
linear functional on Уп, and if
A) uL = Loi
then the continuity of / (Theorem 7.10) shows that uLeQ}\Rn)\ the denseness of
Q){Rn) in У>п shows that two distinct L's cannot give rise to the same u. Thus (L)
describes a vector space isomorphism between the dual space $f'n of Sfn, on the one
hand, and a certain space of distributions on the other. The distributions that arise
in this way are called tempered:
The tempered distributions are precisely those и е Q}\Rn) that have continuous
extensions to Sfn.
In view of the preceding remarks, it is customary and natural to identify uL
with L. The tempered distributions on Rn are then precisely the members ofSf'n.
The following examples will explain the use of the word "tempered" in this
connection; it indicates a growth restriction at infinity. (See also Exercise 3.)
7.12 Examples (a) Every distribution with compact support is tempered. Suppose
К is the compact support of some и e @'(Rn\ fix \j/ e Q}{Rn) so that \j/ = 1 in some open
set containing K, and define
A) п(/) = и(ф/) (feSTJ.
If/г -* 0 in Уп, then all Daft -> 0 uniformly on R", hence all D'tyft) -> 0 uniformly on
R", so that ф/i -> 0 in 3){Rn). It follows that й is continuous on &n. Since й(ф) = и(ф)
for ф е Q){Rn), й is an extension of u.
(b) Suppose \i is a positive Borel measure on Rn such that
B) f (l+\x\2rkdtix)<*>
JRn
for some positive integer к. Then \x is a tempered distribution. The assertion is, more
explicitly, that the formula
C) Л/= f
defines a continuous linear functional on &\.
To see this, suppose fi -> 0 in £fn. Then
D) e,= sup(l+ |x|2)fc|y;-(x)| ->0.
xeRn
Since \Aft\ is at most et- times the integral in B), Aft -> 0. This proves the continuity
ofA.

FOURIER TRANSFORMS 175
(с) Suppose \<p < со, N>0, and g is a measurable function on R" such that
E) f \(l + \x\2)-»g(x)\>dmn(x) = C<<x>.
J R,l
Then g is a tempered distribution.
As in (b), define
F) '" . A/=f fgdm,.
Assume first that p > 1; let q be the conjugate exponent. Then Holder's inequality
gives
G) |Л/| < CI/P{JJA + 1*12№) I* rfm.
where M is taken so large that
f (
The inequality G) proves that Л is continuous on Sfn. The case p = I is even easier.
(</) It follows from (c) that every g eW{Rn) A < p < oo) is a tempered distribu-
distribution. So is every polynomial and, more generally, every measurable function whose
absolute value is majorized by some polynomial.
7.13 Theorem // a is a multi-index, P is a polynomial, g e £fn, and и is a tempered
distribution, then the distributions Dhi, Pu, and gu are also tempered.
proof This follows directly from (b) of Theorem 7.4 and the definitions
(Pu)(f) = u(Pf),
(gu)(f) = u(gf). Illl
7.14 Definition For и е £f'n, define
A) й{ф) = и(ф) (феУп).
Since ф -» ф is a continuous linear mapping of Sfn into Sfn [(d) of Theorem 7.4],
and since и is continuous on £fn, it follows that й е 9"п.
We have thus associated with each tempered distribution и its Fourier transform
й, which is again a tempered distribution. Our next theorem will show that the formal

176 DISTRIBUTIONS AND FOURIER TRANSFORMS
properties of Fourier transforms of rapidly decreasing functions are preserved in the
larger setting of tempered distributions.
But first there arises a consistency question that ought to be settled. If/e Ll{Rn),
then/may also be regarded as a tempered distribution, say uf, so that two definitions
of the Fourier transform are available, namely, (c) of Section 7.1 and Definition 7.14.
The question is whether they agree, i.e., whether the distribution {ufY corresponds
to the function/. The answer is affirmative, because
(и,Г(ф) = uf(f) = Iff = 1/ф = (и/КФ)
for every фе^п. The third of these equalities is the identity C) of Section 7.7; the
others are definitions.
Since L2(Rn) c«S^, the same question arises for the Fourier-Plancherel trans-
transform. The answer is again affirmative, by the same proof, since the identity Iff = I /ф
persists for/e L2(Rn) and феУп.
7.15 Theorem
(a) The Fourier transform is a continuous, linear, one-to-one mapping of £f'n onto £f'n,
of period 4, whose inverse is also continuous.
(b) If ue <?'n and P is a polynomial, then
(P{D)uT = Pu and {PuT = P(- D)u.
Note that these are the analogues of (b) of Theorem 7.7 and (c) of Theorem 7.4.
The topology to which (a) refers is the weak*-topology that £fn induces on Sf'n. Note
also that the differential operators P(D) and P(—D) are defined in terms of Z>a, not
D*; see (d) of Section 7.1.
proof Let W be a neighborhood of 0 in Sf'n. Then there exist functions
ф1, ..., фк е £fn such that
A) {иеУ'п:\и{ф?)\ <1 for 1 < / < k} a W.
Define
B) у={иеУ'п: \и(ф()\ <1 for 1 </</:}.
Then К is a neighborhood of 0 in Sf'n, and since
C)
we see that й e ^whenever ue V. This proves the continuity of Ф, where we write
Фи = п. Since Ф has period 4опУ„, C) shows that Ф has period 4 on Sf'n, that
is, that Ф4и = и for every mg^. Hence Ф is one-to-one and onto, and since
Ф = Ф3, Ф is continuous.

) FOURIER TRANSFORMS 177
Statement (b) follows from (c) of Theorem 7.4 and from Theorem 7.13,
by the computations
(P(D)un<f>) = (P(D)u){$) = и(Р(-В)Ф)
and
= и(Рф)=(Ри)(ф)=(Риу(ф),
where ф is an arbitrary function in Sfn. ////
7.16 Examples We saw in (d) of Section 7.12 that polynomials are tempered dis-
distributions. Their Fourier transforms are easily computed. We begin with the poly-
polynomial 1; regarded as a distribution on Rn, 1 acts on test functions ф by the formula
A) l«0=f 1ф*тя=\ ф<1т„.
JRn JRn
Hence
B) •:. Цф) = 1 (<£) = f $dmn = ф(д) = 8(ф),
where 5 is the Dirac measure on Rn. Likewise,
C) $(ф) = д(ф) = Ф@) = f ф dmn = \(ф).
JRn
Thus B) and C) give the results
D) T = <5 and E = 1.
If P is now an arbitrary polynomial on Rn, and if we apply (b) of Theorem 7.15
with и = S and with и = 1, the results in D) show that
E) (P(D)d)л = P and P=P(- D)S.
The two formulas in D) [as well as those in E)] can also be derived from each
other by the inversion theorem, which may be stated for tempered distributions in the
following way: /*
//we £f'n, then (й)л = и, where й is defined by
F) й(ф) = и(ф)
The proof is trivial, since (ф)л = ф, by (a) of Theorem 7.7:
(йУ(ф) = й(ф) = и(($У) = i#) = й(ф).
Note that ^ = 5.

178 DISTRIBUTIONS AND FOURIER TRANSFORMS
If we combine E) with Theorem 6.25, we find that a distribution is the Fourier
transform of a polynomial if and only if its support is the origin (or the empty set).
The following lemma will be used in the proof of Theorem 7.19. Its analogue,
with <3{Rn) in place of £fn, is much easier and was used without comment in the proof
of Theorem 6.30.
7.17 Lemma //w = A,0, .. .,0)e Rn, if ф еУп, and if
A) фХх) = Ф(Х+Ш)е-ф(Х) (*e*»,a>0),
then фЕ -> дф/дхг in the topology of У n, as г -» 0.
proof The conclusion can be obtained by showing that the Fourier transform
of ф£ — дф1дхх tends to 0 in Уп, that is, by showing that
B) фЕф-+0тУп, as e -> 0,
where
C) ^expOy-l^ (^g>0)
If P is a polynomial and a is a multi-index, then
D) P ■ Щфе $) = £ cP ■ (Р-'ф) ■ (M/O.
P<<x
A simple computation shows that
(ey\ if|j8|=0,
E) \Щ1У)\ < U\yA if|j5|=l,
U1^'-1 if|j5|>l.
The left side of D) tends therefore to 0, uniformly on R", as г -»■ 0. The definition
of the topology of Sfn (Section 7.3) shows now that B) holds. ' ////
7.18 Definition Vue&"n and феУп, then
(и*фХх) = и(гхф) (xeR").
Note that this is well defined, since тх ф e ^n for every x e Rn.
7.19 Theorem Suppose ф е Sfn and и is a tempered distribution. Then
(а) и*фе C™(Rn), and
D\u *ф) = (D*u) * ф = и *
for every multi-index a,

FOURIER TRANSFORMS 179
(b) и* ф has polynomial growth, hence is a tempered distribution,
(c) (и*фГ = фп,
(d) (w * ф) *ф = и*(ф * ф),/ог every ф е Sfn,
(e) й*$ = (фиГ.
proof The second equality in (a) is proved exactly as in Theorem 6.30, since
convolution obviously still commutes with translations. This also shows that
Lemma 7.17 now gives D\u * ф) = и * (£>аф) if a = A, 0, ..., 0). Iteration of
this special case gives (a).
Let/?N(/) denote the norm A) of Section 7.3, for/e £fn. The inequality
B) 1 + \x + y\2<2(l + \x\2)(l + \y\2) - (x,yeR")
shows that
C) PN(txf) < 2»A + | x 12)%(Л (* e R", /6 ST,\
Since w is a continuous linear functional on Уп and since the norms pN determine
the topology of £fn, there is an N and а С < oo such that
D)
see Chapter 1, Exercise 8. By C) and D),
E) | (и * ф)(х) | = | и(тх ф) | < 2*Ср„
which proves (b).
Thus w * ф has a Fourier transform, in £f'n. If ф e @(Rn), with support K,
then
(и * ФПФ) = (« * ф)(^) = f (и * ФХхЖ-х) dmn{x)
JRn
= f w[i/r(-x)xxф] dmn(x) = u \ ф(~х)тхф dmn(x)
J-K J-K
so that
F) (и*фГ(Ф)
In the preceding calculation, Theorem 3.27 was applied to an ^„-valued
integral, when и was moved across the integral sign. So far, F) has been proved
for ф e <3(Rn). Since <3(Rn) is dense in S?n, the Fourier transforms of members
of @(Rn) are also dense in Уи, by (b) of Theorem 7.7. Hence F) holds for every
ф e £fn. The distributions (u * ф)л and фй are.therefore equal. This proves (c).

180 DISTRIBUTIONS AND FOURIER TRANSFORMS
In the computation that precedes F), the two end terms are now seen to be
equal for any ф e £fn. Hence
G) (и * фХф) = и((ф * ФУ),
which is the same as
(8) ((u * ф) * ф)@) = (и * (ф • iA))@).
If we replace ф by xx ф in (8), we obtain (d).
Finally, (w * $)A = фй = (фиУ, by (с) above and F) of Section 7.16; this
gives (e\ since (фиу = ((#)T- I III
Paley-Wiener Theorems
One of the classical theorems of Paley and Wiener characterizes the entire functions
of exponential type (of one complex variable), whose restriction to the real axis is in
L2, as being exactly the Fourier transforms of L2-functions with compact support; see,
for instance, Theorem 19.3 of [23]. We shall give two analogues of this (in several
variables), one for C00-functions with compact support, and one for distributions with
compact support.
7.20 Definitions If п is an open set in <£", and if /is a continuous complex func-
function in Q, then / is said to be holomorphic in Q if it is holomorphic in each variable
separately. This means that if (аъ ..., an) e Q. and if
/(«i, ■ • ■, а*-1, «,- + К ai+l9 ..., an),
each of the functions gu ..., gn is to be holomorphic in some neighborhood of 0 in
С A function that is holomorphic in all of 0" is said to be entire.
Points of <Zn will be denoted by z = (z1? ..., zn), where zkeC If zk =
xk + % > x = (*i> • • • > XJ> J = (Уь • • •, З'п)» tnen we write z = jc + /7. The vectors
x = Re z and у = Im z
are the real and imaginary parts of z, respectively; R" will be thought of as the set of
all z e <Zn with Im z = 0. The notations
z- t = zltl + ••• + zw
^@ = exp (iz • t)
will be used, for any multi-index a and any t e Rn.

FOURIER TRANSFORMS 181
7.21_ Lemma Iff is an entire function in Cn that vanishes on R", then f = 0.
proof We consider the case n = I as known. Let Pk be the following property
of/: If ze 0n has at least к real coordinates, then f(z) = 0. Pn is given; Po is to
be proved. Assume 1 < i < n and Pt is true. Take a{, ..., a{ real. The function
#,- considered in Section 7.20 is then 0 on the real axis, hence is 0 for all Ae 0.
It follows that PiTl is true. ////
In the following two theorems,
rB = {xeRn: \x\ <r}.
7.22 Theorem
(a) If ф e@(Rn) has its support in rB, and if
A) f(z)= f 4>(t)e-iztdmn(t) (zen
then f is entire, and there are constants yN < oo such that
B)\ |/(z)| <у^A + |z|)-VIlm2' (Z6C",N = O, 1,2,...).
(Z>) Conversely, if an entire function f satisfies the conditions B), then there exists
ф e @(Rn), with support in rB, such that A) holds.
proof (a) If t erB then
The integrand in A) is therefore in L\Rn), for every z e (£", and/is well defined
on Cn. The continuity of/is trivial, and an application of Morera's theorem, to
each variable separately, shows that /is entire. Integrations by part give
z7B)= f
Hence
C) |z
and B) follows from the inequalities C).
(b) Suppose / is an entire function that satisfies B), and define
D) #0= f /(хУ'-Лп„(х) (teR").
Rn
Note first that A + \x\)Nf(x) is inL1^") for every TV, by B). Hence ф е C°°(/?"),
by the argument that proved (c) of Theorem 7.4.

182 DISTRIBUTIONS AND FOURIER TRANSFORMS
Next, we claim that the integral
-00
E) J f(£ + щ, z2 , ..., zn) exp {i[f ,(<!; + irj) + t2 z2 + • • • + tn zn]} dl
is independent of rj, for arbitrary real ft, ..., fn and complex z2 , ..., zn. To
see this, let Г be a rectangular path in the (£ + /*^)-plane, with one edge on the
real axis, one on the line rj = rjx, whose vertical edges move off to infinity. By
Cauchy's theorem, the integral of the integrand E) over Г is 0. By B), the con-
contributions of the vertical edges to this integral tend to 0. It follows that E) is the
same for r\ = 0 as for ц = r\x. This establishes our claim.
The same can be done for the other coordinates. Hence we conclude from
D) that
F) <Kt)= f
for every у е R".
Given teR\ t Ф 0, choose j> = Af/|f|, where Я > 0. Then t -y = X\t\,
and therefore
G) l«0l sSywe"-|f|)i f A + \x\YN dmn{x),
where N is chosen so large that the last integral is finite. Now let A-> oo. If
\t\ > r, G) shows that ф(г) = 0. Thus ф has its support in rB.
Now A) follows, for real z, from D) and the inversion theorem. Since
both sides of A) are entire functions, they coincide on 0", by Lemma 7.21. This
completes the proof. ■////
The following remarks will motivate the next theorem.
Let и be a distribution in R", with compact support. Then й is defined, as a
tempered distribution, by и(ф) = и(ф). However, the definition f(x) = \fe-xdmn,
made for feL}(Rn), suggests that й ought to be a function, namely,
й(х) = и(е-х) (xeRn),
because e_xe С°°(Л") and ы(ф) makes sense for every ф е C00^"), as shown by (d)
of Theorem 6.24. Moreover, e_z e С°°(Л") for every z e <£", and u(e-2) therefore looks
like an entire function, whose restriction to Rn is u.

FOURIER TRANSFORMS 183
That all this is correct is part of the content of the next theorem, which also
characterizes the resulting entire functions by certain growth conditions.
7.23 Theorem
(a) Ifue &(Rn) has its support in rB, if и has order N, and if
then f is entire, the restriction offto Rn is the Fourier transform ofu, and there is
a constant у < oo such that
(b) Conversely, iffis an entire function in Cn which satisfies B) for some N and some
y, then there exists и e ^'(i?n), with support in rB, such that A) holds.
Note: The notation й will sometimes be used to denote the extension to Cn given
by A). Thus
for z e <£\ This extension is sometimes called the Fourier-Laplace transform of u.
proof (a) Suppose и e <2)r(Rn) has its support in rB. Pick ф е <2)(Rn) so that
ф = 1 on (r + 1J?. Then и = фи, and (e) of Theorem 7.19 shows that
C) й = (фиУ = й * ф.
Thus и е С°AГ). Pick ф е 6fn so that ф = ф. Then
(и * ф)(х) = (и * ф)(х) = й(тхф) = и((тхфУ)
= и(е_х ф) = и(фе_х) = и(е-х),
so that C) gives
Our next aim is to show that the function/defined by A) is entire. Choose
aeCn,be <Zn, and put
E) g(X) =f(a + ЛЬ) = u(e_a_Xb) (X e <£).
The continuity of /poses no problem: If w -> z in (En, then e-w -> e_z in C°°(Rn),
and и is continuous on C°°(i?n). To prove that/is entire it is therefore enough
to show that each of the functions g defined by E) is entire.
Let Г be a rectangular path in С Since X -> e_a-.Xb is continuous, from С
to C*(Rn), the C°(Rn)-valued integral
F) F-

184 DISTRIBUTIONS AND FOURIER TRANSFORMS
is well defined. Evaluation at any t e Rn is a continuous linear functional on
It therefore commutes with the integral sign. Hence
F(t) = \e_a_kb(i)dX= \е-1а'<е-«ь'»ЧЛ
Thus F = 0, and F) gives
0 = u(F)= \u(e-a-»)d*.= \g(X)dL
By Morera's theorem, g is entire.
The proof of part (a) will be completed by proving B). Choose an auxiliary
function h on the real line, infinitely differentiable, such that h(s) = 1 when
s < 1 and h(s) = 0 when s > 2, and associate with each z e 0n (z Ф 0) the function
G) W0 = <rfa"ft(UIM -r\z\) (teR").
Then ф2 e 3){Rn). Since the support of и is in rB and h(\ t\ \z\ - r\z\) = 1 if
| f| < \z\ ~l 4- r, comparison of A) and G) shows that
(8) f(z) = и(ф2).
Since и has order TV, there is a y0 < oo such that \и(ф)\ < уо11ФНлг f°r a^
ф e @(Rn), where ||0||N is as in A) of Section 6.2; see (d) of Theorem 6.24. Hence
(8) gives
(9) I/001 <Уо\\ФЛм.
On the support of ф2, \t\ < r 4- 2/|z|, so that
If we now apply the Leibniz formula to the product G) and use A0), (9)
implies B).
This completes the proof of part (a),
(b) Since/now satisfies B), we have
A1) |/(*)| <y(l + \x\)N (xeR").
The restriction of/to Rn is therefore in Sf'n and is the Fourier transform of some
tempered distribution u.
Pick a function he@(Rn), with support in B9 such that J /г = 1, define
he{i) = e~nh(t/e), for e > 0, and put
A2) №=f(z)Uz) (zz<Zn\
where hE now denotes the entire function whose restriction to Rn is the Fourier
transform of h£. Statement (a) of Theorem 7.22, applied to h£, leads to the
conclusion that/£ satisfies B) of Theorem 7.22 with r + e in place of r. Therefore

FOURIER TRANSFORMS 185
Xb) of Theorem 7.22 implies that /£ = фе for some фЕ е @(R") whose support
lies in (r -f- s)B.
Consider some \p e £?n such that the support of ф does not intersect rB.
Then ффе = 0 for all sufficiently small e > 0. Since /ф e L^R") and he(x) =
/г(вх) -> 1 boundedly on Rn, we conclude that
- и(ф) = йДО) = f # </mM = lim f/E^ </mn
= lim f 4*A ^^n = \ФФв dmn = 0.
Hence и has its support in rB.
Now we see that z -> u(e__z) is an entire function, and since A) holds for
zei?" (by the choice of w), Lemma 7.21 completes the proof of (b). ////
Sobolev's Lemma
If О is a proper open subset of Rn, no Fourier transform has been defined for functions
whose domain is Q. or for distributions in Q. Nevertheless, Fourier transform tech-
techniques can sometimes be used to attack local problems. Theorem 7.25, known as
Sobolev's lemma, is an example of this.
7.24 Definitions A complex measurable function/, defined in an open set Q с Rn,
is said to be locally L1 in Q. if JK \f\2 dmn < oo for everyxompact КczQ.
Similarly, a distribution и e @'(Q) is locally L2 if there is a function g, locally
L2 in п, such that и(ф) = jn дф dmn for every ф е @(п). То say that a function/has
a distribution derivative Daf which is locally/,2 refers to the distribution /)a/and means,
explicitly, that there is a function g, locally L2, such that
= (-1I" f
for every ф е @(Q). A priori, this says nothing about the existence of Daf m the classi-
classical sense, in terms of limits of quotients.
On the other hand, the class Cip)(Q) consists, for each nonnegative integer p, of
those complex functions/in п whose derivatives D*/exist in the classical sense, for
each multi-index a with |a| <£/?, and are continuous functions in Q.
We shall write D\ for the differential operator (d/dxi)k.
7.25 Theorem Suppose n, p, r are integers, n > 0, p >0, and
о) r>p+\-

186 DISTRIBUTIONS AND FOURIER TRANSFORMS
Suppose f is a function in an open set Q. с Rn whose distribution derivatives Dff are
locally L2 in п, for 1 < / < n, 0 < к < г.
Then there is a function f0 e С(р)(п) such that fo(x) =f(x) for almost every
Note that the hypothesis involves no mixed derivatives, i.e., no terms like DXD2 f
The conclusion is that/can be " corrected " so as to be in C(p)(Q), by redefining it on a
set of measure 0.
Note also, as a corollary, that if all distribution derivatives of/are locally L2
in 12, then/oeC°°(ft).
proof By hypothesis, there are functions gik, locally L2 in п, that satisfy
B) f gikф dmn = (-l)fc f fDtt dmn [ф е®Ш
for 1 < / < /2, 0 < к < г.
Let со be an open set whose closure К is a compact subset of Q. Choose
ф е 9{п) so that ф = 1 on K, and define F on Rn by
In D, the Leibniz formula gives
5=0 \S/ s=0
In the complement Qo of the support of ф, D\F' = 0. These two distributions
coincide infin(l0. Hence D\F, originally defined as a distribution in Яп, is
actually in L2(Rn), for 1 < i < n, because the functions (Ц~*ф)дь are in L2(Q).
[Having compact support, ЩFis therefore also in //(iT).]
The Plancherel theorem, applied to F and to D\F,..., Drn F, shows now
that
D) f \F\2dmn<K
JRn
and
E) f yf | F{y) |2 dmn(y) < oo A </<«).
J Rn
Since
F) A + b|Jr < Bя + 2)'A +y? + ~-+ У2Л

FOURIER TRANSFORMS 187
where \y\=XA+--+ У2пI/2> D) and E) imP'y
G) f (\ + \y\J'\F(y)\2dmn(y)<a>.
J Rn
If J denotes the integral G), and if an is the (n — l)-dimensional volume of the
unit sphere in JRn, the Schwarz inequality gives
\y\Y\F(y)\ dmn(y)J <J f A + \y\Jp~lr dmn(y)
j JRn
1 + typ-2rp-ldt<O09
since 2/7 — 2r + n — 1 < — 1. We have thus proved that
(8) f {\ + \y\y\F(y)\dmn(y)<<b.
J Rn
Define
(9) FJLx) = f F(y)elx ■" dmn(у) {х е R").
Rn
By (c) of the inversion theorem 7.7, F^ — F a.e. on R". Moreover, (8) implies
that yaF(y) is in L1 whenever | a | < p. Iteration of the proof of (c) of Theorem
7.4 leads therefore to the conclusion
A0) F»e C(P)(R").
Our given function/coincides with Fin со. Hence/= Гш a.e. in со.
If со' is another set like со, the preceding proof gives a function F^, e
C(p)(Rn), which coincides with / a.e. in со'. Hence /v = ^ ш °>' п со- The
desired function /0 can therefore be defined in Q by setting fo(x) = Fj^x) if
xeco. ////
Exercises
/ Suppose A is an invertible linear operator on Rn,feLi(Rn), and g{x) =f(Ax). Express
g in terms of/. This generalizes (d) of Theorem 7.2.
2 Is the topology of £fn induced by some invariant metric which turns the Fourier trans-
transform into an isometry of £fn onto Sfn ?
3 Suppose f(x) = ex, g{x) = ex cos (ex\ on the real line. Show that g is a tempered dis-
distribution but that/is not.
4 By Exercise 3 there exist distributions in Rn which are not tempered. Such distributions
are continuous linear functionals on 3)(Rn) which have no continuous linear extension
to Sfn. Explain why this does not contradict the Hahn-Banach theorem.

188 DISTRIBUTIONS AND FOURIER TRANSFORMS
5 (a) Construct a sequence in Qi{Rn) which converges to 0 in the topology of Sfn but not
in that of <2)(Rn).
(b) Construct a sequence of polynomials which converges in the topology of Qi\Rl) but
not in that of Sf\.
6 Prove that the operations listed in Theorem 7.13 are continuous mappings of Sf'n into
7 If и е 9"n, prove that
(rx uT =e-xu and (ex u)* =rxu
for every x g Rn.
8 Suppose/e V(Rn),f^ О, Л is a complex number, and/= Л/ What can you say about Л ?
9 Prove (a) of Theorem 7.8 directly (without using Fourier transforms).
10 The Fourier transform of a complex Borel measure /x on 7?" is customarily defined to be
the function (L given by
(x g Rn).
Of course, /x is also a tempered distribution, and as such its Fourier transform was de-
defined in Section 7.14. Show that these two definitions are consistent. Prove that each /x
is bounded and uniformly continuous.
77 Suppose Л: Sf n -> C(Rn) is continuous, linear, and тхА = Атх for every x g Rn. Does it
follow that there exists иеУ; such that
Аф = u * ф
for every фе£Г„1
12 If {hj} is an approximate identity, as in Definition 6.31, and «g^, does it follow that
и * hj -> и as j -> со, in the weak*-topology of Sf'n ?
13 Suppose X and Y are complete metric spaces, Л is dense in X9 f: A -> Y is uniformly
continuous.
(a) Prove that/has a unique continuous extension F: X-> Y.
(b) If/is an isometry, prove that the same is true of F, and prove that F(X) is closed
in Y.
(This was used in the proof of the Plancherel theorem; see also Exercise 19, Chapter 1.)
14 Suppose Fis an entire function in <?", and suppose that to each e > 0 there correspond an
integer N(e) and a constant y(e) < со such that
|F(z)|<y(e)(l-f IzD^V1"" (zeC").
Prove that Fis a polynomial.
15 Suppose/is an entire function in Cn, N is a positive integer, r > 0, and
| f(z)\ < A-f |z|)V|Im z| for all z g <?",
|/W|<1 for all x g R\
Prove that then
|/(z)|<er|Imz| for all z e <£\

FOURIER TRANSFORMS 189
Suggestion: Fix z = x + zy e C"; define
0*(X) = A - &Л)- *- V""y(;t + Ay)
for A g (Z1, s > 0, and apply the maximum modulus theorem to a large semicircular
region in the upper half-plane to deduce that \gs(:)\ < 1. Let s ->0.
76 In F) of Theorem 7.23 it is not asserted that и has order N. The following example
shows that this is not always true.
Let fx be the Borel probability measure on R* which is concentrated on the unit
sphere S2 and which is invariant under all rotations of S2. Compute (by using spherical
coordinates) that
sin \x\
fi(x)=-^ (хеП
Put и = Difi. Then
\u(x)\=\xifL(x)\<l
Deduce from Exercise 15 that
although и is not a distribution of order 0. (Its order is 1.) Find an explicit formula for
the entire function u(e-z), z e (Z3.
17 Suppose и is a distribution in Rn, with compact support K, whose Fourier transform и is
a bounded function on Rn.
(a) Assume n = 1 or n = 2, and prove that фи = 0 for every ф е C°°G?n) that vanishes
on K.
(b) Assume n — 2, and assume that there is a real polynomial P, in two variables, that
vanishes on K. Prove that Pu — 0 and that й therefore satisfies the partial differential
equation P(—D)u = 0. For example, when К is the unit circle, then
и + Аи = 0,
where A = d2jdx\ + д2/дх22 is the Laplacian. /^-s
(c) Show, with the aid of Exercise 16 and the polynomial 1 — x\ — x\ — x\, thatj(&),
hence also (a), becomes false with я = 3 in place of n = 2.
id) Assume n = l,feLl(R),f= 0 on K, and/satisfies a Lipschitz condition of order J,
that is, |/@ -/COI < C\ t -s\i/2. Prove that then
f f(x)u(x)dx = 0.
J -00
Suggestion: For any n, let //^ be the set of all points outside К whose distance
from К is less than e > 0. Let {he} be an approximate identity, as in the proof of (b) of
Theorem 7.23, use the Plancherel theorem to obtain
and show that therefore
||«llcoЦЛ1IU liminf Lr« f \ф\2Фп} '
for any ф e <3(Rn) that vanishes on K.
This yields (a). A slight modification yields (d); (b) follows from (a).

190 DISTRIBUTIONS AND FOURIER TRANSFORMS
18 Was it necessary to introduce the function ф into the proof of Theorem 7.25 ? Could the
proof have been simplified by setting F(x) =f(x) on K, F(x) = 0 off Kl
19 Show that the hypotheses of Theorem 7.25 imply that Da/is locally L2 for every multi-
index a with | a | < r.
20 Let feL2(R2) be the continuous function whose Fourier transform is
/OO = A + M)-40og B + И)} (У е R2).
Since \y\ 3/(x) is in L2(R2), Theorem 7.25 implies that/e Cil\R2). Show that the strong-
stronger conclusion /e CB)(i£2) is false, by proving that
/№, 0)+/(-*, 0)-2/@,0)
— > —со a.sh ->0.
This shows that > cannot be replaced by > in A) of Theorem 7.25.
21 Suppose и is a distribution in R" whose first derivatives Diu,..., Dnu are functions in
L2(Rn). Prove that и is also a function and that и is locally L2. (Show that "locally"
cannot be omitted in the conclusion.) Hint: и is in fact the sum of an L2-function and an
entire function.
When n = 1, show that и is actually a continuous function. Show that this stronger
conclusion is false when n = 2. For example, consider the function
F ixeR)- -
See Exercise 11, Chapter 8, for the same result under weaker hypotheses.
22 Periodic distributions, or distributions on a torus Tn, have Fourier series whose theory
is somewhat simpler than that of Fourier transforms. This is mainly due to the compact-
compactness of Tn: Every distribution on Tn has compact support. In particular, tempered
distributions are nothing special.
Prove the various assertions made in the following basic outline.
Tn={(eXi,...,eXn):xjrea\}.
Functions ф on Tn can be identified with functions ф on Rn that are 27r-periodic in each
variable, by setting
ф(хи...,хп)=ф(е1х\...,е(х").
Zn is the set (or additive group) of «-tuples к = (ku ..., kn) of integers kj. For к e Z",
the function ek is defined on Tn by
ek(eix\ ..., eiXn) =eik'x = exp {/(/ел + ••• + *„*„)}.
on is the Haar measure of Tn. If ф е L\vn), the Fourier coefficients of ф are
= f e-4dvn (keZ").
is the space of all functions ф on Tn such that ф e C°°(i?n). If ф е @{Тп) then
L/2
< 00

FOURIER TRANSFORMS 191
.for N — 0,1, 2,.... These norms define a Frechet space topology on <2){Tn\ which coin-
coincides with the one given by the norms
max sup |(/>•#)(*)| (N = 0,1,2,...).
|a|^7V xeRn
is the space of all continuous linear functional on @(Tn); its members are the
distributions on Tn. The Fourier coefficients of any и е <2t'(Tn) are defined by
й(к) = и(е.к) (keZn).
To each и e* @'(Tn) correspond an N and а С such that
Conversely, if g is a complex function on Zn that satisfies \g(k)\ < C(l + \k\)N for some
С and JV, then g = u for some w e ^'(T1).
There is thus a linear one-to-one correspondence between distributions on Tn,
on one hand, and functions of polynomial growth on Z\ on the other.
If^c^c^c-- are finite sets whose union is Z", and if и е Я)'(Тп\ the
"partial sums"
converge to и asy-> oo, in the weak*-topology of <&'(Tn).
The convolution u*vofue @'(Tn) and t; g !3'(Tn) is most easily defined as having
Fourier coefficients u(k)v(k). The analogues of Theorems 6.30 and 6.37 are true; the
proofs are much simpler.
23 Modify the proof of Theorem 7.25 so that Fourier series are used in place of Fourier
transforms, by replacing F by a suitable periodic function.
24 Put с = B/тгI/2. For у = 1, 2, 3,..., define gj on the real line by
c/t ifl/j<\t\<j
a,(t)-lC/t 1П//<|/|
gAV ~ \0 otherwise.
Prove that {gj} is a uniformly bounded sequence of functions which converges pointwise,
as/-> oo. If/eJL2^/?1), it follows thaXf*gj converges, in the Z2-metric, to a function
Hfe L2. This is the Hilbert transform off; formally,
(The integral exists, in the principal value sense, for almost every x, but this is not so
easy to prove; if /satisfies a Lipschitz condition of order 1, for instance, the proof is
trivial.) Prove that
\\Hf\\2=\\f\\2 and
for every /g L^R1). Thus H is an L2-isometry, of period 4.
Is it true that

8
APPLICATIONS TO DIFFERENTIAL EQUATIONS
Fundamental Solutions
8.1 Introduction We shall be concerned with linear partial differential equations
with constant coefficients. These are equations of the form
A) P(D)u = v
where P is a nonconstant polynomial in n variables (with complex coefficients), P(D)
is the corresponding differential operator (see Section 7.1), v is a given function or
distribution, and the function (or distribution) и is a solution of A).
A distribution E e @'(Rn) is said to be a fundamental solution of the operator
P(D) if it satisfies A) with v = 5, the Dirac measure:
B) . P(D)E = 8.
The basic result (Theorem 8.5, due to Malgrange and Ehrenpreis) that will be proved
here is that such fundamental solutions always exist.
Suppose we have an E that satisfies B), suppose v has compact support, and put
C) ч u = E*v.

APPLICATIONS TO DIFFERENTIAL EQUATIONS 193
Then и is a solution of A), because
D) P(D)(E * v) = (P(D)E) *v = S*v = v,
by Theorems 6.35 and 6.37. \
The existence of a fundamental solution thus leads to a general existence theorem
for the equation A); note also that every solution of A) differs from E * v by a solution
of the homogeneous equation P(D)u = 0. Moreover, C) gives some additional infor-
information about u. For instance, if v e Q){Rn), then и е C°°(Rn).
It may of course happen that the convolution E * v exists for certain v whose
support is not compact. This raises the problem of finding E so that its behavior at
infinity is well under control. The best possible result would of course be to find an E
with compact support. But this can never be done. Jf it could, Ё would be an entire
function, and B) would imply РЁ = 1. But the product of an entire function and a
polynomial cannot be 1 unless both are constant.
However, the equation РЁ = 1 can sometimes be used to find E, namely, when
\jP is a tempered distribution; in this case, the Fourier transform of l/P furnishes a
fundamental solution which is a tempered distribution. For examples of this, see
Exercises 5 to 9.
Another related question concerns the existence of solutions of A) with compact
support if the support of v is compact. The answer (given in Theorem 8.4) shows very
clearly that it is not enough to study P on R" in problems of this sort but that the
behavior of P in the complex space €n is highly significant.
8.2 Notations Tn is the torus that consists of all points
A) * = (ем\...У'")
in <£", where 9U ... ,9n are real; an is the Haar measure of T", that is, Lebesgue meas-
measure divided by Bn)n.
A polynomial in Cn, of degree N, is a function
B) P(z)= X ф>* (zef),
\*\<N
where a ranges over multi-indices and c(oc) e €. If B) holds and if c(oc) ф 0 for at least
one a with | a| = TV", P is said to have exact degree N.
8.3 Lemma If P is a polynomial in €", of exact degree N, then there is a constant
A < oo, depending only on P, such that
A) \f(z)\<Ar~N\ | (fP)(z + rw)| dan(w)
for every entire function f in €n,for every z e Cn, and for every r > 0.

194 DISTRIBUTIONS AND FOURIER TRANSFORMS
proof Assume first that F is an entire function of one complex variable and
that
N
B) Q(X) = с f] (A + at) (А е С).
Put боМ^ПО +fl,-A). Then cF@) = (FQ0)(Q). Since | <20| = | Q| on the
unit circle, it follows that
C) |cF@)| < — f \(FQ)(ew)\ d6.
The given polynomial Pcan be written in the formP = Po + Pi + • •• + PN,
where each Pj is a homogeneous polynomial of degree/ Define A by
D) 4= f \p»\
This integral is positive, since P has exact degree N. [See part (b) of Exercise 1].
If z e C" and w e T", define
E) F(X) = /(z + rXw\ Q(X) = P(z + rlw) (X e C).
The leading coefficient of Q is rNPN(w). Hence C) implies
F) r" I pn(w) i |/(z) i < ~ f" i cfpxz + w'M i de.
If we integrate F) with respect to an, we get
G) |/(z)| < Ar~N -1-f d0\ |(/P)(z + reiew)\ don{w).
The measure an is invariant under the change of variables w-+el9w. The
inner integral in G) is therefore independent of в. This gives A). ////
8.4 Theorem Suppose P is a polynomial in n variables, v e Q}'{Rn), and v has
compact support. Then the equation
A) P(D)u = v
has absolution with compact support if and only if there is an entire function g in €n
such that
B) " Pg = Ь.
When this condition is satisfied, A) has a unique solution и with compact support;
the support of this и lies in the convex hull of the support ofv.

APPLICATIONS TO DIFFERENTIAL EQUATIONS 195
-PROOi: If A) has a solution и with compact support, (a) of Theorem 7.23
shows that B) holds with g = й.
Conversely, suppose B) holds for some entire g. Choose r > 0 so that v
has its support in rB = {jc e Rn: \x\ < r). By Lemma 8.3, B) implies
C) Ш\<Л\ \t(z + W)\dan(w) (zeC").
By (a) of Theorem 7.23, there exist N and у such that
D) \i)(z + w)\<y(l + \z + w\)Nexp{r\Im(z + w)\}.
There are constants ct and c2 that satisfy
E) l + \z + w\ <q(l + |z|)
and
F) |Im(z + w)| <c2 + |Imz|
for all,2r e Cn and all w e Tn. It follows from these inequalities that
G) \g(z)\ <B{\ + |z|)*exp{r|Imz|} (zef),
where В is another constant (depending on y, A, N, cu c2 , and r). By G) and
(b) of Theorem 7.23, g = й for some distribution w with support in ri?. Hence
B) becomes Рй = i), which is equivalent to A).
The uniqueness of и is obvious, since there is at most one entire function й
that satisfies Рй = i).
The preceding argument showed that the support Su of и lies in every
closed ball centered at the origin that contains the support Sv of v. Since A)
implies
(8) P(Dyxxu) = Txv (xeR"),
the same statement is true of jc + Su and x + Sv. Consequently, Su lies in the
intersection of all closed balls (centered anywhere in Rn) that contain Sv. Since
this intersection is the convex hull of Sv, the proof is complete. ////
8.5 Theorem If P is a polynomial in €n, of exact degree N, then the differential
operator P(D) has a fundamental solution E that satisfies
A) \Щ)\ < Ar~N f dan(w) f \ф(г + rw)\ dmn(t)
for every ф e Q>(Rn) and for every r > 0.

196 DISTRIBUTIONS AND FOURIER TRANSFORMS
Here A is the constant that appears in Lemma 8.3. The main point of the
theorem is the existence of a fundamental solution, rather than the estimate A) which
arises from the proof.
proof Fix r > 0, and define
B) №11 = f dan(w) f |#(t + rw)| dmn(t).
JTn JRn
In preparation for the main part of the proof, let us first show that
C) lim ||^|| =0 if ф] ->0 in @(Rn).
j-*CO
Note that $(t + w) = (e_w^)A(f) if / e Rn and w e €n. Hence
D) №||= f dan(w) f \(е_гЖ\<1тп.
JTn JRn
If ф] -> 0 in 3)(Rn), all ^7- have their supports in some compact set K. The func-
functions erw (w e Tn) are uniformly bounded on K. It follows from the Leibniz
formula that
E) II ЛЧе-™ WIL ^ C(^' °0 max
The right side of E) tends to 0, for every a. Hence, given e > 0, there exists j0
such that
F) ||(/ - A)^-rw^)ll2 < в (j >j0, w 6 T"),
where A = D\ + • * • + D2n is the Laplacian. By the Plancherel theorem, F) is
the same as
G) JjA + 1112y$j(t + rw)\2 dmn@ <e2,
from which it follows, by the Schwarz inequality and B), that \\ф^\ < Се for all
J>Jo-> where
(8) C2= f A + |?|2)-2«
This proves C).
Suppose now that ф е 9(Rn) and that
(9) ф
Then i^ = P^, ф and i^ are entire, hence ф determines ф. In particular, ф@) is
a linear functional of ф, defined on the range of P(D). The crux of the proof

APPLICATIONS TO DIFFERENTIAL EQUATIONS 197
consists in showing that this functional is continuous, i.e., that there is a distri-
distribution и е @'(Rn) that satisfies
A0) и(Р(В)ф) = ф@) (ф
because then the distribution E = й satisfies
= Е(Р(-В)ф) = и((Р(~В)фУ)
so that P{D)E = 5, as desired.
Lemma 8.3, applied to Рф =■ ф, yields
A1)
By the inversion theorem, ф@) = \Кпф dmn. Thus A1), B), and (9) give
A2) | «0)| < Аг-»\\Рф)ф\\ (ф е
Let Y be the subspace of 3)(Rn) that consists of the functions Р(Я)ф,
ф е @(Rn). By A2), the Hahn-Banach theorem 3.3 shows that the linear func-
functional that is defined on Y by Р(В)ф -> ф@) extends to a linear functional и on
that satisfies A0) as well as
A3)
By C), и е @'(Rn). This completes the proof. ////
Elliptic Equations
8.6 Introduction If и is a twice continuously differentiable function in some open
set Q a R2 that satisfies the Laplace equation
then it is very well known that и is actually in С°°(П), simply because every real harmonic
function in П is (locally) the real part of a holomorphic function. Any theorem of
this type—one which asserts that every solution of a certain diiferential equation has
stronger smoothness properties than is a priori evident—is called a regularity theorem.
We shall give a proof of a rather general regularity theorem for elliptic partial
differential equations. The term "elliptic" will be defined presently. It may be of
interest to see, first of all, that the equation
B) *

198 DISTRIBUTIONS AND FOURIER TRANSFORMS
/
behaves quite differently from A), since it is satisfied by every function и of the form
u(x, y) =f(y), where/is any differentiable function. In fact, if B) is interpreted to
mean
C) -д
then/can be a perfectly arbitrary function.
8.7 Definitions Suppose п is open in Rn, N is a positive integer, fa e C°°(O) for
every multi-index a with | a | < N, and at least one/a with | a | = N is not identically 0.
These data determine a linear differential operator
О) L= £ faDa
\*\<N
which acts on distributions и е Q)'(Q) by
B) Lu= £ /. Д.И.
|a|<JV
The order of L is iV. The operator
C) E Л А
|a|=JV
is the principal part of L. The characteristic polynomial of L is
D) р(дг, у) = £ /t(JC)j;a (x e Q, у e Rn).
This is a homogeneous polynomial of degree N in the variables у = (yu ..., yn), with
coefficients in C°°(O). N
The operator L is said to be elliptic if p(x, у) ф 0 for every хеп and for every
у e Rn, except, of course, for у = 0. Note that ellipticity is defined in terms of the
principal paVt of L; the lower-order terms that appear in A) play no role.
For example, the characteristic polynomial of the Laplacian
is ^(jc, у) = — (у\ + • • • + yl), so that A is elliptic.
On the other hand, if L = d2/^ dx2, then ^(jc, у) = -У\У2^ an(i ^ is not
elliptic.
The main result that we are aiming at (Theorem 8.12) involves some special
spaces of tempered distributions, which we now describe.
8.8 Sobolev spaces Associate to each real number s a positive measure jis on Rn
by setting
A) '

APPLICATIONS TO DIFFERENTIAL EQUATIONS 199
If/eL2(X), that is, if J|/|2 djis < oo,then/is a tempered distribution [Example (c) of
7.12]; hence/is the Fourier transform of a tempered distribution u. The vector space
of all и so obtained will be denoted by Hs; equipped with the norm
/r X1/2
B) (J2)
Hs is clearly isometrically isomorphic to L2(//s).
These spaces Hs are called Sobolev spaces. The dimension n will be fixed through-
throughout, and no reference to it will be made in the notation.
By the Plancherel theorem, H° = L2.
It is obvious that Hs с Нг if t <s. The union X of all spaces Hs is therefore a
vector space. A linear operator Л: X -» X is said to have order t if the restriction of Л
to each Hs is a continuous mapping of Hs into Я5"'; note that / need not be an integer.
Here are the properties of the Sobolev spaces that will be needed.
8.9 Theorem
(a) Every distribution with compact support lies in some Hs.
(b) If — oo < t < oo, the mapping и -> v given by
is a linear isometry of Hs onto Hs~l and is therefore on operator of order t whose
inverse has order — t.
(c) Jfb e L?(Rn\ the mapping u~>v given by v = bit is an operator of order 0.
(d) For every multi-index a, Da is an operator of order | a |.
(e) Iffe 9yn, then и ->fu is an operator of order 0.
proof If и е <3}'{Rn) has compact support, (a) of Theorem 7.23 shows that
A) Г
for some constants С and N. Hence и e Hs if s < —N — и/2. This proves part
(a); (b) and (c) are obvious. The relation
1(л«иГО01 =J/l|fiO0l ^A + М2)|в|/21<Ю01
implies
B) I|.d*«L-w<IHL
so that (d) holds.
The proof of (e) depends on the inequality
C) A + |x + .И2M < 2lJl(l + W2)s(l + Ы2)м,

200 DISTRIBUTIONS AND FOURIER TRANSFORMS
valid for jc e Rn, у е Rn, -co < s < oo. The case s = 1 of C) is obvious. From
it the case s = — 1 is obtained by replacing jc by jc — у and then у by — y. The
general case of C) is obtained from these two by raising everything to the power
\s\. It follows from C) that
D) f | h(x - y) |2 diis(x) < 2>"A + | у 12)l*l f | h |2 dp,
JRn JRn
for every measurable function h on Rn.
Now suppose ueHs,fe^n, *>|.у|+л/2. Since fe^n, \\f\\t < go.
Put у = /iN_,(#"). Then у < oo. Define F= \й\ * |/|. By Theorem 7.19,
E) |(/мГ| = |й*/|<|й|*|/| = ^.
By the Schwarz inequality,
F) \F(x)\2 < f |/(y)|2 dft(y) f |U(x - j;)|2 rf/z-.O»)
for every д: е Rn. Integrate F) over Rn, with respect to fis. By D), the result is
G) f |F|2d/i,<2<"y|l/ll?ll«ll.2.
It follows from E) and G) that
(8) IIML<BNyI/2ll/IUI«L.
This proves (e). /'///
8.10 Definition Let Q. be open in R". A distribution м е @f(Q) is said to be locally
Hs if there corresponds to each point xeQ a distribution veHs such that и = v in
some neighborhood со of jc. (See Section 6.19.)
8.11 Theorem If и е Q)'(Q) and —oo<s<oo, the following two statements are
equivalent:
(a) u is locally Hs.
(b) фи е Hs for every ф е 0(Q).
Moreover, if s is a nonnegative integer, (a) and (b) are equivalent to
(c) Da и is locally L2 for every a with \ a \ < s.
Statement (b) may need some clarification, since и acts only on test functions
whose supports lie in п. However, фи is the functional that assigns to each ф е
the number
ЦгиКф) = и{фф).
Note that ij/ф e &(U), so that и(фф) is defined.

APPLICATIONS TO DIFFERENTIAL EQUATIONS 201
proof Assume и is locally Hs. Let К be the support of some ф е @(п). Since
К is compact, there are finitely many open sets cot с Q, whose union covers K,
and in which и coincides with some vteHs. There exist functions фге@(со1)
such that £ ф1 = 1 on K. If ф е $(Rn) it follows that
since" фсффе Щ(со(). Thus фи = ^фг фиг. By 0) of Theorem 8.9, ^, ^ e Hs
for each f. Thus фи е Hs, and (a) implies (b).
If F) holds, if x e п, and if ф e @(Q) is 1 in a neighborhood со of x, then
и = фи in a), and ^w e Hs by assumption. Thus (b) implies (a).
Assume again that (b) holds. If ф е @(п), then фи е Hs, hence Ва(фи) е
#*->!, by {d) of Theorem 8.9. If |a| < s, then
Hs-\*\ dH0 =
Thus Ва(фи) e l3(Rn). Taking ф = 1 in some neighborhood of a point л: е Q.
shows that Z)a и is locally L2 in п. Thus F) implies (c).
Finally, assume Dau is locally L2 for every a with |a| < s. Fix ^ e
The Leibniz formula shows that Ва(фи) e l}(Rn) if |a| < 51. Hence
(I) f |
If ^ is a nonnegative integer, A) holds with the monomials y\, ...,y*in place of
ya. It follows, as in the proof of Theorem 7.25, that
B) J A + \у\У\(фиГ(у)\2с1тп(У)<со.
Rn
Thus фи e Hs, (c) implies (b), and the proof is complete. ////
8.12 Theorem Assume Q is an open set in Rn, and
(a) L = Y^fa^a & a linear elliptic differential operator in Q, of order N> 1, with
coefficients fa e C00(Q\
(b) for each a with \a\ = N,fais a constant,
(c) и and v are distributions in Q that satisfy
A) #. Lu = v,
and v is locally Hs.
Then и is locally Hs+N.
Corollary If L satisfies (a) and (b) and if v e C00^), then every solution и
of (I) belongs to C°°(Q). In particular, every solution of the homogeneous equation
Lu = 0 is in С°°(П).

202 DISTRIBUTIONS AND FOURIER TRANSFORMS
For ifve C°°(O), then \pv e $(Rn) for every ф е 9(п)\ hence v is locally Hs
for every s, and the theorem implies that и is locally Hs for every s; it follows
from Theorems 8.11 and 7.25 that не С°°(Д).
Assumption (b) can be dropped from the theorem, but its presence makes the
proof considerably easier.
proof Fix a point хеп, let Bo с п be a closed ball with center at jc, and let
ф0 e <2>(п) be 1 on some open set containing Bo. By (a) of Theorem 8.9, фоиеН*
for some t. Since W becomes larger as t decreases, we may assume that / =
s + N — k, where к is a positive integer. Choose closed balls
each centered at jc, and each properly contained in the preceding one. Choose
ф1, ..., фк e @(Q) so that фк = 1 on some open set containing Bt, and фг — 0
off Bi^_l. Since фоие Н\ the following "bootstrap" proposition implies that
ф&еН'*1, ...,фкиеHt+k.
It therefore leads to the conclusion that и is locally Hs+N, because t + к = s + N
and фк = 1 on Bk.
Proposition If, in addition to the hypotheses of Theorem 8.12, фи е Н* for
some t < s + N — 1 and for some ф e @(Q) which is 1 on an open set containing
the support of a function ф е @(п), then фи е Ht+1.
proof We begin by showing that
B) Цфи)еН*-"+1.
Consider the distribution
C) Л = Цфи) - фЬи = Цфи) - фv.
Since its support lies in the support of ф, и can be replaced by фи in C), without
changing Л:
D) А = Цффи)-фЬ(фи)= X Ъ-ШФЫ-ФЯ*(Ы1
If the Leibniz formula is applied to Ва(ф • фи), one sees that the derivatives of
order N of фи cancel in D). Therefore Л is a linear combination [with coeffi-
coefficients in Q)(Rn)\ of derivatives of фи, of orders at most N - 1. Since фи е H\
parts (d) and (e) of Theorems 8.9 imply that AeHf~N+1. By Theorem 8.11,
фveHs, and since t — N + I < s, we have фveHt~N+1. Now B) follows
from C).

APPLICATIONS TO DIFFERENTIAL EQUATIONS 203
Since L is elliptic, its characteristic polynomial
/4H=
|a|=JV
has no7 zero in Rn, except at у = 0. Define functions
F) ■ . q(y)=\y\ ~NP(y\ r{y) = A + |у\N)q(y),
for у e Rn, у #0,' and define operators Q, R, S on the union of the Sobolev
spaces by
and
■ - |a|<JV
Since p is a homogeneous polynomial of degree N, q{ky) = q{y) if X > 0,
and since p vanishes only at the origin, the compactness of the unit sphere in Rn
implies that both q and \\q are bounded functions. It follows from (c) of
Theorem 8.9 that both Q and Q~x are operators of order 0.
Since both A + \y\2)~N/2(l + \y\N) and its reciprocal are bounded func-
functions on R", it follows from the preceding paragraph, combined with (b) and
(c) of Theorem 8.9, that R is an operator of order N whose inverse R~x has order
-N.
Since \j/fa e T3(R") it follows from (d) and (e) of Theorem 8.9 that S is an
operator of order N — 1.
Sincep = r — q, and sincep is assumed to have constant coefficients^, we
have
(9) Z faOawf = pw = (r~q)w = (Rw-Qw)"
\\«\=N I
if w lies in some Sobolev space. Hence
By B), Цфи)еН<-»+\
Since фи e H* and фф =/j), (e) of Theorem 8.9 implies that фи = ффи е Hf.
Hence
(И) (й
because Q has order 0 and S has order N - 1 > 0. It now follows from A0) that
A2) R№u)eH'-n+1,
and since R'1 has order —N, we finally conclude that фи е Ht+i. ////

204 DISTRIBUTIONS AND FOURIER TRANSFORMS
8.13 Example Suppose L is an elliptic differential operator in Rn, with constant
coefficients, and E is a fundamental solution of L. In the complement of the origin,
the equation LE = 5 reduces to LE = 0. Theorem 8.12 implies therefore that, except
at the origin, E is an infinitely differentiable function. The nature of the singularity of
E at the origin depends, of course, on L.
8.14 Example The origin in R2 is the only zero of the polynomial^) = yt + iy2.
If п is open in R2, and if и е 9)'(Щ is a distribution solution of the Cauchy-Riemann
equation
д д
2)
Theorem 8.12 implies that и e С°°(П). It follows that и is a holomorphic function of
2 — xx + ix2 in Q. In other words, every holomorphic distribution is a holomorphic
function.
Exercises
/ The following simple properties of holomorphic functions of several variables were
tacitly used in this chapter. Prove them.
(a) If /is entire in <£ny if w e <£", and if </>(Л) =/(Aw), then </> is an entire function of one
complex variable.
(b) If P is a polynomial in <£" and if
J ТП
then P is identically 0. Hint: Compute JTn |P|2 dun.
(c) If P is a polynomial (not identically 0) and g is an entire function in <£", then there is
at most one entire function / that satisfies Pf— g.
Find generalizations of these three properties.
2 Prove the statement about convex hulls made in the last sentence of the proof of Theo-
Theorem 8.4.
3 Find a fundamental solution for the operator dzjdx1 dx2 in R2. (There is one that is the
characteristic function of a certain subset of R2.)
4 Show that the equation
is satisfied (in the distribution sense) by every locally integrable function и of the form
u(xi, x2) =/(*i + x2) or u{xu x2) =/(*i - x2)
and that even classical solutions (i.e., twice continuously differentiable functions) need
not be in C00. Note the contrast between this and the Laplace equation.

APPLICATIONS TO DIFFERENTIAL EQUATIONS 205
5 For x e R3, define f(x) = (i + (xl2). Show that feL2(R3) and that /is a fundamental
solution of the operator /— A in R3. Find / by direct computation and also by the
following reasoning:
(a) Since / is a radial function (i.e., one that depends only on the distance from the
origin) the same is true of/; see Exercise 1 of Chapter 7.
(b) Away from the origin, (/— A)/= 0, and/e С ".
(c) If F(\y\) =f(yX (b) implies that F satisfies an ordinary differential equation in
@, oo) that can easily be solved explicitly. Ans. f(y) = (тг/2I/2М "* exp ( - \y\).
Do the same with R" in place of /?3; you will meet Bessel functions.
6 For 0 < A < n and x e /?", define
K,(x)=\x\~\
Show that
(a) K,(y) ---= ф, Х)КП-ЛУ) (У е /?"),
where
I Suggestion: If n < 2Л < 2«, /G is the sum of an L1-function and an L2-function.
For these A, Equation (a) can be deduced from the homogeneity condition
K,(tx) - г"К,(х) (х е /?-,/> 0).
The case 0 < 2Л < // follows from the inversion theorem (for tempered distributions).
A passage to the limit gives the case 2Л = n. The constants c(n, A) can be computed
from//<£ = J7& with ф(х) =ехр (- \x\ 2/2).
7 Take // > 3 and A = 2 in Exercise 6, and deduce that — c(n, 2)Kn-2 is a fundamental
solution of the Laplacian A in Rn. For example, if v has compact support in R3, show
that a solution of Аи — i; is given by
1
-— f
4тг JR3
Identify Я2 and ^ (so that z = xt-\- ix2)\ put
а д д з
O\ .* О I •
dxi дх2 ' dxi дх2
Show that the Fourier transform of 1/z (regarded as a tempered distribution) is —//z.
Show that this result is equivalent4o the Cauchy formula
-!.
Since д log | vv| = 1/iv and A = дд, deduce that
ф(г) = f (Д$(иО log | w - z| ^/т2(и^) [^ е
Thus log |z| is a fundamental solution of the Laplacian in R2.

206 DISTRIBUTIONS AND FOURIER TRANSFORMS
9 Use Exercise 6 to compute that
lim [e-1 - b - R2-£y)] = log \y\ (у е R2),
£-0
where b is a certain constant. Show that this leads to another proof of the last statement
in Exercise 8.
10 Suppose P(D) = D2 + aD + bl. (We are now in the case /2 = 1.) Let/and g be solutions
of P(D)u = 0 which satisfy
/@)=*@) and
Define
and put
С c) dx
J -
Prove that Л is a fundamental solution of P(D).
11 Suppose и is a distribution in Rn whose first derivatives D^u, ..., Dnu are locally 1Л
Prove that м is then locally L2. Hint: If ф e <3(Rn) is 1 in a neighborhood of the origin
and if AE = 8, then А(фЕ) ~8е Q)(Rn). Hence
is in C°°(i?n). Each D^E) is an L^function with compact support.
12 Suppose и is a distribution in Rn whose Laplacian Аи is a continuous function. Prove
that и is then a continuous function. Hint: As in Exercise 11,
и - (фЕ) * (Аи) g C™(Rn).
13 Prove analogues of Exercises 11 and 12, with R" replaced by an arbitrary open set O.
14 Show, under the hypotheses of Exercise 12, that
(a) d2u/dxl is locally L2, but
(b) d2ujdx\ need not be a continuous function.
Outline of (b) for periodic distributions in R2 (Exercise 22, Chapter 7): If g e C(T2)
has Fourier coefficients g{m, n) and if/is defined by
f(m9 n) = A + m2 + n2ylg(m, n),
then/e C(T2) and A/-/- <? e С(Г2), since 2 |/(™, л)| < oo. The Fourier coefficients
of d2fjdx\ are -m2f(m,n). If d2//d;t? were continuous for every geC(T2\ then
(d2f/dx2)(Ot 0) would be a continuous linear functional of g. Hence there would be a
complex Borel measure ju, on T2, with Fourier coefficients
n, n) —
1 + m + n2
The next exercise shows that no such measure exists.

APPLICATIONS TO DIFFERENTIAL EQUATIONS 207
15 If /л is a complex Borel measure on Г2, and if
prove that
lim lim y(A, B) = lim lim y(A, B) .
Suggestion? If DA(t) = BЛ -f I) 2-<* eM9 then /)^(x) = 1 if x = 0, /)д(х) ->0
otherwise, and
y{Ay B) = f DA(x)D,(y) dfi(x, y).
JT2
Conclude that each of the two iterated limits exists and that both are equal to /x({0, 0}).
If fi were as in Exercise 14, one of the iterated limits would be 1, the other 0.
16 Suppose L is an elliptic linear operator in some open set п <= R", and suppose that the
order of L is odd.
(а) Prove that then n = 1 or n = 2.
(б) If л = 2, prove that the coefficients of the characteristic polynomial of X cannot all
*be real.
In view of (a), the Cauchy-Riemann operator is not a very typical example of an
elliptic operator.

У
TAUBERIAN THEORY
Wiener's Theorem
9.1 Introduction A tauberian theorem is one in which the asymptotic behavior
of a sequence or of a function is deduced from the behavior of some of its averages.
Tauberian theorems are often converses of fairly obvious results, but usually these
converses depend on some additional assumption, called a tauberian condition. To see
an example of this, consider the following three properties of a sequence of complex
numbers sn = a0 + • • • + an.
(a) lim sn — s.
(b) If/(r) = f an г", 0 < r < 1, then lim/(r) = s.
(c) lim nan = 0.
n-*ao
Since/(r) = A — r) ]T sn rn and A - r) £ rn = 1, /(r) is, for each r e @, 1), an average
of the sequence {sn}. It is extremely easy to prove that (a) implies (b). The converse is
not true, but (b) and (c) together imply (a); this is also quite easy and was proved by
Tauber. The tauberian condition (c) can be replaced by the weaker assumption that

TAUBERIAN THEORY 209
{nan} is bounded (Littlewood). It is remarkable how much more difficult this weaken-
weakening of (c) makes the proof.
Wiener's tauberian theorem deals with bounded measurable functions, originally
on the real line. If ф e L°°(jR) and if ф(х) -» 0 as jc-> + oo, then it is almost trivial that
(К*ф)(х)-+0 as x-*+oo for every Kel}(R). The convolutions К*ф may be
regarded as averages of ф, at least when J К = 1. Wiener's converse [(a) of Theorem
9.7] states that if (К* ф)(х) -> 0 for one Ke I}(R) and if the Fourier transform of this
Evanishes at no point of R, then (/* ф)(х) -* 0 for every fe l}(R); the stronger con-
conclusion that ф(х) -* 0 need not hold under these hypotheses, but it does hold if a
slight additional condition (slow oscillation) is imposed on ф [(b) of Theorem 9.7].
The unexpected tauberian condition—the nonvanishing of К—enters the proof
in the following manner: If (К* ф)(х) -> О, the same is true if ATis replaced by any of
its translates, hence also if. К is replaced by any finite linear combination g of trans-
translates of K. When К has no zero, it turns out that the set. of these functions g is dense
in 1} (Theorem 9.5). One is thus led to the study of translation-invariant subspaces
ofL1.
9.2 Lemma Suppose fe Lx(Rn), t e Rn, and e > 0. Then there exists h e 1}(R"),
with \\Щ1 < 6, such that
A) h(s)=f(t)-f(s)
for all s in some neighborhood of t.
The lemma states that/is approximated, in the Z^-norm, by a function /+ h
whose Fourier transform is constant in a neighborhood of the point t.
proof Choose gel}(Rn) so that g = 1 in some neighborhood of the origin.
For X > 0, put
B) 9x(x) = eu'*X-»g{xlX) (x e R»)
and define
C) hx(x)=f(t)gl(x)-(f*gl)(x).
Since gk(s) — 1 in some neighborhood Vk of t, C) shows that A) holds for
s e Vx, with hx in place of h. Next,
D) hk{x) = f Ky)[e-U'>gx(x) - дл(х - у)] dmn{y).
JRn
The absolute value of the expression in brackets is
E) \Г»д(Г1х)-Г"д(А-\х-УУ)\.

210 DISTRIBUTIONS AND FOURIER TRANSFORMS
It follows that
F) IIMi < f I/Ml dmn(y) f \g(O ~ g(Z - Г V)l dmn@,
JRn JRn
by the change of variables x = A£. The inner integral in F) is at most 2\\g\\i9
and it tends to 0 for every yeRn,asl-+ oo. Hence \\кл\\г -> 0, as X -> со, by
the dominated convergence theorem. ////
9.3 Theorem If ф e L»(Rn), Y is a subspace ofl}(Rn), and
A) /* ф = Ofor every fe Y,
then the set
B) Z(Y)= {){seR":f(s)=0}
feY
contains the support of the tempered distribution ф.
proof Fix a point t in the complement of Z(Y). Then f(t) = 1 for a certain
fe Y. Lemma 9.2 furnishes /r e L\Rn), with 11% < 1, such that h(s) = 1 -/(я)
in some neighborhood V of f.
To prove the theorem, it suffices to show that ф = 0 in V, or, equivalently,
that ф(ф) = 0 for every ф е 9>n whose Fourier transform ф has its support in V.
Since
C) ф(ф) = <KW = (Ф * Ш0),
it suffices to show that ф * ф =0.
Fix such а ф. Put <70 = ^r,^m = h * ^^ form > 1. Then \\дт\\г < II^IITII'Alli,
and since 11% < 1, the function G = ]Г#Ш is in L1^")- Since fi(j) = 1 -f(s)
on the support of ф, we have
D) A - h(sM(s) = ФШ^) (s e Rn),
or
Thus ф = G *f and A) implies
F) ^*Ф = а*/*^ = о. ////
9.4 Wiener's theorem If Y is a closed translation-invariant subspace ofl}{Rn) and
ifZ(Y) is empty, then Y = I}(Rn).

TAUBERIAN THEORY 211
- proof To say that Fis translation-invariant means that rxfe Yiffe У and
x e Rn. If ф e L°°(i?n) is such that J/0 = 0 for every /e Y, the translation-
invariance of Y implies that/* ф = 0 for every/e Y. By Theorem 9.3, the
support of the distribution ф is therefore empty, hence ф = 0 (Theorem 6.24),
and since the Fourier transform maps Sf'n to &"n in a one-to-one fashion
(Theorem 7.15), it follows that ф = 0 as a distribution. Hence ф is the zero
element of П°(К").
Thus Y\ = {0}. By the Hahn-Banach theorem, this implies that Y = l}(Rn).
Ill/
9.5 Theorem Suppose Ke l}{Rn) and Yis the smallest closed translation-invariant
subspace ofl}{Rn) that contains K. Then Y=Ll(Rn) if and only if K(t) Ф 0 for every
tER".
proof Note that Z{Y) = {t e R": K(t) = 0}. The theorem thus asserts that
Y=Ll(Rn) if and only if Z(Y) is empty. One-half of this is Theorem 9.4; the
other half is trivial. ////
9.6 Definition A function ф е L?(Rn) is said to be slowly oscillating if to every
e > 0 correspond an Л < oo and a S > 0 such that
A) [Ф(х)-ф(у)\ <s if |x| >A, \y\ >A, \x-y\ <S.
If n = 1, one can also define what it means for ф to be slowly oscillating at + oo:
the requirement A) is replaced by
B) \Ф(х)-ф(у)\ <s ifx>A,y>A, \x-y\ <d.
The same definition can of course be made at — oo.
Note that every uniformly continuous bounded function is slowly oscillating but
that some slowly oscillating functions are not continuous.
We now come to Wiener's tauberian theorem; part (b) was added by Pitt.
9.7 Theorem
(a) Suppose ф е П°(ВГ)9 Ke Ll(R"), R(t) ф 0 for every t e Rn, and
A) lim (K * ф)(х) = aK@).
M-oo
Then
B) lim (/* ф)(х) = я/@),
for every fel){Rn).

212 DISTRIBUTIONS AND FOURIER TRANSFORMS
(b) If, in addition, ф is slowly oscillating, then
C) lim ф(х) = а.
proof Put ф(х) = ф(х) - a. Let Y be the set of all/e L}(Rn) for which
D) lim (/* ф)(х) = О.
W-00
It is clear that У is a vector space. Also, Y is closed. To see this, suppose
/e У, ||/-/-||i->0. Since
ft * i/^ ->/* i^ uniformly on /Г; hence D) holds. Since
F) ((*,/) * i/OO) = (*,(/* ФЖх) = (/* i/OO - j),
У is translation-invariant. Finally, A^g У, by A), since K* a = #K@).
Theorem 9.5 now applies and shows that У = l}(Rn). Thus every/e l}(Rn)
satisfies D), which is the same as B). This proves part (a).
If ф is slowly oscillating and if e > 0, choose A and S as in Definition 9.6,
and choose /e L^tf") so that/> 0,/@) = 1, and/(x) = 0 if |jc| > 8. By B),
G) lim (/* ф)(х) = а.
M-00
Also,
(8) ФМ - (/* 0)W = f [ф(х) - ф(х - y)]f(y) dmn{y).
J\y\<d
If I jc| > Л + ^, our choice of A, S, and/shows that
(9) \Ф(х)-(/*ф)(х)\ <e.
Now C) follows from G) and (9).
This completes the proof. ////
9.8 Remark If n = 1, Theorem 9.7 can be modified in an obvious fashion, by
writing x -> + oo in place of | x \ -> oo wherever the latter occurs and by assuming in
(b) that ф is slowly oscillating at +oo. The proof remains unchanged.
The Prime Number Theorem
9.9 Introduction For any positive number x, n(x) denotes the number of primes
p that satisfy p < x. The prime number theorem is the statement that
A) . lim*(

TAUBERIAN THEORY 213
We shall prove this by means of a tauberian theorem due to Ingham, based on that of
Wiener. The idea is to replace the rather irregular function ж by a function F whose
asymptotic behavior is very easily established and to use the tauberian theorem to
draw a conclusion about ж from knowledge of F.
9.10 Preparation, The letter/? will now always denote a prime; m and n will be
positive integers; x will be a positive number; [x] is the integer that satisfies x — 1 <
[x] < x; the symbol d\ n means that d and n/d are positive integers. Define
A) AdO-ft j
@ otherwise,
B)
The following properties of \j/ and F will be used:
\ ф(х) < ж(х) log x ^ 1 ^(jc)
log x x log (%/log2 x)
if x > e, and
E) F(x) = x log л: — x + £(x) log л:,
where £(;c) remains bounded as x -» oo.
By D), the prime number theorem is a consequence of the relation
F) lim
which will be proved from C) and E) by a tauberian theorem.
proof of D) [log x/logp] is the number of powers of/7 that do not exceed x. Hence
= Z Г— I
P<x U°g PJ
P<x
This gives the first inequality in D). If 1 < у < х, then
y<P<x y<p<x
Hence n(x) < у + \j/(x)/log y. With >> = x/log2 л:, this gives the second half of D).

214 DISTRIBUTIONS AND FOURIER TRANSFORMS
proof of E) If n > 1, then
The mth summand is 0 except when njm is an integer, in which case it is A(n/m).
Hence
F(n) - F(n - 1) = £ л(-) = £ A(d) = log n.
The last equality depends on the factorization of n into a product of powers of distinct
primes. Since F(l) = 0, we have computed that
G) F(n) = £ log m = log (и!) (я = 1, 2, 3,...),
m= 1
which suggests comparison of F(x) with the integral
rx
(8) /(*) - log t dt = x log x - x + 1.
Ji
If n <x <n -\- 1 then
(9) J(n) < F(n) < F(x) < F(n + 1) < J(n + 2)
so that
A0) |F(x)-/(jc)| <21og(x + 2).
Now E) follows from (8) and A0).
9.11 The Riemann zeta function As is the custom in analytic number theory,
complex variables will now be written in the form s = о + it. In the half-plane о > 1,
the zeta function is defined by the series
Since | n~s| =n~°, the series converges uniformly on every compact subset of this
half-plane, and С is holomorphic there.
A simple computation gives
JV+l N „n+l N
s Mx^Jx^s^n x-'-'dx^ 5>~s - JV(iV 4-l)'s.
When a > 1, iV(JV + l)~s -> 0 as iV-> oo. Hence
B) ^CW =

TAUBERIAN THEORY 215
If b(x) = [x] - x, it follows from B) that
C) as) = ^~,+s Гь(х)х~1 ~s dx {a > 1).
Since b is bounded, the last integral defines a holomorphic function in the half-plane
о > 0. Thus C) furnishes an analytic continuation of С to с > 0, which is holomorphic
except for a simple pole at s = 1, with residue 1. The most important property we shall
need is that С has no zeros on the line a = 1 :
D) C(l+//)^0 (-oo</<oo).
The proof of D) depends on the identity
Since A - у?) = 1 +p~s + p ~2s + * • *, the fact that the product E) equals the
series A) is an immediate consequence of the fact that every positive integer has a
unique factorization into a product of powers of primes. Since J]p~a < oo if a > 1,
E) shows that CO) ф 0 if о > 1 and that
F). log as)=i 5m-vms (*>i).
p m— 1
Fix a real / Ф 0. If о > 1, F) implies that
G) log KVXV + 'OCO + 2ft)I = £ ™~ V"m<T Re {3 + 4/T/m' + p~2imt} > 0,
because Re {3 + 4ег0 + e2i9} = 2A + cos вJ for all real в. Hence
(8)
C(<7 + it)
о - 1
1
a- 1
If C(l + г7) were 0, the left side of (8) would converge to a limit, namely,
| CXI + '01 41 CO + 2ft)l» as a decreases to 1. Since the right side of (8) tends to
infinity, this is impossible, and D) is proved.
9Л2 Ingham's tauberian theorem Suppose g is a real nonde creasing function on
@, со), g(x) = 0 if x<l,
A) <Kx)=t
and
B) G(x) = ax log x + bx + xs(x),
where a, b are constants and s(x) -> 0 as x -> oo. Then
C) lim x~1g(x) = a.
X-+OO

216 DISTRIBUTIONS AND FOURIER TRANSFORMS
) !
Ifg is the function ф defined in Section 9.10, Ingham's theorem implies, in view
of Equations C) and E) of Section 9.10, that F) of Section 9.10 holds, and this, as we
saw there, gives the prime number theorem.
proof We first show that xg(x) is bounded. Since g is nondecreasing,
= xla log 2 + e(x) - el-I < Ax,
where A is some constant. Since
it follows that
D) g(x)<
We now make a change of variables that will enable us to use Fourier
transforms in a familiar setting. For — oo < x < oo, define
E) h(x) = g(e*\ Щх) = f>(* - log и).
Then h(x) = 0 if x < 0, H(x) = G{e*)\ hence B) becomes
F) H(x) - ex(ax + b + e^x))
where ег(х) -> 0 as x -> oo. If
G) ф(х) = е-х/1(х) (-оо<х<сх)),
then ф is bounded, by D). We have to prove that
(8) lim ф(х) - а.
x-+ oo
Put k(x) = [ex]e~x, let Я be a positive irrational number, and define
(9) K(x) - Щх) - fc(x - 1) - k(x - A) (- oo < x < oo).
Then Kel}(—ao, oo); in fact, e*K{x) is bounded. (See Exercise 8). If s =
<7 + rt, <r > 0, then formula B) of Section 9.11 shows that
f°° -xs Г
:= С
+

TAUBERIAN THEORY 217
Repeat this with k(x — 1) and k(x — X) in place of k(x), use (9), and then let
a -> 0. The result is
r°° t(\ 4- it)
A0) Г K(x)e~itx dx = B- e~u - g-"') Л
J-oo 1 + it
Since C(l + it) Ф 0 and since X is irrational, K@ # 0 if t Ф 0. Since С has a
pole with residue 1 at s = 1, the right side of A0) tends to 1 + X as f -> 0. Thus
K@) Ф 0.
To apply Wiener's theorem, we have to estimate К * ф. То do this, put
u(x) = [ex], let v be the characteristic function of [0, oo), and let /x be the measure
that assigns mass 1 to each point of the set {log n: n = 1, 2, 3, ...} and whose
support is this set. By E), H = h * pi. Also, и = v * /x. Hence
A1) (h * u)(x) = (h*v* MX) = (H * i;)(jc) = Сн{у) dy.
(Note that we now take convolutions with respect to Lebesgue measure, not
with respect to the normalized measure mx.) Since
(ф * k)(x) = Г ey~xh(x - y)[e>]e~y dy = e~x(h * u)(x),
^ -oo
F) and A1) imply that
A2) (ф * k){x) = e~x Гн(у) dy = ax + b-a + e2(jc),
Jo
where s2(x) -> 0 as x -> oo. By A2) and (9),
A3) lim (K * ф)(х) = (\+k)a = a f K(y) dy.
x~*-oo — oo
Therefore Wiener's theorem 9.7 (see also Remark 9.8) implies that
A4) lim (/* <£)(x) = « Г f(y)dy
x~* oo — oo
for every fe l}{— oo, oo).
Let /i and /2 be nonnegative functions whose integral is 1 and whose
supports lie in [0, e] and [ — e, 0], respectively. By G), ехф(х) is nondecreasing.
Thus ф(у) < еЕф(х) if x - г < у < х, and ф(у) > е~Еф(х) if x < у < x + s.
Consequently,
A5) e~\U * ф)(х) < ф(х) < e\f2 *
It follows from A4) and A5) that the upper and lower limits of ф(х), as x -> со,
lie between ae~e and ae£. Since e > 0 was arbitrary, (8) holds, and the proof is
complete. ////

218 DISTRIBUTIONS AND FOURIER TRANSFORMS
The Renewal Equation
As another application of Wiener's tauberian theorem we shall now give a brief dis-
discussion of the behavior of bounded solutions ф of the integral equation
4>(x)-f <Mx-t)d0(t)=f(x)
which occurs in probability theory. Here fi is a given Borel probability measure^is a
given function, and ф is assumed to be a bounded Borel function, so that the integral
exists for every x e R. The equation can be written in the form
Ф-Ф*Р=/,
for brevity.
We begin with a uniqueness theorem.
9.13 Theorem If ц is a Borel probability measure on R whose support does not lie
in any cyclic subgroup of R, and if ф is a bounded Borel function that satisfies the homo-
homogeneous equation
A) ф(х)-(ф*ц)(х) = 0
for every x e R, then there is a constant A such that ф(х) = A except possibly in a set
of Lebesgue measure 0.
proof Since \i is a probability measure, ju(O) = 1. Suppose that j)(t) = 1 for
some t Ф 0. Since
B) №= С
J-0
it follows that ц must be concentrated on the set of all x at which e~iXt = 1,
that is, on the set of all integral multiples of 2n/t. But this is ruled out by the
hypothesis of the theorem.
If о = S — pi, where S is the Dirac measure, then о = 1 — ft. Hence a(t) = 0
if and only if t = 0, and A) can be written in the form
C) ф * <t = 0.
Put g(x) = exp {-x2); put К = g * a. Then К el}, K(t) = 0 only if
t = 0, and C) shows that К * ф = 0. By Theorem 9.3 (with the one-dimensional
space generated by К in place of Y) the distribution ф has its support in {0}.
Hence ф is a finite linear combination of S and its derivatives (Theorem 6.25),
so that ф is a polynomial, in the distribution sense. Since nonconstant poly-
polynomials are not bounded on R, and since ф is assumed to be bounded, we have
reached the desired conclusion. ////

TAUBERIAN THEORY 219
9.14 Convolutions of measures If ц and к are complex Borel measures on Rn,
then
A) /->f f
is a bounded linear functional on C0(R"), the space of all continuous functions on Rn
that vanish at infinity. By the Riesz representation theorem, there is a unique Borel
measure \i * к on Rn that satisfies
B) f fd(ji*k)=\ \ J(x + y)dfi(x)dX(y) [feC0(Rn)].
A standard approximation argument shows that B) then holds also for every bounded
Borel function /. In particular, we see that
C) " ' Qi*Ar = fiX.
Two other consequences of B) will be used in the next theorem. One is the
almost obvious inequality
D) \Ы*Ц<Ы\\Ц,
where the norm denotes total variation. The other is the fact that \i * к is absolutely
continuous (relative to Lebesgue measure mn) if this is true of /л; for in that case,
E) f f(x+y)dix{x)=Q
JRn
for every у e Rn, if /is the characteristic function of a Borel set E with mn(E) — 0, and
B) shows that Qi * k)(E) = 0.
Recall that every complex Borel measure /x has a unique Lebesgue decomposi-
decomposition
F) V=Va + Vs,
where \xa is absolutely continuous relative to mn and jas is singular.
The next theorem is due to Karlin.
9.15 Theorem Suppose \x is a Borel probability measure on R, such that
A) На Ф 0,
B) Г |*| «//!(*)< 00,
J -oo
C) M= Г xdii(x)?0.
* — an

220 DISTRIBUTIONS AND FOURIER TRANSFORMS
Suppose thatfe L}(R), thatf(x) -> 0 as x -> ±00, and that ф is a bounded function
that satisfies
D) ф(х) - (ф * jOOc) =/(*) (- 00 < x < 00).
Then the limits
E) ф(оо) = \im ф(х), ф(— оо) = lim (/>(%)
F) 0@0) - ф(- 00) = -J- f
proof Put с = S — /i, as in the proof of Theorem 9.13. Define
The assumption B) guarantees that T^g L}(R). A straightforward computation,
whose details we omit, shows that
and that
(9) f/(*) dx = (K* 0)(j) - (X * ф)(г) (- oo < r < s < 00),
since /= ф * (T.
By A), ju is not singular. The argument used at the beginning of the proof
of Theorem 9.13 shows therefore that 6(i) Ф 0 if ХФ 0. Hence (8) and C) imply
that К has no zero in R.
Since fel}(R), (9) implies that К* ф has limits at ±00, whose difference
isj^oo/
We shall show that ф is slowly oscillating. Once this is done, E) and F)
follow from the properties of К and К * ф that we just proved, by Pitt's theorem
(b) of 9.7.
Repeated substitution of ф =/+ ф * ц into its right-hand side gives
A0) ф =/ + /* 11 + -• +/* j^ + Ф * ^
=h + 9n + K (« = 2,3,4,...),
where p1 = p,/л" = p. * \in~xJn =/+ • • • +/* /i", and
01) ' д, = Ф*№а, Ья = ф

TAUBERIAN THEORY 221
Foreach n,fn(x) -» 0 as x -» ± oo, and gn is uniformly continuous. Hence
fn + 9n ls slowly oscillating. Since the total variations satisfy
A2) ||(/x")J < HOiJ-ll < ЦК
we have
A3) \hn(x)\ < \\ф\\ • ll^ll- (-oo <*< oo),
where \\ф\\ is the supremum of \ф\ on R. By A), ||juJ < 1. Hence /*„->(),
uniformly oni?. Consequently, ф is the uniform limit of the slowly oscillating
functions fn+ gn. This implies that ф is slowly oscillating, and completes the
proof. ////
Exercises
/ Prove the theorem of Tauber stated in Section 9.1.
2 Suppose ф е L°°(R") and the support of the distribution <j> consists of к distinct points
su ... ;sk. Construct suitable functions ф1у ..., фк such that (ф * ф^~ has the singleton
{sj} as support, and conclude that ф is a trigonometric polynomial, namely,
ф(х) = atf15! 'x-\ \-ak eisk •x (a.e.).
(The case к = 1 is done in the proof of Theorem 9.13.)
3 Suppose У is a closed translation-invariant subspace of Ll(Rn) such that Z(Y) consists
of к distinct points. (The notation is as in Theorem 9.3.) Use Exercise 2 to prove that
Fhas codimension к in Ll(Rn), and conclude from this that Y consists of exactly those
feL\Rn) whose Fourier transforms are 0 at every point of Z(Y).
4 Prove the following analogue of {a) of Theorem 9.7: If феЬ™№п\ and if to every
teRn corresponds a function KteViR") such that Kt(t) Ф 0 and (Кг*ф)(х)->0 as
|*| -> oo, then (/* ф)(х)-*0 as |x| -> oo, for every feL^R").
5 Assume KeL1(Rn) and К has at least one zero in Rn. Show that then there exists
феЬ'*'^) such that (К* ф)(х) = 0 for every xe Rn, although ф does not satisfy the
conclusion of (a) of Theorem 9.7.
6 If ф(х) = sin (x2), — oo < x < oo, show that
lim (/*<£)(*) = 0
|л|-юо
for every feL 1(R)y although the conclusion of (b) of Theorem 9.7 does not hold.
7 For a > 0, let fa be the characteristic function of the interval [0, a]. Define fp in the
same way; put g =fa +fp. Prove that the set of all finite linear combinations of trans-
translates of g is dense in Ь^{К) if and only if pi a is irrational.
8 If a >0 and ax — 1, prove that
and deduce from this that exK{x) is bounded, as asserted in the proof of Theorem 9.12.

222 DISTRIBUTIONS AND FOURIER TRANSFORMS
9 Let Q denote the set of all rational numbers. Lef/x be a probability measure on R that
is concentrated on Q, and let ф be the characteristic function of Q. Show that ф(х) =
(ф * /х)(х) for every x e R, although ф is not constant. (Compare with Theorem 9.13.)
What other sets could be used in place of Q to achieve the same effect ?
10 Special cases of the following facts were used in Theorem 9.15. Prove them.
(a) If <£ e L°°(i?n) and к eLl(Rn), then к * ф is uniformly continuous.
(b) If tyj} is a sequence of slowly oscillating functions on Rn that converges uniformly
to a function фу then ф is slowly oscillating.
(c) If /x and Л are complex Borel measures on R", then
Put ф(х) = cos (I x 11/3) and define
1 r1
f{pc) = ф(х) - - J^O ~y)dy (- со < x < oo).
Prove that /e (L1 n C0)(i?) but that no bounded solution of the equation
has limits at + oo or at -co. (This illustrates the relevance of the condition M Ф 0 in
Theorem 9.15.)
12 Let /x be a probability measure concentrated on the integers. Prove that every function
ф on R which is periodic with period 1 satisfies ф — ф * /x = 0. (This is relevant to
Theorems 9.13 and 9.15.)
13 Assume^ e L°°@, oo),
fV(*)|-<oo, Г\Щх)\ — < со,
Jo X Jo X
f K(x)x~lt — Ф 0 for -oo < t < oo,
Jo x
and
Prove that
(
This is an analogue of (a) of Theorem 9.7. How would "slowly oscillating" have
to be defined to obtain the corresponding analog of (b) of Theorem 9.7 ?
14 Complete the details in the following outline of Wiener's proof of Littlewood's theorem.
Assume \nan\ <1, f{r)=^o anrn, and /(r)->0 as r->l. If jn = a0H Л-ап, it is
to be proved that sn ->0 as n -» oo.

TAUBERIAN THEORY 223
(a) \sn -/A - l//i)| < 2. Hence {sn} is bounded.
(Z>) If ф(х) = 5-n on [n, n + 1) and 0 < x < j>, then
(c) J? xe~"<f>{t) dt =f(e-*) -> 0 as x -> 0. Hence
if
Гк(х)х~и — = ГA + it) Ф 0 if t is real.
Jo x
Put H{x) = l/(ejc) if A + e) < X < 1, ад = О otherwise. Conclude that
\
1
J лA+£)Л
lim - \ ^O) Jy = 0.
(/) By (b) and (<?), lim ф(х) = О.
Note: If nan->0 is assumed to hold, then a modification of Step (a) is all that is
needed for the proof.
15 Let Y be a closed subspace of Ll(R"). Prove that F is translation-invariant if and only if
f*ge Y whenever /e Y and g e Ll(Rn).
The closed translation invariant subspaces of Ll(R") are thus exactly the same as
the closed ideals in the convolution algebra Ll(Rn).

PART THREE
Banach Algebras and
Spectral Theory

10
BANACH ALGEBRAS
Introduction
10.1 Definition A complex algebra is a vector space A over the complex field С
in which a multiplication is defined that satisfies
A) x(yz) = (xy)z,
B) (x + y)z = xz + yz, x{y + z) = xy + xz,
and
C) cc(xy) = (ccx)y = x(ccy)
for all x, y, and z in A and for all scalars a.
If, in addition, A is a Banach space with respect to a norm that satisfies the
multiplicative inequality
D) 1кИ1<МЫ1 (хеА,уеА)
and if A contains а шн7 element e such that
E) jte = ex — x (x e A)

228 BANACH ALGEBRAS AND SPECTRAL THEORY
and
F) И = l,
then A is called a Banach algebra.
Note that we have not required that A be commutative, i.e., that xy = yx for
all x and у in A, and we shall not do so except when explicitly stated.
It is clear that there is at most one e e A that satisfies E), for if e' also satisfies
E), then e' = e'e = e.
The presence of a unit is very often omitted from the definition of a Banach algebra.
However, when there is a unit it makes sense to talk about inverses, so that the spectrum
of an element of A can be defined in a more natural way than is otherwise possible.
This leads to a more intuitive development of the basic theory. Moreover, the resulting
loss of generality is small, because many naturally occurring Banach algebras have a
unit, and because the others can be supplied with one in the following canonical
fashion.
Suppose A satisfies conditions A) to D), but A has no unit element. Let At
consist of all ordered pairs (x, a), where x e A and a e <£. Define the vector space
operations in Ax componentwise, define multiplication in A1 by
G) (x, oc)(y, P) = (xy + *y + fix, a/0,
and define
(8) ||(*, a)|| = \\x\\ + |a|, e = @,l).
Then A{ satisfies properties A) to F), and the mapping x -> (x, 0) is an isometric
isomorphism of A onto a subspace of At (in fact, onto a closed two-sided ideal of At)
whose codimension is 1. If x is identified with (x, 0), then A± is simply A plus the
one-dimensional vector space generated by e. See Examples }03(d) and 11.13(e).
The inequality D) makes multiplication a continuous operation in A. This means
that if xn -> x and yn ->у then xnyn -> xy, which follows from the identity
(9) xnyn ~xy = (xn - x)yn + x(yn - y).
In particular, multiplication is left-continuous and right-continuous:
A0) xny-+xy and xyn-+xy
if xn-+x and yn->y.
Jt is interesting that D) can be replaced by the (apparently) weaker requirement
A0) and that F) can be dropped without enlarging the class of algebras under con-
consideration.
10.2 Theorem Assume that A is a Banach space as well as a complex algebra
with unit element e Ф 0, in which multiplication is left-continuous and right-continuous.

BANACH ALGEBRAS 229
Then there is a norm on A which induces the same topology as the given one and which
makes A into a Banach algebra.
(The assumption e Ф 0 rules out the uninteresting case A = {0}.)
proof Assign to each x e A the left-multiplication operator Mx defined by
A) , Mx{z)=xz {zeA).
Let A be the set of all Mx. Since right multiplication is assumed to be continuous,
A cz &(A), the Banach space of all bounded linear operators on A.
It is clear that x -> Mx is linear. The associative law implies that Mxy =
Mx My. If x e A, then
B) Ы = \\хе\\ = \\Мхе\\<\\Мх\\\\е\\.
These facts can be summarized by saying that x -> Mx is an isomorphism of A
onto the algebra A, whose inverse is continuous. Since
C) \\MxMy\\ < ||М,||||М,|| and \\Me\\ = \\I\\ = 1, '
A is a Banach algebra, provided it is complete, i.e., provided it is a closed sub-
\ space of &(A), relative to the topology given by the operator norm. (See
Theorem 4.1.) Once this is done, the open mapping theorem implies that x ~> Mx
is also continuous. Hence ||jc|| and \\MX\\ are equivalent norms on A.
Suppose ТеЩА), ТгеА, and T{-±T in the topology of ЩА). If Tt
is left multiplication by xt e A, then
D)
As i->oo, the first term in D) tends to T(y), and Ть{е)->Т{е). Since multi-
multiplication is assumed to be left-continuous in A, it follows that the last term of
D) tends to T{e)y. Put x = T(e). Then
E) T(y) = T(e)y = xy = Mx(y) (у е A),
so that T= MxeA, and A is closed. ////
10.3 Examples (a) Let C(JC) be the Banach space of all complex continuous
functions on a nonempty compact Hausdorff space K, with the supremum norm.
Define multiplication in the usual way: (fg)(p) =f(p)g(p). This makes C{K) into
a commutative Banach algebra; the constant function 1 is the unit element.
If К is a finite set, consisting of, say, n pointy, then C{K) is simply Cn, with
coordinatewise multiplication.
In particular, when n — 1, we obtain the simplest Banach algebra, namely (£,
with the absolute value as norm.

230 BANACH ALGEBRAS AND SPECTRAL THEORY
(b) Let X be a Banach space. Then 08(X), the algebra of all bounded linear
operators on X, is a Banach algebra, with respect to the usual operator norm. The
identity operator /is its unit element. If dim X = n < oo, then 08{X) is (isomorphic to)
the algebra of all complex n-by-n matrices. If dim X > 1, then 08{X) is not com-
commutative. (The trivial space X = {0} must be excluded.)
Every closed subalgebra of 08 (X) that contains / is also a Banach algebra. The
proof of Theorem 10.2 shows, in fact, that every Banach algebra is isomorphic to one
of these.
(c) If К is a nonempty compact subset of <£, or of 0", and if A is the subalgebra
of C(K) that consists of those fe C{K) that are holomorphic in the interior of K,
then A is complete (relative to the supremum norm) and is therefore a Banach algebra.
When К is the closed unit disc in 0, then A is called the disc algebra.
(d) ^(R"), with convolution as multiplication, satisfies all requirements of
Definition 10.1, except that it lacks a unit. One can adjoin one by the abstract
procedure outlined in Section 10.1 or one can do it more concretely by enlarging
to the algebra of all complex Borel measures ц on jR" of the form
where feL1(R"), S is the Dirac measure on R", and A is a scalar.
(e) Let M(Rn) be the algebra of all complex Borel measures on Rn, with con-
convolution as multiplication, normed by the total variation. This is a commutative
Banach algebra, with unit 3, which contains (d) as a closed subalgebra.
10.4 Remarks There are several reasons for restricting our attention to Banach
algebras over the complex field, although real Banach algebras (whose definition
should be obvious) have also been studied.
One reason is that certain elementary facts about holomorphic functions play
an important role in the foundations of the subject. This may be observed in Theorems
10.9 and 10.13 and becomes even more obvious in the symbolic calculus.
Another reason—one whose implications are not quite so obvious—is that 0
has a natural nontrivial involution (see Definition 11.14), namely, conjugation, and
that many of the deeper properties of certain types of Banach algebras depend on the
presence of an involution. (For the same reason, the theory of complex Hilbert
spaces is richer than that of real ones.)
At one point (Theorem 10.44) a topological difference between' С and R will
even play a role.
Among the important mappings from one Banach algebra into another are the
homomorphisms. These are linear mappings h that are also multiplicative:
h{xy) = h(x)h(y).

BANACH ALGEBRAS 231
Of particular interest is the case in which the range is the simplest of all Banach algebras,
namely, <£ itself. Many of the significant features of the commutative theory depend
crucially on a sufficient supply of homomorphisms onto С
Complex Homomorphisms
10.5 Definition . Suppose A is a complex algebra and ф is a linear functional on A
which is not identically 0. If
A) - ф(ху) = ф(х)ф(у)
for all x e A and у e A, then ф is called a complex homomorphism on A.
(The exclusion of ф = 0 is, of course, just a matter of convenience.)
An element x e A is said to be invertible if it has an inverse in A, that is, if there
exists an element лс" е A such that
B) x~lx = xx~l =e,
where e is the unit element of A.
Note that no x e A has more than one inverse, for if yx = e = xz then
у = ye = y(xz) = (yx)z = ez = z.
10.6 Proposition If ф is a complex homomorphism on a complex algebra A with
unit e, then ф(е) = 1, and ф(х) Ф Ofor every invertible x e A.
proof For some у е А, ф(у) Ф 0. Since
Ф(у) = Ф(Уе) = Ф(у)Ф(е),
it follows that ф(е) = 1. If jc is invertible, then
so that ф(х) Ф 0. ////
Parts (a) and (c) of the following theorem are perhaps the most widely used
facts in the theory of Banach algebras; in particular, (c) implies that all complex
homomorphisms of Banach algebras are continuous.
10.7 Theorem Suppose A is a Banach algebra, x e A, \\x\\ < 1. Then
(a) e — x is invertible,
m K.-*>--.-*u7Mlf.
(с) \ф(х)\ < 1 for every complex homomorphism ф on A.

232 BANACH ALGEBRAS AND SPECTRAL THEORY
proof Since ||jc"|| < \\x\\" and ||x|| < 1, the elements
A) sn = e
form a Cauchy sequence in A. Since A is complete, there exists s e A such that
sn -> s. Since xn -> 0 and
B) j^*-*)^-*"*1 = (*-*)•■*,,,
the continuity of multiplication implies that s is the inverse of e — x. Next,
A) shows that
\\s-e- x\\ = ||x2 + x3 + • • -|| < £ Ц*1Г =
n=2
~1
Finally, suppose XeC, \X\ > 1. By/(a), e — l~1x is invertible. By
Proposition 10.6, I
1 - Г>(х) = ф(е - Я^) ^ 0.
Hence ф(л:) ^ Я. This completes the proof. ////
We now interrupt the main line of development and insert a theorem which
shows, for Banach algebras, that Proposition 10.6 actually characterizes the complex
homomorphisms among the linear functionals. This striking result has apparently
found no interesting applications as yet.
10.8 Lemma Suppose f is an entire function of one complex variable, /@) = 1,
/'@) = 0, and
A) 0< |/(A) | <e^ (Ae<£).
Thenf(X)= I for all X eg.
proof Since/has no zero, there is an entire function g such that/= exp{#},
g@) =g'@) = 0, and Re [д(Щ < |Я|. This inequality implies
B) \9W\<\2r-g(X)\
The function
is holomorphic in {A: |A| < 2r}, and |/rr(A)| < 1 if | A| = r. By the maximum
modulus theorem,
D)
Fix Д and let r -> oo.- Then C) and D) imply that #(A) = 0. ////

BANACH ALGEBRAS 233
10.9 Theorem (Gleason, Kahane, Zelazko) If ф is a linear functional on a
Banach algebra A, such that ф(е) = 1 and ф(х) Ф 0 for every invertible x e A, then
A) Ф(ху) = ф(х)ф(у) (хеА,уе А).
Note that the continuity of ф is not part of the hypothesis.
proof Let N be the null space of ф. If x e A and у е A, the assumption
ф{е) = 1 shows that
B) ' * = * + <£(*)<?, J> = * + <K>0e,
where a e N, b e N. If ф is applied to the product of the equations B), one
obtains
C) ф(ху) = ф(аЪ) + ф{х)ф(у).
The desired conclusion A) is therefore equivalent to the assertion that
. D) abeN if a e N and b e N.
Suppose we had proved a special case of D), namely,
. E) a2 e N if a e N
Then C), with x = y, implies
F) Ф(х2) = [ф(х)]2 (хеА).
Replacement of x by x + у in F) results in
G) ф(ху + Ух)=2ф(х)ф(у) (хеА,уеА).
Hence
(8) xy-\-yxeN if x'eN, ye A.
Consider the identity
(9) (xy - yxJ + (xy + yxJ = 2[x(yxy) + (yxy)x].
IfxeN, the right side of (9) is in N, by (8), and so is (xy + yxJ, by (8) and F).
Hence (xy — yxJ is in N, and another application of F) yields
A0) xy+*-yxeN if x e N, у е A.
Addition of (8) and A0) gives D), hence A).
Thus E) implies A), for purely algebraic reasons. The proof of E) uses
analytic methods.
By hypothesis, N contains no invertible element of A. Thus \\e — x\\ > 1
for every x e N9 by (a) of Theorem 10.7. Hence
A1) Пе-х\\> |Я| = \ф(Ле-х)\ (x eN, A e <Z).

234 BANACH ALGEBRAS AND SPECTRAL THEORY
We conclude that ф is a continuous linear functional on A, of norm 1.
To prove E), fix a e N, assume ||a|| = 1 without loss of generality, and
define
A2) ,,-V^,
Since |<Ka")| < ||a"|| < ||оЦи - 1,/is entire and satisfies \f{X)\ < ехр |A| for
all X e <Z. Also,/@) = ф{е) = 1, and/'@) - ф(а) = О.
If we can prove that f{X) Ф 0 for every X e 0, Lemma 10.8 will imply
that/40) = 0; hence ф(я2) = 0, which proves E).
The series
00 Xn
A3) £(A) X
converges in the norm of A, for every X e С The continuity of ф shows that
A4) f(X) = ф(Е(Х)) (X e C).
The functional equation E(X + /л) = E(X)E(fi) follows from A3) exactly as in the
scalar case. In particular,
A5) E(X)E(-X) = E@) = e (keC).
Hence E(X) is an invertible element of A, for every Xe<£. This implies, by hy-
hypothesis, that ф(Е(Х)) Ф 0, and therefore f(X) ф 0, by A4). This completes the
proof. ////
Basic Properties of Spectra
10.10 Definitions Let A be a Banadi algebra; let G = G{A) be the set of all
invertible elements of A. If x e G and у e G, then j-1x is the inverse of x~V; thus
x~xy e G, and G is a group.
If x eA, the spectrum o(x) of x is the set of all complex numbers X such that
Xe — x is nor invertible. The complement of a(x) is the resolvent set of jc; it consists.of
all X e С for which {Xe — x)~x exists.
The spectral radius of x is the number
( Л\ n(-V-\ С-11ТЛ /I 3 I • 3 CT rr(-b-W
\L) r\^) — bUp \ A . Л t UyXjj.
It is the radius of the smallest closed circular disc in 0, with center at 0, which contains
o(x). Of course, A) makes no sense if o(x) is empty. But this never happens, as we
shall see.

BANACH ALGEBRAS 235
10.11 Theorem Suppose A is a Banach algebra, x e G(A), Л е Л, || A || < i || x" * 1Г * •
then x + h e G{A), and
A) ||(x + /2)-1-^-1+jc-1Ax-1||<2||x-1||3P||2.
proof Since x + h = x(e + jc/?) and \\x~1h\\<%, Theorem 10.7 implies
that x + h e G{A) and that the norm of the right member of the identity
(x + hy1 - x-1 + x-'hx'1 = [(e + x-'h)'1 -ел- х'Щх'1
is at most 2||jc-1A||2||jc-1||. ////
10.12 Theorem If A is a Banach algebra, then G(A) is an open subset of A, and the
mapping x-^x'1 is a homeomorphism of G(A) onto G(A).
proof That G(A) is open and that x -> x~* is continuous follows from Theorem
10.11. Since x ~> x~x maps G(A) onto G(A) and since it is its own inverse, it is a
homeomorphism. ////
10.13 Theorem If A is a Banach algebra and x e A, then
(a) the spectrum a(x) of x is compact and nonempty, and
(b) the spectral radius p(x) of x satisfies
A) p(x) = lim ||xn|l1/n = inf ||jc1 1/и.
n->oo n> 1
Note that the existence of the limit in A) is part of the conclusion and that the
inequality
B) P«<IMI
is contained in the spectral radius formula A).
proof If | Я| > ||jc|| thence — l~1x lies in G(A), by Theorem 10.7, and so does
Xe — x. Thus X ф a(x). This proves B). In particular, o(x) is a bounded set.
To prove that a{x) is closed, define g: <Z -> A by g(X) = Xe — x. Then g is
continuous, and the complement Q of o{x) is g~1(G{A)), which is open, by
Theorem 10.12. Thus a(x) is compact.
Now define/: Q -> G(A) by
C)

236 BANACH ALGEBRAS AND SPECTRAL THEORY
Replace x by le — x and h by (ц — l)e in Theorem 10.11. If A e Q and \i is
sufficiently close to A, the result of this substitution is
D) ||/(/i) -/(A)
so that
E) li
Thus/is a strongly holomorphic Л-valued function in Q.
If | A | > ||jc||, the argument used in Theorem 10.7 shows that
F) /(A) = f Я-»" V = Г1е + Г2х +-".
This series converges uniformly on every circle Гг with center at 0 and radius
r> ||jc||. By Theorem 3.29, term-by-term integration is therefore legitimate.
Hence
If a{x) were empty, П would be 0, and the Cauchy theorem 3.31 would
imply that all integrals in G) are 0. But when n = 0, the left-hand side of G) is
e Ф 0. This contradiction shows that a(x) is not empty.
Since Q contains all A with | A| > p(x), an application of C) of the Cauchy
theorem 3.31 shows that the condition r > \\x\\ can be replaced in G) by r >p{x).
If
(8) M(r) = max ||/(rel'0)|| (r > p(x)),
в
the continuity of/implies that M{r) < oo. Since G) now gives
we obtain
A0) lim sup ||x"||1/n < r (r > p(x))
n-+ oo
so that
A1) lim sup
On the other hand, if A e o{x), the factorization
A2) Xne - x" = (Ae -

BANACH ALGEBRAS 237
shows that kne - xn is not invertible. Thus 1" e o(xn). By B), \Xn\ < \\x"\\ for
n = 1, 2, 3, Hence
A3) p(x)<M\\xn\\i/n,
and A) is an immediate consequence of A1) and A3). ////
The nonemptiness of a(x) leads to an easy characterization of those Banach
algebras that are division algebras. )
10.14 Theorem (Gelfand-Mazur) If A is a Banach algebra in which every
nonzero element is invertible, then A is (isometrically isomorphic to) the complex field.
proof If oc e A and lx Ф A2, then at most one of the elements Xxe — x and
l2 e — x is 0; hence at least one of them is invertible. Since a(x) is not empty, it
follows that a{x) consists of exactly one point, say Цх), for each x e A. Since
A(x)e — x is not invertible, it is 0. Hence x = X(x)e. The mapping x -> l(x) is
therefore an isomorphism of A onto 0, which is also an isometry, since \Цх)\ =
Theorems 10.13 and 10.14 are among the key results of this chapter. Much of
the content of Chapters 11 to 13 is independent of the remainder of Chapter 10.
10.15 Remarks (a) Whether an element of A is or is not invertible in A is a
purely algebraic property. The spectrum and the spectral radius of an x e A are
thus defined in terms of the algebraic structure of A, regardless of any metric (or
topological) considerations. On the other hand, lim ||x"||1/n depends obviously on
metric properties of A. This is one of the remarkable features of the spectral radius
formula: It asserts the equality of certain quantities which arise in entirely different
ways.
(b) Our algebra A may be a subalgebra of a larger Banach algebra B, and it
may then very well happen that some x e A is not invertible in A but is invertible in B.
The spectrum of x depends therefore on the algebra. The inclusion <?A(x) :=> oB{x)
holds (the notation is self-explanatory); the two spectra can be different. The spectral
radius is, however, unaffected by^ the passage from A to B, since the spectral radius
formula expresses it in terms of metric properties of powers of x, and these are in-
independent of anything that happens outside A.
Theorem 10.18 will describe the relation between aA(x) and aB{x) in greater
detail.
10.16 Lemma Suppose V and W are open sets in some topological space X,
V cz W, and W contains no boundary point of V. Then V is a union of components of W.

238 BANACH ALGEBRAS AND SPECTRAL THEORY
Recall that a component of ^is, by definition, a maximal connected subset of W.
proof Let п be a component of J^that intersects V. Let U be the complement
of V. Since W contains no boundary point of V, Q is the union of the two
disjoint open sets D n V and Q n U. Since Q is connected, fi n I/ is empty.
Thus QcK. ////
10.17 Lemma Suppose A is a Banach algebra, xn e G(A) for n = 1, 2, 3, ..., x is
a boundary point of G{A), and xn —> x as n —> oo.
Then \x~11| -> oo as n -+ oo.
proof If the conclusion is false, there exists M < oo such that ||jc^~x || <M
for infinitely many n. For one of these, ||д:и — jc|| < 1/M. For this л,
Hz, __ y-^H = Цу-^у — JC^II < 1
|| t: л,п л\\ || л,п \J^n луц -v. x,
so that x~lx e G(A). Since x = лгДлс ~! jc) and <7(Л) is a group, it follows that
x e G(A). This contradicts the hypothesis, since G(A) is open.
10.18 Theorem
(a) If A is a closed subalgebra of a Banach algebra B, and if A contains the unit
element ofB, then G(A) is a union of components of A n G(B).
(b) Under these conditions, if x e A, then ga(x) is the union of <jb{x) and a {possibly
empty) collection of bounded components of the complement ofaB(x). In particular,
the boundary ofaA(x) lies in eB{x).
proof (a) Every member of A that has an inverse in A has the same inverse
in B. Thus G(A) с G(B). Both G(A) and A n G(B) are open subsets of A. By
Lemma 10.16, it is sufficient to prove that G(B) contains no boundary point ;> of
G(A).
Any such у is the limit of a sequence {xn} in G(A). By Lemma 10.17,
ll^n!! -* °o- If у were in G(B), the continuity of inversion in G(B) (Theorem
10.12) would force x'1 to converge to y~l. In particular {Цх!!} would be
bounded. Hence у $ G(B), and (a) is proved.
(b) Let QA and QB be the complements of ga(x) and of (?B(x), relative
to d. The inclusion QA с QB is obvious, since XeQa if and only if Xe — x e
G(A). Let Xo be a boundary point of QA. Then Aoe — x is a boundary point of
G(A). By (a), loe - x ф G(B). Hence Яо $ пв. Lemma 10.16 implies now that
QA is the union of certain components of QB. The other components of QB
are therefore subsets^of aA{x). This proves (b). ////

BANACH ALGEBRAS 239
Corollary IfaB(x) does not separate C, that is, if its complement QB is connected,
then ga(x) = aB(x).
For then QB has no bounded components.
The most important application of this corollary occurs when gb(x)
contains only real numbers.
As another application of Lemma 10.17 we now prove a theorem whose con-
conclusion is the same as that of the Gelfand-Mazur theorem, although its consequences
are not nearly so important.
10.19 Theorem If A is a Banach algebra and if there exists M < oo such that
A) 11*11 IMI<M||xy|| (xeA,yeA),
then A is (isometrically isomorphic to) С
proof Let у be a boundary point of G{A). Then у = lim yn for some sequence
{yn} in G{A). By Lemma 10.17, Ц^!! -> oo. By hypothesis,
"'•B) WynWbn'W^MWeW (/i=l,2,3,...).
Hence \\yn\\ -> 0 and therefore у = 0.
If x e A, each boundary point X of a{x) gives rise to a boundary point
Xe — x of G(A). Thus x = Xe. In other words, A = {Xe: X e C). ////
It is natural to ask whether the spectra of two elements x and у of A are close
together, in some suitably defined sense, if x and у are close to each other. The next
theorem gives a very simple answer.
10.20 Theorem Suppose A is a Banach algebra, x e A, Q is an open set in <Z, and
o(x) cz Q. Then there exists S > 0 such that a(x + y) <= Qfor every у sA with ||>>|| < 6.
proof Since ||{Xe — x)~l\\ is a continuous function of X in the complement
of o(x), and since this norm tends to 0 as X -» oo, there is a number M < oo
such that
for all X outside Q. If у e A, \\y\\ < 1/M, and X ф Q, it follows that
Xe - (x + y) = (Де - *)[<? - (Де - x)" xy]
is invertible in Л, since \\(Xe — хУху\\ < 1; hence X $ a(x + y). This gives the
desired conclusion, with S = 1/M. ////

240 BANACH ALGEBRAS AND SPECTRAL THEORY
Symbolic Calculus
10.21 Introduction If x is an element of a Banach algebra A and if/(/l) =
a0 + • • • + ocnln is a polynomial with complex coefficients oct, there can be no doubt
about the meaning of the symbol/(x); it obviously denotes the element of A defined by
f(x) = ccoe + && + ••• + anxn.
The question arises whether/(x) can be defined in a meaningful way for other functions
/. We have already encountered some examples of this. For instance, during the proof
of Theorem 10.9 we came very close to defining the exponential function in A. In
fact, if f{X) = £алЛ* is any entire function in <£, it is natural to define f{x) e A by
f(x) = Ха*х*> this series always converges. Another example is given by the mero-
morphic functions
oc-X
In this case, the natural definition of f{x) is
which makes sense for all x whose spectrum does not contain a.
One is thus led to the conjecture that/(x) should be definable, within A, when-
whenever/is holomorphic in an open set that contains o(x). This turns out to be correct
and can be accomplished by a version of the Cauchy formula that converts complex
functions defined in open subsets of С to v4-valued ones defined in certain open sub-
subsets of A. (Just as in classical analysis, the Cauchy formula is a much more adaptable
tool than the power series representation.) Moreover, the entities/(jc) so defined (see
Definition 10.26) turn out to have interesting properties. The most important of these
are summarized in Theorems 10.27 to 10.29. \
In certain algebras one can go further. For instance, if x is a bounded normal
operator on a Hilbert space Я, the symbol f(x) can be interpreted as a bounded
normal operator on H when/is any continuous complex function on a(x), and even
when/is any complex bounded Borel function on a(x). In Chapter 12 we shall see
how this leads to an efficient proof of a very general form of the spectral theorem.
10.22 Integration of A-valued functions If A is a Banach algebra and / is a
continuous ^-valued function on some compact Hausdorff space Q on which a
complex Borel measure ц is defined, then \ f d\x exists and has all the properties that
were discussed in Chapter^, simply because A is a Banach space. However, an addi-

BANACH ALGEBRAS 241
tional property can be added to these and will be used in the sequel, namely: If x e A,
then
(i)
and
B)
To prove A), let Mx be left multiplication by x, as in the proof of Theorem
10.2, and let Л be a bounded linear functional on A. Then AMX is a bounded linear
functional. Definition 3.26 implies therefore that
AMX f fdp = f (AMxf) dp = Л f (Mx/) dii9
for every Л, so that
Mxf fd[i = J (Mxf)dn,
which is just another way of writing A). To prove B), interpret Mx to be right multi-
multiplication by x.
10.23 Contours Suppose К is a compact subset of an open па 0, and Г is a
collection of finitely many oriented line intervals yl9 ..., yn in Q, none of which inter-
intersects K. In this situation, integration over Г is defined by
A) Г ф(Х) dl = У Г ф(Х) dL
It is well known that Г can be so chosen that
B) Indr(C
and that the Cauchy formula
C)
then holds for every holomorphic function/in п and for every С е К. See, for instance,
Theorem 13.5 of [23].
We shall describe the situation B) briefly by saying that the contour Г surrounds
К in U.
Note that neither К nor Q nor the union of the intervals yt has been assumed to
be connected.

242 BANACH ALGEBRAS AND SPECTRAL THEORY
10.24 Lemma Suppose A is a Banach algebra, x e A, a e (Z, а ф a(x), Q is the
complement of a in C, and Г surrounds o(x) in Q. Then
A)
-^ f (a - X)\Xe -хуил = (ае-х)" (л = 0, ± 1, ±2, ...).
proof Denote the integral by yn. When I $ a(x), then
(le - хУ1 = (осе - хУ1 + (a - l)(oce - х)~\Хе - лг).
By Section 10.22, yn is therefore the sum of
B) (ae*)(
27П ^r
since Indr (a) = 0, and
C) (ae - x)~l • — f (a - l)n+1(le - x)'1 dL
2ni ^r
Hence
D) (*e-x)yn=yn + 1 (/i = 0, ±1, ±2,...)-
This recursion formula shows that A) follows from the case n = 0.
We thus have to prove that
E) J-.
Zni
Let Гг be a positively oriented circle, centered at 0, with radius r > \\x\\. On
Tr, (le — хУ1 = Л /Г"*". Termwise integration of this series gives E),
with Гг in place of Г. Since the integrand in E) is a holomorphic Л-valued
function in the complement of a(x) (see the proof of Theorem 10.13), and since
F) Indrr(C)=l=Indr(C)
for every С е a(x), the Cauchy theorem 3.31 shows that the integral E) is
unaffected if Г is replaced by Гг. This completes the proof. ////
10.25 Theorem Suppose
A) *
myk
is a rational function with poles at the points am. [P is a polynomial, and the sum in A)
has only finitely many terms.] If x e A and if a(x) contains no pole of R, define
B) , R(x) = P(x) + X стЛ (x - ocm eyk.
m,k

BANACH ALGEBRAS 243
If SI is an open set in 0 ihat contains a(x) and in which R is holomorphic, and if Г
surrounds o(x) in Q, then
C) R(x) = — f R(k)(ke -x)'1 dL
2ni Jr
proof Apply Lemma 10.24. ////
Note that B) is certainly the most natural definition of a rational function of
x e A. The conclusion C) shows that the Cauchy formula achieves the same result.
This motivates the following definition.
10.26 Definition Suppose A is a Banach algebra, п is an open set in 0, and H(Q)
is the algebra of all complex holomorphic functions in п. By Theorem 10.20,
A) AQ={xeA:<7(x)aU}
is an open subset of A.
We define H(AQ) to be the set of all ^-valued functions/, with domain AQ, that
arise from an/e H(Q) by the formula
B) /W = r
2
where Г is any contour that surrounds a(x) in Q.
This definition calls for some comments.
(a) Since Г stays away from a(x) and since inversion is continuous in A, the
integrand is continuous in B), so that the integral exists and defmes/(X) as an element
of A.
(b) The integrand is actually a holomorphic v4-valued function in the comple-
complement of a(x). (This was observed in the proof of Theorem 10.13. See Exercise 3.)
The Cauchy theorem 3.31 implies therefore that/(x) is independent of the choice ofT,
provided only that Г surrounds a{x) in Q.
(c) If x = осе and otefi, B) becomes
C) -7(а*)=/(ф.
Note that осе e AQ if and only if осе Q. If we identify X e 0 with le e A, every/ e Н(п)
may be regarded as mapping a certain subset of AQ (namely, the intersection of An
with the one-dimensional subspace of A generated by e) into A, and феп C) shows
that / may be regarded as an extension off
In most treatments of this topic,/(x) is written in place of our/(x). The notation
/is used here because it avoids certain ambiguities that might cause misunderstandings.

244 BANACH ALGEBRAS AND SPECTRAL THEORY
(d) If S is any set and A is any algebra, the collection of all Л-vaIued functions
on S is an algebra, if scalar multiplication, addition, and multiplication are defined
pointwise. For instance, if и and v map S in to A, then
(uv)(s) = u(s)v(s) (s e S).
This will be applied to Л-valued functions defined in Aa.
10.27 Theorem Suppose A, H(Q\ and H(Aa) are as in Definition 10.26. Then
H(Aa) is a complex algebra. The mapping f-+J is an algebra isomorphism of H(Q)
onto H(AQ) which is continuous in the following sense:
Jffn e Н(п) (и = 1, 2, 3, ...) andfn ->/ uniformly on compact subsets of п, then
(I) f(x)=\imfn(x) (xEAa).
n~*ao
Ifu(X) = I and v(l) = 1 in Q, then u(x) = x and v{x) = e for every x e Aa.
proof The last sentence follows from Theorem 10.25. The integral represen-
representation B) in Section 10.26 makes it obvious that/-»/ is linear. If/= 0, then
B) /(a)e=JV) = 0 (aeH),
so that/= 0. Thus/-^/is one-to-one.
The asserted continuity follows directly from the integral B) in Section
10.26, since ||(Яг - jc)~~ 11| is bounded on Г. (Use the same Г for all/,, and apply
Theorem 3.29.)
It remains to be proved that/-»/is multiplicative. Explicitly, if/e H(Q),
g e H(Q), and h(l) =f(X)g(X) for all X e U, it has to be shown that
C) ад=/(хЖх) (хе4).
If/and g are rational functions without poles in Q, and if h =fg, then
h(x) =f(x)g(x), and since Theorem 10.25 asserts that R(x) = R(x), C) -holds.
In the general case, Runge's theorem (Th. 13.9 of [23]) allows us to approximate
/and g by rational functions/, and gn, uniformly on compact subsets of п.
Then fngn converges to h in the same manner, and C) follows from the continuity
of the mapping/-»/.
Note that H(Aa) is a commutative algebra, since H(Q) is obviously com-
commutative. ////
10.28 Theorem Suppose x e An andf e Щп).
(a) j'(x) is invertible in A if and only iff(X) ф 0 for every X e o(x).
(b) o(?(x)) =/(<r«).
Part (b) is called the spectral mapping theorem.

BANACH ALGEBRAS 245
proof (a) If/has no zero on a(x), then g = l/fis holomorphic in an open set
Qt such that a{x) cfijcfi. Since fg = 1 in Qu Theorem 10.27 (with ut in
place of Q) shows that f(x)g(x) = e, and thus f(x) is invertible. Conversely, if
/(a) = 0 for some a e a{x) then there exists h e #(Q) such that
A) (A-a)h(A)=f(A) (ХеП),
which implies
B) , (x - oce)h(x) =/(*) = Я(х)(* - ae),
by Theorem 10.27. Since x — осе is not invertible in A, neither is f(x), by B).
(b) Fix pe£. By definition, jS e o{f(x)) if and only if f(x) - fie is not
invertible in A. By (a), applied to f—'fl is place of/, this happens if and only if
f—P has a zero in a(x), that is, if and only if /? g/((t(jc)). ////
The spectral mapping theorem makes it possible to include composition of
functions among the operations of the symbolic calculus.
10.29 Theorem Suppose x e Aa, fe #(Q), Qt is an open set containing f(o(x)),
g e Я<А), and h(X) = g(f(Xj) in Qo , the set of all Лей withf(X) e Пи
Then f{x) e Aai and h(x) = g(f{x)).
Briefly, h=g of if h=gof.
proof By (b) of Theorem 10.28, a(f(x)) с Qu and therefore g(f(x)) is defined.
Fix a contour Г\ that surrounds f(o(x)) in fij. There is an open set Й7,
with a(x) с ^cQ0,so small that
A) Indri (/(A)) = 1 (ЯеЖ).
Fix a contour Го that surrounds <j(x) in Ж. If С е Г1? then 1/(C — /) e H(W).
Hence Theorem 10.27, with W in place of Cl, shows that
B) Ke-J{x)Yl=~\ Ц-№]-\Хе-х)-иХ (С е Г,).
Since Tj surrounds a(J(x)) in fi,, A) and B) imply
S(Kx)) = -J- f
27сг Jr0 2tii Jfi
= ^ f 0Cf№X*e - *ГХ ^ = h \
2ttj Jr,, 27ti Jr

246 BANACH ALGEBRAS AND SPECTRAL THEORY
We shall now give some applications of this symbolic calculus. The first one
deals with the existence of roots and logarithms. To say that an element x e A has an
nth root in A means that x = yn for some у e A. If x = exp (y) for some у е A, then
у is a logarithm of x.
Note that exp (у) = £оУУл! but that the exponential function can also be
defined by contour integration, as in Definition 10.26. The continuity assertion of
Theorem 10.27 shows that these definitions coincide (as they do for every entire
function).
10.30 Theorem Suppose A is a Banach algebra, x e A, and the spectrum a(x) of
x does not separate Ofrom oo. Then
(a) x has roots of all orders in A,
(b) x has a logarithm in A, and
(c) ife>0, there is a polynomial P such that Цх — P(x)\\ < e.
Moreover, if a(x) lies in the positive real axis, the roots in (a) can be chosen so as
to satisfy the same condition.
proof By hypothesis, 0 lies in the unbounded component of the complement
of (t(x). Hence there is a function/, holomorphic in a simply connected open
set Q =э а(х), which satisfies
exp (ДА)) = A.
It follows from Theorem 10.29 that
exp (/(*)) = x,
so that у = f(x) is a logarithm of x. If 0 < X < oo for every X e c(x),/can be
chosen so as to be real on o(x), so that a(y) lies in the real axis, by the spectral
mapping theorem. If z = exp (y/n), then zn = x, and another application of the
spectral mapping theorem shows that a(z) c= @, oo) if o(y) с ( —oo, oo). This
proves (a) and (b); of course (a) could have been proved directly, without
passing through (b).
To prove (c), note that I/A can be approximated by polynomials, uniformly
on some open set containing o(x) (Runge's theorem), and use the continuity
assertion of Theorem 10.27. ////
These results are not quite trivial even when A is a finite-dimensional algebra.
For example, it is a special case of (b) that a complex n-Ъу-п matrix M is the exponential
of some matrix if and only if 0 is not an eigenvalue of M, that is, if and only if M is
invertible. To deduce this from (b), let A be the algebra of all complex п-Ъу-п matrices
(or the algebra of all bounded linear operators on Cn).

BANACH ALGEBRAS 247
10.31 Theorem
(a) Suppose A is a Banach algebra, x e A, P is a polynomial in one variable, and
P(x) = 0. Then a(x) lies in the set of zeros of P.
(b) In particular, if x is idempotent, i.e., ifx2 = x, then a(x) a {0, 1}.
(c) If the spectrum of some element of A is not connected, then A contains a non-
trivial idempotent.
The trivial idempotents are 0 and e, of course.
proof By the spectral mapping theorem,
This gives (a) and (b). If o{x) is not connected, there are disjoint open sets Qo
and Ql9 both of which intersect a(x) and whose union Q covers a(x). Put
f(X) = 0 in Uo, f(X) ='l in пх. Then fe H(U). Put у =/(*). Since f2 =f
Theorem 10.27 implies that y2 = y, and so у is idempotent. By the spectral
mapping theorem,
*СИ) =/(*(*)) = {0,1}.
Hence у is nontrivial, since 0 and e have one-point spectra. ////
10.32 Definition Let &(X) be the Banach algebra of all bounded linear operators
on the Banach space X. The point spectrum ap(T) of an operator Те &(X) is the
set of all eigenvalues of T Thus X e op{T) if and only if the null space Ж(Т — U)
of T — XI has positive dimension.
When A = $(X), the spectral mapping theorem can be refined in the following
way.
10.33 Theorem Suppose Те ЩХ), U is open in V, a(T) с Q, andf e H(Q).
(a) ,IfxeX,aeu, and Tx = ocx, then f(T)x =/(офс.
(b) f(*P(T))czap(j(T)).
(c) //(kg GP(J(T)) andf— oc does not vanish identically in any component of п, then
oc ef(ap(T)).
(d) If f is not constant in any component of Q, then f(ap(T)) = ap(f(T)).
Part (a) states that every eigenvector of T, with eigenvalue oc, is also an eigen-
eigenvector of/(r), with eigenvalue/(a).
proof (a) If x = 0 there is nothing to be proved. Assume x Ф 0 and Tx = ocx.
Then a e o{T), and there exists g e H(Q) such that
A) f(X)-f{a)=g{X){X-a).

248 BANACH ALGEBRAS AND SPECTRAL THEORY
By Theorem 10.27, A) implies
B) J{T)
Since (T- al)x = 0, B) proves (a).
Thus/(a) is an eigenvalue of /(T) whenever a is an eigenvalue of T. It
follows that (a) implies (b).
Under the hypotheses of (c),
C) ос Е ap(f(T)) a a(?(T)) =/(tr(T)),
so that
D) f-\oc)na(T)^0.
Moreover, the set D) is finite, because <j(T) is a compact subset of Q and/— a
does not vanish identically in any component of п. Let d, ..., (n be the zeros
off— oc in бг(Т), counted according to their multiplicities. Then
E) /B)-a = ^(A)(A-C1)---(A-a
where g e H(Q) and g has no zero on o-(T), so that
F) f(T) - ocI = g(T)(T- CJ)"'(T- U\
By (a) of Theorem 10.28, g(T) is invertible in &(X). Since a is an eigenvalue of
/(Г),/(Г) — a/is not one-to-one on X. Hence F) implies that at least one of the
operators T— (,tI must fail to be one-to-one. The corresponding C,- is in crp(T),
and since/(CO = oc the proof of (c) is complete.
Finally, (d) is an immediate consequence of (b) and (c). ////
Differentiation
We shall now investigate the extent to which the members of H(An) (see Definition
10.26) behave like holomorphic functions, as far as differentiability, power series
representation, and the open mapping property are concerned. As might be expected,
some of the results are more similar to the classical ones when A is commutative
than when it is not.
10.34 Definition Suppose X and У are Banach spaces, п is an open subset of X,
F maps Q into У, and a e п. If there exists Л е &(X, Y) (the Banach space of all
bounded linear mappings of X into У) such that
=
*->o ||*||
then Л is called the Frechet derivative of F at a. (The uniqueness of Л is trivial.)

BANACH ALGEBRAS 249
The notation (DF)a will be used for the Frechet derivative of F at a.
If (DF)a exists for every a e Q, and if
is a continuous mapping of п into ЩХ, Y), then F is said to be continuously differen-
tiable in Q.
10.35 Difference Quotients If A is a Banach algebra, x e A, and x + heA,
and if both sides of the identity
(Xe ~x)-(Xe-x-h) = h
are multiplied by (le — x — h)~l on the left and by (le — x)~l on the right, one
obtains
A) (Xe-x- hy1 - (Xe - x)~l = (le - x - hylh(Xe - x)~\
provided, of course, that these inverses exist.
Suppose now that Q is open in C, x e Aa, x + h e Aa, and/e H(Q). Choose Г
so that it surrounds a(x) u a(x + h) in Q. Then A) leads to
B) f(x + h) -f(x) = r^ f f(X)(Xe -x- hY'KXe - *)"» dA.
If h and x commute, i.e., if xh — hx, then h can be moved outside the integral B),
as we saw in Section 10.22. This motivates the definition of the difference quotient
C) (SJXx; ft) = -L f /(ЯХЯе - x - ft)" V* -
Z7TI ^r
which satisfies
D) J(x + h)-f(x)=h{Qf){x;h\
provided that xh = hx, an assumption which applies to the remainder of this section.
If ||h|| < 1/M, where M > \\(Ле - jc)~11| for every I on Г, then the series
E) (le-x- h)'1 = £ (Де - x)""^
converges, in the norm topology of A, uniformly for X on Г. Hence C) becomes
F) (Qf)(x; h) = £ -J-. f
„=i Zni ^r
00 1 t
-.?, Л 55

250 BANACH ALGEBRAS AND SPECTRAL THEORY
I
where/(n) is the nth derivative of/, and/(ll) is an abbreviation for [f(n)] ~. The norm
of the coefficient of h"~l in the last power series is dominated by a constant (depending
on/and Г) times M". The series converges, therefore, in norm. By D) and F), the
power series representation
G) Kx + h)=t-,Mx)W
n = 0 nl
holds if xh = hx and \\h\\ is sufficiently small. (See Exercise 9.)
The following facts have now been proved:
10.36 Theorem Suppose A is a commutative Banach algebra, Qcz С is open,
x e AQ, andfe H(Q). Then there exists <5 > 0 such that
A) /(х + Л)= £ ~Jn\x)hn
for all h e A with \\h\\ < S. Consequently,
B) {Df)x{h)=f\x)h (he A).
In other words, the operator (Dj)x e &(A) is multiplication by f\x).
This is, of course, exactly as in the classical case A = (£. We now consider the
noncommutative situation.
10.37 Commutators Left and right multiplication by an element x of a Banach
algebra A will now be denoted by Lx and Rx, respectively. Since the associative law
y(xz) = (yx)z holds in A, every left multiplier Ly commutes with every right multi-
multiplier Rz. In particular, Lx and Rx commute with each other and with^the operator
A) CX = RX-LX.
Note that Cx(y) — yx — xy, the so-called commutator of у and x.
Of course, Lx, Rx, and Cx are members of @(A). It is easily seen that
B) a{Lx) = *(*) = a(Rx)
and that ||СЛ|| < 2||д:||. Some further information about a(Cx) will be obtained in the
corollary to Theorem 11.23.
10.38 Theorem Suppose A is a Banach algebra, Q is open in С, х е Aa, and fe
H(Q). Then f is a continuously dijferentiable mapping of Aa into A', and
A) (Df)x(y) = ^-. f RWe - хУ1у(Хе - х) dX (у е A)
if Г is any contour that surrounds a(x) in Q.

BANACH ALGEBRAS 251
The operator (Df)x can also be represented by the &(A)-valued integral
B) (£>/), = -L f f{X){XI - Rx) -\XI- Lx) -! dX
Zni Jr
and by the difference quotient
C) (Df)x = (Qj)(Lx; Cx).
Ifu contains all X with \X\ < 3\\x\\, then
D) (Д7)» t
m=l ТП1
The notation used in C) is perhaps not explicit enough. On the left side of C),
/is a function from An into A; on the right side,/stands for a function from &(A)Q
into ЩА); both sides of C) represent members of ЩА).
proof If M > ||O - хУх\\ for all Я on Г and if 2M|b|| < 1, Theorem 10.11
shows that the norm of the difference between f(x + y) —J(x) and the right
side of A) is at most 2M3||j||2, multiplied by the length of Г and the maximum
of |/| on Г. This proves the formula A).
Let g(l) e &(A) be the integrand in B). Since inversion is continuous in
every Banach algebra, and hence in &(A), g is continuous on Г, and Те
can be defined by
E) T
But E) implies
F) Ty=\g{X)ydX {ye A),
and since g(X)y is exactly the integrand in A), F) shows that T=2ni(Dj)x.
This proves B).
It may be worthwhile to indicate in detail how F) follows from E) and
Definition 3.26: If F e A* (the dual space of A), if у e A, and if FtS = F(Sy)
for S e ЩА), then Ft e ЩА)*, and
F(Ty) =FtT=f Fl9{X) dl - f F[g(X)y] dk=F \ g(X)y dL
Let us return to B). If xn -* x, the contour Г used in B) will surround
a(xn) in Q, for all but finitely many n. Discard these. Then (Df)Xn is given by
B), if x is replaced by xn in the integrand. Since
G) (Xe-xny1-^(le-x)~1 as n -> oo,

252 BANACH ALGEBRAS AND SPECTRAL THEORY
uniformly on Г, the integrands in B) converge uniformly. We conclude that
x->(Df)x is continuous. Thus/is continuously differentiable.
Since Rx = Lx + Cx, C) is just another way of writing B).
If Г can be chosen in B) so as to be a circle with radius r > 3||x|| and
center at 0, then
11 00 ll CO 1
ьх) к- ^A Lx ^ Lr IIх II -- T3
for every A on Г, so that
(8)
By (8), the computation F) of Section 10.35 can now be applied to (Qf)(Lx; Cx).
It yields
(9) (e/)(b,;cx)= t -V^
m== 1 fit •
Finally, D) follows from C) and (9), because
A0) g(Lx)y = -^ f g(X){Xl - Lx) ~ ly dX
2711 Jr
= ^-. f 9(Ше - X)" V rfA = flf(x)y
for every j e Л and every ^ e H(Q), hence in particular for every /(m).
This completes the proof. ////
The series D) may actually fail to converge if/has a singularity at distance
3||x|| from the origin. An example of this is described in Exercise 22. The constant 3
that occurs in the last part of Theorem 10.38 is therefore the correct one.
If A is commutative, then Cx = 0. The term with m = 1 is then the only one that
remains in the series D). This agrees with Theorem 10.36.
The following theorem will allow us to extract information about local mapping
properties of the functions / from Theorem 10.36.
10.39 The inverse function theorem Suppose
(a) W is an open subset of a Banach space X,
(b) F: W -> X is continuously differentiable,
(c) for every x e W, (DF)X is an invertible member o

BANACH ALGEBRAS 253
Every point a eW has then a neighborhood U such that
(i) F is one-to-one in U,
(ii) F(U) = V is an open subset of X,
(Hi) jF : V ->U is continuously differentiable.
The conclusion may be stated briefly by saying that F is a local diffeomorphism.
proof If a e W, if T= (DF)a, and if
A) - f(x) = T-1[F(a + x)-F(a)] (x e W - a),
then / satisfies the hypotheses of the theorem, with W — a in place of W. If/
satisfies the conclusion, the same will be true of F. We may therefore replace
F by/. In other words, we assume, without loss of generality, that
B) OeW, jF(O) = O, (BF)o=I,
and we have to prove that 0 has a neighborhood {/that satisfies (/), (ii), and (Hi).
Fix a, 0 < a < 1. Define
C) <Kx) = x-F(x) (xeW).
\Then (Dcf)H = 0, and since ф is continuously differentiable in W, there is an
open ball В с W, centered at 0, such that
D) IIWy<« if xeB.
Suppose x' e B, x" e B, x, = A - t)x' + tx", and
E) Ф@ = ф(х,) @ < / < 1).
Then ф: [0, 1] -^ X is continuously differentiable,
F) ф'0) = iWU*" - jO (* = *,)
by the chain rule, and hence D) implies
G) Ш0И<«11*"-*'11;
note that xt e B, by the convexity of B. (See Exercise 10.) Since
(8) ф(х") - ф(х') = фA) - ф@) = f V(t) dt,
we conclude from G) that ф satisfies the Lipschitz condition
(9) ||ф(х") - ф(х')|| < a||*' - x'|| (x' e В, х" е В).
It now follows from C) that
A0) \\F(x") - F(x')\\ > A - a)||*" - x'|| (x' 6 В, х" е В).

254 BANACH ALGEBRAS AND SPECTRAL THEORY
>
This implies that F is one-to-one in B. Also, if G: F(B) -> В is defined by
G(F(x)) = x, then A0) shows that G is continuous.
Our next aim is to show that F(B) => A — ct)B.
Fix у e A — oc)B. Put x0 =0, x1 = j. Suppose я > 1, and x0, ..., xn
exist so that
A1) *,•= У+ <£(**-1) A <*<")
and
A2) ll*i-*,-ill<«"'~1lb'll A </<«).
(These conditions hold when n = 1.) By A2),
A3) Hxjs; Е1|х(-х;_,||< i>i-1llyll<(i-«r1lb'll,
i=\ i=l
so that xn e В, ф(хп) exists, and one can define
A4) хп + 1==у + ф(хп).
It follows from A4), A1), and (9) that
A5) ||xn+1 - xj| = \\ф(хп) - ф(хп_1)\\ < oc\\xn - x^l
Our induction hypotheses hold now with n + 1 in place of n, and the
construction can proceed, to yield a sequence {xn} that satisfies A1) and A2)
for all n. Since a < 1, A2) shows that {xj is a Cauchy sequence, which converges
to some x e B, by A3). Now A1) and C) imply thatF(x) = y.
If к=A -oc)B and U=G(V) = BnF-\V), then conclusions (f) and
(//) hold. To complete the proof, we now show that G is continuously differen-
tiablein V.
Suppose yeV, у + к е V, к Ф 0, x = G(y), x + h = G(y + k)!, put
. Then
k)- G(y) - S~xk = A - S*
= S^ - ^) = ~S~\F{x + A) - F(x) -
ByA0),(l -a)||A||<||/c||. Hence
\\G(y + k) - G{y) - S~lk\\ ц \\Fjx + A) - F(x) - Sh\\
11*11 0-
As /:->0, A0) implies that A->0, and since S = (DF)X, the last inequality
shows that S'1 =(DG)y. In other words,
A6) (DG^^KDF)^]-1 (yeV).

BANACH ALGEBRAS 255
Since G maps V continuously into &(X), and since inversion is continuous in
ЩХ) (Theorem 10.12), A6) shows that y-*(DG)y is a continuous mapping of
V into ЩХ).
This completes the proof. ////
10.40 ' Theorem Suppose A is a commutative Banach algebra, Q is open in <?,
x e Aa, and the derivative f ofsomefeH(Q) has no zero on a(x). Then x has a
neighborhood U с. AQ such that the restriction of f to U is a diffeomorphism whose
range is an open subset of A.
proof By Theorem 10.28, j\x) is invertible in A. By Theorem 10.36, (Df)x
is therefore invertible in &(A). Since у -> (Df)y maps AQ continuously into
ЩА), and since the invertible members of ЩА) form an open set, x has a
neighborhood in which (Df)y is invertible. The conclusion follows therefore
.from Theorem 10.39. ////
Note that the theorem just proved does not assert what one might expect to be
true, namely, that/is an open mapping of Aa into A whenever fe H(Q) is not constant
in any component of Q. This is actually false; see Exercise 13 for an example. The
theorem does prove that/is open near points x at which (Df)x is invertible.
The hypothesis that/' has no zero on a(x) means that/is locally one-to-one
in some open set that contains a(x). Theorem 10.42 will show that this local condition
does not imply that/is open at x if commutativity of A is dropped from the assump-
assumptions. An analogous global theorem does, however, turn out to be true:
10.41 Theorem Suppose A is a Banach algebra, Q is open in (?,/e//(Q), and f
is one-to-one in Q. Then / is a diffeomorphism of AQ onto Af(Q).
proof Let#:/(Q)-»Q be the inverse of/. Since g of and fog are the identity
mappings in D and in/(Q), respectively, Theorem 10.29 shows that g о/is the
identity in AQ, so that/is one-to-one, and/°$ is the identity in Af(Q), so that
Af{Q) is the range of/. Since both/and its inverse $ are continuously differen-
tiable (Theorem 10.38), the proof is complete. ////
10.42 Theorem Suppose A *± &(X), where X is a complex Banach space and
dim X > I. If Q. is open in <£, iffe H(Q), and iff is not one-to-one in Q, then some
To e Aa has a neighborhood U such thatJ(U) contains no neighborhood of f(T0).
Thus / is not open at To .
proof By assumption, Q contains two points а ф ft at which /(a) =f(P) = c,
say. Let У be a closed subspace of X, of codimension 1; choose xl9 x2
Xi Ф 0, such that x2 is in У but xx is not; and define To e A by
A) ToXl=aocl9 Toy = Py if yeY.

256 BANACH ALGEBRAS AND SPECTRAL THEORY
)
If Я ф ос and l ф ft, multiplication of xx by (Я - a) and of у by (A - P)
defines {XI-Tq)'1. Thus a(T0) = {a, jS}, and Гое^. By (л) of Theorem
10.33,
B) /(To) = cL
Put x3 = x, + x2, let M be the one-dimensional subspace of X generated
by x3, and let S be the distance from T0x3 to M. Then S > 0, since Tox3 =
a*! + /fo2 and a # /?. Let Do be the union of the components of Q that contain
a and p. (There are either one or two of these components.) Let U be the set
of all Те A such that
C) ||T-ro||<-^~ and cf(T)czQ0.
\\хз\\
Then U is a neighborhood of To. We shall prove that/(t/) does not
contain /(To) + rjS if ^/ # 0 and if S e A is defined by
D) SXi =x3, Sy = 0 for у e Y.
We argue by contradiction. Suppose a(T) a Qo, rj Ф0, and
E) f(T)=j(T0) + r1S = cJ+r]S.
Then
F) /(T)x3=(c + i7)x3, /(T)^ = c^ for yeY.
Thus c + >/ is an eigenvalue of/(T), with eigenspace M. Since/— (c + ?y)
vanishes neither at a nor at /J, it does not vanish identically in any component of
п0, and (c) of Theorem 10.33 implies that с + rj =/(y) for some eigenvalue
у of T. By (л) of Theorem 10.33, the corresponding eigenspace lies in M,
hence equals M, since dim M = 1. Thus Tx3 e M. Our choice of S implies
now that
G) 3
Hence Tis not in U. • ////
10.43 The exponential function To illustrate the preceding results, let us see
what they tell about the exponential function, defined by the power series
со I
exp (x) = X - xn
in every Banach algebra A. (See also Theorem 10.30.)

BANACH ALGEBRAS 257
(a) If a(x) contains no two points whose difference is an integral multiple of
2ni, the compactness of a(x) shows that exp is one-to-one in some open set Q => a(x);
hence exp is a diffeomorphism of the neighborhood An of x into A, by Theorem 10.41.
(b) The Frechet derivative of exp at x is
where Ф is the entire function defined by
This follows from the last part of Theorem 10.38, since /(m) =/ for m > 1 when
/(A) = exp (Л).
The zeros of Ф are at 2kni, к = ± 1, ±2, If none of these lies in ff(Cx),
then Ф(СХ) is invertible, by the spectral mapping theorem, and so is (D exp)x, and
exp is again a diffeomorphism near x.
(c) We shall see later (Theorem 11.23) that
(?(CX) cz a(x) — a(x).
This provides a link between the preceding paragraphs (a) and (b).
(d) If A is commutative, then (Z> exp)x is invertible, for every x e A, since it is
simply multiplication by exp (x), an invertible member of A. (Theorem 10.36.)
Hence exp is a local diffeomorphism, as in the familiar case A = 0. However, if
A = £%(X), as in Theorem 10.42, then exp is not an open mapping of A into A.
The Group of Invertible Elements
We shall now take a closer look at the structure of G = G(A), the multiplicative group
of all invertible elements of a Banach algebra A.
Gx will denote the component of G that contains e, the identity element of G.
Sometimes G± is called the principal component of G. By the definition of component,
Gx is the union of all connected subsets of G that contain e.
The group G contains the set
exp (A) = {exp (x): x e A},
the range of the exponential function in A, simply because exp ( — x) is the inverse of
exp (x). In fact, the power series definition of exp (x) (see Section 10.43) yields the
functional equation
exp (x + y) = exp (x) exp (y),
provided that xy = yx; also, exp @) = e.

258 BANACH ALGEBRAS AND SPECTRAL THEORY
Note also that G is a topological group (see Section 5.12) since multiplication and
inversion are continuous in G.
10.44 Theorem
(a) Gt is an open normal subgroup ofG.
(b) Gx is the group generated by exp (A).
(c) If A is commutative, then Gx = exp (A).
(d) If A is commutative, the quotient group G/Gt contains no element of finite order
{except for the identity).
proof {a) Theorem 10.11 shows that every x e Gx is the center of an open
ball U a G. Since U intersects Gx and U is connected, U с Gv Therefore
Gt is open.
If x e G1 then x~1G1 is a connected subset of G which contains x~1x = e.
Hence x~1G1 с Gl9 for every x e Gx. This proves that G1 is a subgroup of G.
Also, y~1G1y is homeomorphic to Gu hence connected, for every у е G, and
contains e. Thus y~1G1y с Gt. By definition, this says that G1 is a normal
subgroup of G.
(b) Let Г be the group generated by exp (A). For n = 1, 2, 3, ..., let En
be the set of all products of n members of exp (A). Since y~1 e exp (A) whenever
у e exp (А), Г is the union of the sets En. Since the product of any two connected
sets is connected, induction shows that each En is connected. Each En contains
e, and so En a Gv Hence Г is a subgroup of Gu
Next, exp(v4) has nonempty interior, relative to G (see Theorem 10.30);
hence so has Г. Since Г is a group and since multiplication by any x e G is a
homeomorphism of G onto G, Г is open.
Each coset of Г in Gt is therefore open, and so is any union of these cosets.
Since Г is the complement of a union of its cosets, Г is closed, relative to Gx.
Thus Г is an open and closed subset of Gv Since Gx is connected, Г = Gv
(c) If A is commutative, the functional equation satisfied by exp shows
that exp (A) is a group. Hence (b) implies (c).
(d) We have to prove the following proposition:
If A is commutative, if x eG, and if xn e Gx for some positive integer n,
then x eGv
Under these conditions, jc" = exp (a) for some a e A, by (c). Put у = exp
{п~га) and z = xy'1. Since у e Gl9 it suffices to prove that z e Gv
The commutativity of A shows that
zn = xny~n = exp (a) exp ( — a) = e.

BANACH ALGEBRAS 259
Put /(A) = Iz - (A - l)e, and let £ = {1е&-/B)е(?}. If ос е a(z), then
ocn e a(zn) = o(e) = {1}. If А фЕ, it follows that (A - 1)" = A". This equation
has only n - 1 solutions in 0. Hence E is connected. Consequently, /(£) is a
connected subset of G which contains/@) = e. Thus/(£) с Gx. In particular,
This completes the proof. ////
Theorem 12.38 will show that exp (A) is not always a group.
Exercises
Throughout this set of exercises, A denotes a Banach algebra.
1 Suppose xeA, ye A.
(a) If x and xy are invertible in A, prove that у is invertible.
(b) If xy and yx are invertible in A, prove that x and jy are invertible in A. (The com-
commutative case of this was tacitly used in the proofs of Theorems 10.13 and 10.28.)
(c) Show that it is possible to have xy = e Ф yx. For example, consider the right and
left shifts SR and Sl , defined on some Banach space of functions / on the non-
. negative integers by
f 0 if n = 0
<*■>*■>=U-d if-^i,
(Sl f)(n) =f(n + 1) for n > 0.
(d) If xy = e Ф yx, show that yx is a nontrivial idempotent.
(e) If dim A < oo, show that jy* = <? whenever лгу = <?.
2 Suppose x£^,j;£^.
(a) Prove that e — yx is invertible if e — xy is invertible. ///л/: If z inverts e — xy,
consider e + jzjc.
(b) If Л e B*, Л 9* 0, and Л e o(xy), prove that Л е a(yx). Show, however, that o(xy)
may contain 0 although o(yx) does not.
(c) If x is invertible, show that a(xy) = сгСих).
5 Suppose Q. is open in 0; /: Q, -> ^ and <£: П -> 0 are holomorphic. Prove that ф/:С1-^А
is holomorphic. [This was used in the proof of Theorem 10.13, with <£(Л) = Л".]
4 Another proof of the theorem that o(x) is never empty can be based on Liouville's
theorem 3.32 and the fact that (\e - x)'1 ->0 as Л -> oo. Complete the details.
5 Call x e A a topological divisor of zero if there is a sequence {у„} in /4, with \\yn\\ = 1,
such that
lim Jtyn = 0 = lim jM x.
n-*oo
(a) Prove that every boundary point x of the set of all invertible elements of A is a
topological divisor of zero. Hint: Take yn = jc^ VIIjuT j ||, where xn -> x.
(b) Which Banach algebras have no topological divisors of zero other than 0?

260 BANACH ALGEBRAS AND SPECTRAL THEORY
Suppose K = {\e(£: 1 < |A| <2}; put /(Л) = Л. Let A be the smallest closed sub-
algebra of C(K) that contains 1 and/. Let В be the smallest closed subalgebra of C(K)
that contains/and 1// Describe the spectra aA(f) and oB{f).
Do the same when К is a circle.
Strengthen the continuity assertion in A) of Theorem 10.27 in the following way: If К
is any compact subset of Cl, and if
then fn(x) ->/O) uniformly on AK.
8 {a) Fubini's theorem was applied to vector-valued integrals in the proof of Theorem
10.29. Justify this.
(b) Construct another proof of Theorem 10.29, that uses no contour integrals, as
follows: Prove the theorem first for polynomials g and then for rational functions
g e Н@ч), and obtain the general case from Runge's theorem.
9 In the computation F) of Section 10.35, integration by parts was applied to a vector-
valued integral. Justify this.
10 Prove the version of the chain rule that was used in the proof of Theorem 10.39, and
prove the fundamental theorem of calculus for vector-valued integrals, as used in (8) of
Theorem 10.39.
// Prove in detail that the convergence in G) of Theorem 10.38 is indeed uniform on Г.
12 Suppose к is a positive integer, a> = exp (Im/k), and/: A -> A is defined by f(x) = xk.
(a) Prove that/is a diffeomorphism in some neighborhood of x0 e A if x0 satisfies the
condition
ct(xo) n ojna(x0) = 0 for n = 1,..., к — 1.
(b) Prove that the same conclusion holds if A is commutative and x0 is invertible in A.
13 Let A be the algebra of all matrices of the form
\0 a)
with ae^ /?e <Z\ Show that |a| -f |j8| is a Banach algebra norm on A. Define
fix) = x2 for x e A. Find /04). Is /(Д) open in A ? Is/an open mapping? [Compare
with part (Jb) of Exercise 12.]
14 Show that every two-dimensional complex algebra A with unit e is isomorphic either
to (I1 with coordinatewise addition and multiplication or to the algebra described in
Exercise 13. Hint: In one case there exists x Ф ±e with x2 = e; in the other, there
exists x Ф 0 with хг = 0. Prove that one of these must occur.
Show that there exists a three-dimensional noncommutative Banach algebra.
15 , Prove the relation
exp (C) = exp (Rx) exp (—£,),
and use it to derive the formula
exp (-x)y exp (*) - [exp iCx)]y,
valid for x and у in any Banach algebra A. (The notation is as in Section 10.37.)

BANACH ALGEBRAS 261
16 Suppose A = C(T), the algebra of all continuous complex functions on the unit circle T,
with the supremum norm. Show that two invertible members of C(T) are in the same
coset of Gi if and only if they are homotopic mappings of T into the set of all nonzero
complex numbers. Deduce from this that G/Gi is isomorphic to the additive group of
the integers. (The notation is as in Theorem 10.44.)
17 Suppose A = M{R\ the convolution algebra of all complex Borel measures on the real
line; see (e) of Example 10.3. Supply the details in the following proof that G/Gi is
uncountable: If a e R, let 8a be the unit mass concentrated at a. Assume Ba e Gi. Then
Sa = exp (/xa) for some /xa e M(R); hence, for — oo < / < oo,
—Ш = /xa@ + 2km,
where к is an integer. Since /xa is a bounded function, a = 0. Thus 80 is the only 8a in Gi.
No coset of Gi in G contains therefore more than one Ba.
18 Suppose П is open in <2T, a is an isolated boundary point of П,/: О -> Xis a holomorphic
X-valued function in О (where X is some complex Banach space), n is a nonnegative
integer, and
is bounded as A -> a. Then /is said to have a pole (of order <ri) at a.
.(a) Suppose xe A and (A^ — x)~l has a pole at every point of o(x). [Note that this
can happen only when o(x) is a finite set.] Prove that there is a nontrivial polynomial
P such that P(x) = 0.
(b) As a special case of (a), assume o(x) = {0} and (A<? — x) has a pole of order n at 0.
Prove that x" = 0.
19 Let £я be the right shift, acting on /2, as in Exercise 1. Let {cn} be a sequence of complex
numbers such that cn Ф 0 but с -> 0 as n -> oo. Define M e Щ£2) by
and define Ге W2) by T=MSR.
(a) Compute ИГ" ||, for /я = 1, 2, 3,
(« Show that <jG) = {0}.
(c) Show that Г has no eigenvalue. (Its point spectrum is therefore empty, although its
spectrum consists of a single point!)
(d) Show that (A/ — Г) does not have a pole at 0.
(e) Show that Г is a compact operator.
20 Suppose x e A, xn e A, and lim xn = x. Suppose О is an open set in С that contains a
component of o(x). Prove that a(xn) intersects О for all sufficiently large n. (This
strengthens Theorem 10.20.) Hint: If ст(дг) <= О u O0, where O0 is an open set disjoint
from O, consider the function/that is 1 in O, 0 in O0 .
21 Let CR be the algebra of all real continuous functions on [0, 1], with the supremum
norm. This satisfies all requirements of a Banach algebra, except that the scalars are
now real.
(a) If <f>(f) = JJ f(t)dt, then фA) = 1, and <£(/) Ф 0 if /is invertible in C*, but <£ is
not multiplicative.

262 BANACH ALGEBRAS AND SPECTRAL THEORY
(b) If G and Gt are defined in CR as in Theorem 10.44, show that G/Gt is a group of
order 2.
The analogues of Theorem 10.9 and (d) of Theorem 10.44 are thus false for real
scalars. Exactly where would the proof of (d) of Theorem 10.44 break down?
22 Let A be the algebra of all complex 2-by-2 matrices; regard A as &(<Z2), where (I2 is
normed by ||(a, j8)|| = |a| + |j8|. (This determines the norm on A.) Define x e A by
/1 0
* — I _
(a) Find ||*||, a(;c), and a(Cx).
(b) Iftee,t2^l, and /(A) = !/(/- A), so that
compute (Df)x, and show that 2 (лО/00^)^1*-1 converges if and only if
11 — 11 > 2 and | r + 11 > 2. (The number 3 can therefore not be replaced by a
smaller one in the last part of Theorem 10.38.)
Partial answer to (b):
25 What happens if the process of adjoining a unit (described in Section 10.1) is applied
to an algebra A which already has a unit? Clearly, the result cannot be an algebra At
with two units. Explain.
24 Show that xy and yx always have the same spectral radius. Hint: (xy)n = x(yx)n~1y.
25 (a) Prove that A is commutative if there is a constant M < oo such that ||*y|| < M||jx||
for all x and у in A. Hint: \\w~1yw\\ < M \\y\\ if w is invertible in A. Replace w by
exp (Ax), where x e A and A e С Continue as in Theorem 12.16.
(b) Prove that A is commutative if ||jc2|| = ||x||2 for every xeA. Hint: Show that
||*|| = p(x). Use Exercise 24 to deduce that ||и>-1яИ| = |M|. Continue as in (a).
26 Suppose x e A and xn = e for some positive integer n. Prove that x e d. (The nota-
notation is as in Theorem 10.44.) Replace the hypothesis xn = e by more general ones.

11
COMMUTATIVE BANACH ALGEBRAS
This chapter deals primarily with the Gelfand theory of commutative Banach algebras,
although some of the results of this theory will be applied to noncommutative situa-
situations. The terminology of the preceding chapter will be used without change. In
particular, Banach algebras will not be assumed to be commutative unless this is
explicitly stated, but the presence of a unit will be assumed without special mention,
as will the fact that the scalar field is €.
Ideals and Homomorphisms
11.1 Definition A subset J of a commutative complex algebra A is said to be
an ideal if
(a) J is a subspace of A (in the vector space sense), and
(b) xy e J whenever xe A and yeJ.
If J Ф A, J is a proper ideal. Maximal ideals are proper ideals which are not
contained in any larger proper ideal.

264 BANACH ALGEBRAS AND SPECTRAL THEORY
11.2 Proposition
(a) No proper ideal of A contains any invertible element of A.
(b) IfJ is an ideal in a commutative Banach algebra A, then its closure J is also an
ideal.
The proofs are so simple that they are left as an exercise.
11.3 Theorem
(a) If A is a commutative complex algebra with unit, then every proper ideal of A is
contained in a maximal ideal of A.
(b) If A is a commutative Banach algebra, then every maximal ideal of A is closed.
proof (a) Let J be a proper ideal of A. Let 0> be the collection of all proper
ideals of A that contain J. Partially order & by set inclusion, let & be a maximal
totally ordered subcollection of & (the existence of 2, is assured by Hausdorff's
maximality theorem), and let M be the union of all members of &. Being the
union of a totally ordered collection of ideals, M is an ideal. Obviously J c= M,
and M Ф A since no member of 0> contains the unit of A. The maximality
of J implies that M is a maximal ideal of A.
(b) Suppose M is a maximal ideal in A. Since M contains no invertible
element of A and since the set of all invertible elements is open, M contains
no invertible element either. Thus M is a proper ideal of A, and the maximality
of M shows therefore that M = M. ////
11.4 Homomorphisms and quotient algebras If A and В are commutative
Banach algebras and ф is a homomorphism of A into В (see Section 10.4) then the
null space or kernel of ф is obviously an ideal in A, which is closed if ф is continuous.
Conversely, suppose J is a proper closed ideal in A and n: A -> A/J is the
quotient map, as in Definition 1.40. Then A/J is a Banach space, with respect to the
quotient norm (Theorem 1.41). We will show that A/J is actually a Banach algebra
and that n is a homomorphism.
If x — x e J and y'—yeJ, the identity
A) x'y' -xy = (xf - x)y' + x(y' - y)
shows that x'y' — xyeJ; hence n(x'y') = n(xy). Multiplication can therefore be
unambiguously defined in A/J by
B) n(x)n(y) = n(xy) (xeA,ye A).
It is then easily verified that A/J is a complex algebra and that n is a homomorphism.
Since \\n(x)\\ < \\x\\, by the definition of the quotient norm, n is continuous.

COMMUTATIVE BANACH ALGEBRAS 265
Suppose xte A (J = 1,2) and S > 0. Then
C) Wxi+yiW^MxdW+d (/=1,2)
for some yt e J, by the definition of the quotient norm. Since
Oi + yi)(x2 + Уг) e *i*2 + Л
we have
D) W*i*2)ll < \\(xt +уО(х2 +y2)\\ < \\x, +У1\\ \\x2 +y2\\,
so that C) implies the multiplicative inequality
E) IW*iM*2)ll< M*i)ll \\n(x2)\\.
Finally, if e is the unit element of A, then B) shows that n(e) is the unit of A/J,
and since к(е) Ф 0, E) shows that ||7r(e)|| > 1 = |M|. Since ||тг(х)|| < ||x|| for every
xeA, \\n(e)\\ = 1. This completes the proof.
Part (a) of the next theorem is one of the key facts of the whole theory. The
set A that appears in it will later be given a compact Hausdorff topology (Theorem
11.9). The study of commutative Banach algebras will then to a large extent be
reduced to the study of more familiar (and more special) objects, namely, algebras
of continuous complex functions on A, with pointwise addition and multiplication.
However, Theorem 11.5 has interesting concrete consequences even without the
introduction of this topology. Sections 11.6 and 11.7 illustrate this point.
11.5 Theorem Let A be a commutative Banach algebra, and let A be the set of
all complex homomorphisms of A.
(a) Every maximal ideal of A is the kernel of some AeA.
(b) If he A, the kernel ofh is a maximal ideal of A.
(c) An element xe A is invertible in A if and only if h(x) Ф Ofor every he A.
(d) An element xe A is invert ible in A if and only if x lies in no proper ideal of A.
(e) X e o(x) if and only ifh(x) = X for some he A.
proof (a) Let M be a maximal ideal of A. Then M is closed (Theorem 11.3)
and AIM is therefore a Banach algebra. Choose xe A, x ф М, and put
A) J = {ax -\-y:aeA,ye M}.
Then J is an ideal in A which is larger than M, since xe J. (Take a = e, у = 0.)
Thus/ = A, and ax + у = e for some a e А,у e M. If к: A -> A/M is the quotient
map, it follows that я(а)п(х) = п(е). Every nonzero element n{x) of the Banach
algebra A/M is therefore invertible in A/M. By the Gelfand-Mazur theorem,
there is an isomorphism j of A/M onto €. Put h =j ° n. Then he A, and M
is the null space of h.

266 BANACH ALGEBRAS AND SPECTRAL THEORY
! >
(b) If he A, then h~1@) is an ideal in A which is maximal because it has
codimension 1.
(c) If x is invertible in A and he A, then
КхЩх'1) = Кхх'1) = A(£?) = 1,
so that h{x) ФО. If л; is not invertible, then the set {ax: ae A} does not contain
e, hence is a proper ideal which lies in a maximal one (Theorem 11.3) and which
is therefore annihilated by some he A, because of (a).
(d) No invertible element lies in any proper ideal. The converse was
proved in the proof of (c).
(e) Apply (c) to le — x in place of x. ~ ////
Our first application concerns functions on Rn that are sums of absolutely
convergent trigonometric series. The notation is as in Exercise 22 of Chapter 7.
11.6 Wiener's lemma Suppose f is a function on Rn, and
0) f(x)=£ameiM-*, ЕЫ<<ю,
where both sums are extended over all m e Z". If fix) ф 0 for every x e Rn, then
B) 77! = I>m *■•"■•* Wrt H*J<oo.
proof Let A be the set of functions of the form A), normed by ||/|| = J] |#m|.
One checks easily that A is a commutative Banach algebra, with respect to
pointwise multiplication. Its unit is the constant function 1. For each x, the
evaluation/->/(x) is a complex homomorphism of A. The assumption about
the given function/is that no evaluation annihilates it. If we can prove that
A has no other complex homomorphisms, \c) of Theorem 11.5 will imply that
/is invertible in A, which is exactly the desired conclusion.
For r = 1, ..., n, put gr(x) — exp (ixr), where xr is the rth coordinate of x.
Then gr and \\gr are in A and have norm 1. If h e A, it follows from (c) of
Theorem 10.7 that
and
vJ I -L
Hence there are real numbers yr such that
C) h(gr) = exp 0>r) = &G0 A < r < /i),
where j = 0>1?... - yn). If P is a trigonometric polynomial (which means, by

COMMUTATIVE BANACH ALGEBRAS 267
definition, that P is a finite linear combination of products of integral powers
of the functions gr and l/#r), then C) implies
D) h(P) = P(y),
because h is linear and multiplicative. Since h is continuous on A (Theorem 10.7)
and since the set of all trigonometric polynomials is dense in A (as is obvious
from the definition of the norm), D) implies that h(f) =f(y) for every fe A.
Thus h is evaluation at y, and the proof is complete. ////
This lemma was used (with n = 1) in the original proof of the tauberian theorem
9.7. To see the connection, let us reinterpret the lemma. Regard Z" as being embedded
in Rn in the obvious way. The given coefficients am define then a measure ц
on Rn, concentrated on Z", which assigns mass am to each m e Z". Consider the
problem of finding a complex-measure a, concentrated on Zn, such that the convolu-
convolution ц * a is the Dirac measure S. Wiener's lemma states that this problem can be
solved if (and trivially only if) the Fourier transform of/z has no zero on R"; this is
precisely the tauberian hypothesis in Theorem 9.7.
For our next application, let Un be the set of all points z = (zl9 ..., zn) in €n
such that \z-t\ < 1 for \ <i <n. In other words, this polydisc U" is the cartesian
product of n copies of the open unit disc U in С We define A(U") to be the set of
all functions/that are holomorphic in U" (see Definition 7.20) and that are continuous
on its closure Un.
11.7 Theorem Suppose /j, ... ,fk e A(Un), and suppose that to each zeUn there
corresponds at least one i such that ft{z) Ф 0. Then there exist functions ф17 ...,
фк e A(Un) such that
A) Л(*Ж(*) + * * * +/*(*)&(*) =1 (* e 17").
proof A = A(Un) is a commutative Banach algebra, with pointwise multi-
multiplication and the supremum norm. Let J be the set of all sums ^/i^i' w^tn
4>i e A. Then J is an ideal. If the conclusion is false, then J Ф A; hence J lies
in some maximal ideal of A (Theorem 11.3), and some he A annihilates J, by
(a) of Theorem 11.5.
For l<r<n, put*gr(z) = zr. Then ||#r|| = l; hence h(gr) = wr, with
| wr | < 1. Put w = (wt, ..., wn). Then w e U\ and h(gr) = gr(w). It follows
that h(P) = P(w) for every polynomial P, since A is a homomorphism. The poly-
polynomials are dense in A(Un) (Exercise 4). Hence h(f) =f(w) for every fe A,
by essentially the same argument that was used in the proof of Theorem 11.6.
Since h annihilates J, ft(w) = 0 for \ <i <k. This contradicts the hy-
hypothesis. ////

268 BANACH ALGEBRAS AND SPECTRAL THEORY
Gelfand Transforms
11.8 Definitions Let A be the set of all complex homomorphisms of a commuta-
commutative Banach algebra A. The formula
A) x{h) = h{x) (h e A)
assigns to each x e A a function x: Л -> €; we call x the Gelfand transform of x.
Let Л be the set of all x, for x e A. The Gelfand topology of Л is the weak
topology induced by A, that is, the weakest topology that makes every x continuous.
Then obviously A с С(Л), the algebra of all complex continuous functions on A.
Since there is a one-to-one correspondence between the maximal ideals of A
and the members of A (Theorem 11.5), A, equipped with its Gelfand topology, is
usually called the maximal ideal space of A.
The term " Gelfand transform " is also applied to the mapping x -> x of A onto A.
The radical of A, denoted by rad A, is the intersection of all maximal ideals
of A. If rad A = {0}, A is called semisimple.
11.9 Theorem Let A be the maximal ideal space of a commutative Banach algebra A.
(a) A /,y a compact Hausdorff space.
(b) The Gelfand transform is a homomorphism of A onto a subalgebra A of C(A),
whose kernel is rad A. The Gelfand transform is therefore an isomorphism if and
only if A is semisimple.
(c) For each xe A, the range of x is the spectrum a(x). Hence
PL =/>(*)< INI,
where \х\^ is the maximum of \ x(h)\ on A, and x e rad A if and only ifp(x) = 0.
proof We first prove (b) and (c). Suppose x e А, у e A, a e €, h e A. Then
(axYQi) = h(ocx) = och(x) = (ax)(h),
(x + y)\h) = h(x + y) = h(x) + h{y) = *(A) + 9(h) = (x + y)(h),
and
(xyYQi) = h(xy) = h(x)h(y) = x{h)№ = {xp)(h).
Thus л: -> x is a homomorphism. Its kernel consists of those xe A which satisfy
h(x) = 0 for every h e A; by Theorem 11.5, this is the intersection of all maximal
ideals of A, that is, rad A.
To say that Л is in the range of x means that X — x(h) = h(x) for some
he A. By (e) of Theorem 11.5, this happens if and only if X e a(x). This proves
(b) and (c).

COMMUTATIVE BANACH ALGEBRAS 269
To prove (a), let A* be the dual space of A (regarded as a Banach space),
and let К be the norm-closed unit ball of A*. By the Banach-Alaoglu theorem,
К is weak*-compact. By (c) of Theorem 10.7, Ac K. The Gelfand topology
of A is evidently the restriction to A of the weak*-topology of A*. It is therefore
enough to show that A is a weak*-closed subset of A*.
Let Ло be in the weak*-closure of A. We have to show that
A) ' ' Ло(ху) = АохАоу (xeA,yeA)
and
B) Лое=1.
[Note that B) is necessary; otherwise Ло would be the zero homomorphism,
which is not in A.]
Fix xe A, ye A, e > 0. Put
C) W={AeA*: |Л^-Л0^| < г for 1 < i < 4},
where zx — e, z2— x, z3 — y, z4 = xy. Then W is a weak*-neighborhood of
Ло which therefore contains an h e A. For this h,
■D) |1
which gives B), and
AQ(xy) - Ло xAoy = [A0(xy) - h{xy)] + [h(x)h(y) - Ao xAoy]
= [A0(xy) - h(xy)] + [AGO - Ло y]h{x) + [h(x) - Ao x]AQ y,
which gives
E) \A0(xy) -AoxAoy\ < A + ||x|| + \Aoy\)s.
Since E) implies A), the proof is complete. ////
Semisimple algebras have an important property which was earlier proved for €:
11.10 Theorem If ф: В —► A is a homomorphism of a commutative Banach algebra
В into a semisimple commutative Banach algebra A, then ф is continuous.
proof Suppose xn -> x in В and ф(хп) -> у in A. By the closed graph theorem,
it is enough to show that у = ф(х).
Let AA and AB be the respective maximal ideal spaces. Fix he AA; put
ф = h о ф. Then ф е AB. By Theorem 10.7, h and ф are continuous. Hence
h(y) = lim AM*,)) = lim ф(хп) = ф(х) = /#(*)),
for every he AA. Hence у — ф(х) erad A. Since rad A — {0}, у — ф(х). ////

270 BANACH ALGEBRAS AND SPECTRAL THEORY
Corollary Every isomorphism between two semisimple commutative Banach
algebras is a homeomorphism.
In particular, this is true of every automorphism of a semisimple commutative
Banach algebra. The topology of such an algebra is therefore completely determined
by its algebraic structure.
In Theorem 11.9, the algebra A may or may not be closed in C(A), with respect
to the supremum norm. Which of these cases occurs can be decided by comparing
||x2|| with ||x||2, for all xeA. Recall that ||x2|| < ||*||2 is always true.
11.11 Lemma If A is a commutative Banach algebra and
A) r = inf-—~, s = inf , °° (xeA,Xy£0),
\M\2 \\x\\
then s2 < r < s.
proof Since ||x|| да > s \\x||,
for every xeA. Thus s2 < r.
Since ||x2|| > r \\x\\2 for every xeA, induction on n shows that
C) ||*1 > rm-' \\x\\m (m = 2", /i = 1, 2, 3, ...).
Take mth roots in C) and let m -> oo. By the spectral radius formula and (c)
of Theorem 11.9,
D) H^ll^ = p(jc) > r||jc|| (xeA).
Hence r < s. ////
11.12 Theorem Suppose A is a commutative Banach algebra.
(a) The Gelfand transform is an isometry (that is, \\x\\ — \x\^ for every xeA) if
andonlyif\\x2\\ = \\x\\2 for every x e A.
(b) A is semisimple and A is closed in С (A) if and only if there exists К < со such that
\\x\\2 < К \\х21| for every xeA.
proof (a) In the terminology of Lemma 11.11, the Gelfand transform is an
isometry if and only if s — 1, which happens (by the lemma) if and only if r = 1.
(b) The existence of К is equivalent to r > 0, hence to s > 0, by the lemma.
If s > 0, then x -> x is one-to-one and has a continuous inverse, so that A is
complete (hence closed) in C(A). Conversely, if x -> £ is one-to-one and if A
is closed in C(A), the open mapping theorem implies that s > 0. ////

COMMUTATIVE BANACH ALGEBRAS 271
11,13 Examples In some cases, the maximal ideal space of a given commutative
Banach algebra can easily be described explicitly. In others, extreme pathologies
occur. We shall now give some examples to illustrate this.
(a) Let X be a compact Hausdorff space, put A = C(X), with the supremum
norm. For each x e X, f-*f(x) is a complex homomorphism hx. Since C(X) separates
points on X (Urysohn's lemma), x ф у implies hx Ф hy. Thus x-*hx embeds X in A.
We claim that each h e A is an hx. If this is false, there is a maximal ideal M
in C(X) which contains, for each/? e X, a fimction/with/(/?) ф 0. The compactness
of X implies then that M contains finitely many functions fu ...,/„ such that at least
one of them is ^0 at each point of X. Put
Then g e M, since M is an ideal; g > 0 at every point of X; hence g is invertible in
C(X). But proper ideals contain no invertible elements.
Thus x<r+hx is a one-to-one correspondence between Xand A and can be used
to identify A with X. This identification is also correct in terms of the two topologies
that are involved: The Gelfand topology у of X is the weak topology induced by
C(X)and is therefore weaker than т, the original one, but у is a Hausdorff topology;
hence у = т. [See (a) of Section 3.8.]
Summing up, X " is " the maximal ideal space of C(X), and the Gelfand trans-
transform is the identity mapping on C(X).
(b) Let A be the algebra of all absolutely convergent trigonometric series, as
in Section 11.6. We found there that the complex homomorphisms are the evalua-
evaluations at points of Я". Since the members of A are 27i-periodic in each variable, A is
the torus Tn obtained from Rn by the mapping
This is an example in which A is dense in C(A), although А Ф C(A).
(c) In the same way, the proof of Theorem 11.7 contains the result that Un is
the maximal ideal space of A(Un). The argument used at the end of (a) shows that
the natural topology of Un is the same as the Gelfand topology induced by A(On); the
same remark applies to (b).
(d) The preceding example has interesting generalizations. Let A now be a
commutative Banach algebra with a. finite set of generators, say xu ..., xn. This
means that xt e A A < / < n) and that the set of all polynomials in xl9 ..., xn is dense
in A. Define
A) 0(A) = (*!(*),...,*„(*)) (AeA).

272 BANACH ALGEBRAS AND SPECTRAL THEORY
Then ф is a homeomorphism of A onto a compact set Kcz €n. Indeed, ф is continuous
since A c= C(A). If фЦг^) = ф(п2), then hx{x^ = h2(xi) for all /; hence ht(x) = h2(x)
whenever л: is a polynomial in хъ ..., xn, and since these polynomials are dense in A,
hx — h2 . Thus ф is one-to-one.
We can now transfer A from A to К and may thus regard К as the maximal
ideal space of A. To make this precise, define
B) ф(х) = *оф-* (хеА).
Then ф is a homomorphism (an isomorphism if A is semisimple) of A onto a sub-
algebra ф(А) of C(AT). One verifies easily that
C) ф(х№) = г, ifz = (Zl,...,zn)eK,
and therefore
D) IA(P(x1,...,^))(z)=P(z) (zeK)
for every polynomial P in n variables.
It follows that every member of\j/(A) is a uniform limit of polynomials, on K.
The sets К с Сп which arise in this fashion as maximal ideal spaces have a
property known as polynomial convexity:
If \v e €n and w$K, there exists a polynomial P such that \P(z) \ < 1 for every
zeK,but \P(w)\> 1.
To prove this, assume there is no such polynomial. The norm-decreasing
property of the Gelfand transform implies then that
E) \P(w)\<Z\\P(xu...,xM
for every polynomial P; the norm is that of A. Since {xl7 ..., xn} is a set of generators
of A, it follows from E) that there is an h e A such that ф(п) = w. But then w e K,
and we have a contradiction.
The compact polynomially convex subsets of € are simply those whose com-
complement is connected; this is an easy consequence of Runge's theorem. In Cn, the
structure of the polynomially convex sets is by no means fully understood.
(e) Our next example shows that the Gelfand transform is a generalization of
the Fourier transform, at least in the //-context.
Let A be L\Rn) with a unit attached, as described in (d) of Section 10.3. The
members of A are of the form /+ ocS, where feL\Rn\ oceC, and S is the Dirac
measure on Rn; multiplication in A is convolution:
(/+ ocS) * (g + fiS) = (f*g + Pf+ ocg) + ф.
For each t e Rn, the formula
F) * ht(f+ocd)=f(t) + x

COMMUTATIVE BANACH ALGEBRAS 273
defines a complex homomorphism of A; here / is the Fourier transform of/. Jn
addition,
G) hjf+ad) = oc
also defines a complex homomorphism. There are no others. (A proof will be sketched
presently.) Thus A, as a set, is R" и {оо}. Give A the topology of the one-point
compactification of.R". Since/(/)->0 as |f|-»oo, for every feL\Rn), it follows
from F) and G) that A с C(A). Since A separates points on A, the weak topology
induced on A by'A is the same as the one that we just chose.
It remains to be proved that every h e A is of the form F) or G). If h(f) = 0
for every feb\Rn), then h = hao. Assume к(/)Ф0 for some feLl(R"). Then
Kf) = $fP dmn, for some fi e L°°(Kn). Since h(f* g) = h(f)h(g), one can prove that
P coincides almost everywhere with a continuous function b which satisfies
(8) b(x+y)=b{x)b(y) (x,yeRn).
Finally, every bounded solution of (8) is of the form
(9) Ь{х) = е-[х-г (xeRn)
for .some / e R". Thus h(f) =/(/), and h has the form F).
For n = 1, the details that complete the preceding sketch may be found in
Sec. 9.22 of [23]. The case n > 1 is quite similar.
(/) Our final example is £°°(т). Here m is Lebesgue measure on the unit
interval [0, 1], and L^im) is the usual Banach space of equivalence classes (modulo
sets of measure 0) of complex bounded measurable functions on [0, 1], normed by
the essential supremum. Under pointwise multiplication, this is obviously a commuta-
commutative Banach algebra.
If feLm(m) and Gf is the union of all open sets G а С with m(f~\G)) = 0,
then the complement of Gf (called the essential range of/) is easily seen to coincide
with the spectrum o(f) off, hence with the range of its Gelfand transform/. It follows
that/is real if /is real. Hence Z,°°(«i)A is closed under complex conjugation. By the
Stone-Weierstrass theorem, £°°(т)л is therefore dense in C(A), where A is the maximal
ideal space of и°(пг). It also follows that /->/ is an isometry, so that L°°(m)^ is
closed in C(A).
We conclude that f -^ f is an isometry ofU°{m) onto C(A).
Next, /'-> J fdm is a bounded linear functional on C(A). By the Riesz repre-
representation theorem, there is therefore a regular Borel probability measure m on A
that satisfies
A0) f fdm = f fdm [feU>(m)l
JA J0

274 BANACH ALGEBRAS AND SPECTRAL THEORY
If Q is a nonempty open set in Л, Urysohn's lemma implies that there exists
/e C(A),/> 0, such that/= 0 outside Q, and f(p) = 1 at some;? ей. Hence/is not
the zero element of L°°(m) and the integrals A0) are positive.
Thus m(Q) >0ifQ is open and nonempty.
Assume next that ф is a Borel function on А, |ф| < 1. By Lusin's theorem
[23] there are functions/, e C(A), \fn\ < 1, that converge to ф in the norm of L2(m).
Since/-»/preserves complex conjugation (as we saw above) and is a homomorphism,
it follows from A0), applied to (ft - /})(/* ~ //)> that
(П) f \fi-fj\2dm=
Thus {/„} is a Cauchy sequence in L2{m). Also, \fn\ < 1 a.e. [m]. Hence there exists
feL°°(m) such that /и->/ in L2(m), and now A1) implies that/^/in L2(m). The
conclusion is that ф =/a.e. [m]:
Every bounded Borel function ф on A coincides with some fe C(A) a.e. [m].
Thus C(A) and L°°(m) are identical as Banach spaces!
Another consequence of the last result is that A is extremally disconnected.
This means, by definition, that the closure of every open set is open. (Hence disjoint
open sets have disjoint closures.)
To prove this, let Qo be open in A, let Q1 be the complement of the closure
Qo of Qo, let ф be the characteristic function of Qu and choose fe C(A) so that
/ — ф a.e. [til]. Since ф = 0 in Qo and since nonempty open sets have positive measure,
the continuity of/ shows that/(/?) = 0 at every p e Qo . Likewise,/(/?) = 1 if p e п1.
The set on which /is neither 0 nor 1 is open and has measure 0, since/ = ф a.e. [m];
hence it is empty. Let Kt = {p e A: f(p) = /}, / = 0, 1. Then Ko and Kx are disjoint
compact sets whose union is A. They are therefore open; also п0 a Ko, Q2 с Kv
It follows that По = Ko, and the proof is complete.
We have also proved, incidentally, that boundaries of open sets have measure 0,
since m(Q0) = m(K0).
We conclude with an application to measure theory. If E and F are measurable
sets, let us say that F almost contains E if F contains E except for a set of measure 0,
that is, if m{E - F) = 0.
The union of an uncountable collection of measurable sets is not always
measurable. However, the following is true:
If{Ea} is an arbitrary collection of measurable sets in [0, 1], there is a measurable
set E a [0, 1] with the following two properties:
(i) E almost contains every Ea.
(и) If F is measurable and F almost contains every Ea, then F almost contains E.

COMMUTATIVE BANACH ALGEBRAS 275
Thus E is the least upper bound of {Ea}. The existence of E implies that the
Boolean algebra of measurable sets (modulo sets of measure 0) is complete.
With the machinery now at our disposal, the proof is very simple.
Let/, be the characteristic function of Ea. Its Gelfand transform/a is then the
characteristic function of an open (and closed) set Qa с A. Let п be the union of all
these Qa. Then п is open, so is its closure U, and there exists /e L°°(m) such that
/ is the characteristic function of U. The desired set E is the set of all x e [0, 1] at
which /(*) = 1.
Involutions
11.14 Definition A mapping x -> x* of a complex (not necessarily commutative)
algebra A into A is called an involution on A if it has the following four properties,
for all x e А, у е A, and XeC:
A) (x + y)* = x* + y*.
B) Qjc)* - Xx*.
C) , (xy)*=y*x*.
D) ' jt** = x.
In other words, an involution is a conjugate-linear antiautomorphism of period 2.
Any x e A for which x* = л; is called hermitian, or self-adjoint.
For example, /->/ is an involution on C(X). The one that we will be most
concerned with later is the passage from an operator on a Hilbert space to its adjoint.
11.15 Theorem If A is a Banach algebra with an involution, and if xe A, then
(a) x + x*, i(x — x*), and xx* are hermitian,
(b) x has a unique representation x = и + iv, with и e A, v e A, and both и and v
hermitian,
(c) the unit e is hermitian,
(d) x is invertible in A if and only ifx* is invertible, in which case (л;*) = (x)*,
and
(e) X e o(x) if and only iflE g(x*).
proof Statement (a) is obvious. If 2w = x + x*, 2v = i(x* - x), then x = и + iv
is a representation as in (b). Suppose x = и + iv' is another one. Put w = v' — v.
Then both w and iw are hermitian, so that
/vv = (/и7)* = — /w* = — /w.
Hence w = 0, and the uniqueness follows.

276 BANACH ALGEBRAS AND SPECTRAL THEORY
Since e* =ee*, (a) implies (c); (d) follows from (c) and (xy)* =j*x*.
Finally, (e) follows if (d) is applied to Xe — x in place of x. ////
11.16 Theorem If the Banach algebra A is commutative and semisimple, then
every involution on A is continuous.
proof Let h be a complex homomorphism of A, and define ф(х) = й(х*).
Properties A) to C) of Definition 11.14 show that ф is a complex homomor-
homomorphism. Hence ф is continuous. Suppose xn -> x and x* -+y in A. Then
й(х*) = ф(х) = lim 0(хя) = lim Я(хи*) = %).
Since ^4 is semisimple, j = jc*. Hence x -> x* is continuous, by the closed graph
theorem. ////
11.17 Definition A Banach algebra A with an involution x -> x* that satisfies
A) 11***11 = Ml2
for every x e A is called a J5*-algebra.
Note that ||x||2 = ||*jc*|| < ||x|| ||**|| implies ||x|| < ||x*||, hence also
Thus
B)
in every i?*-algebra. It also follows that
C) 11***11 = 11*11 ||д*||.
Conversely, B) and C) obviously imply A).
The following theorem is the key to the proof of the spectral theorem that will
be given in Chapter 12.
11.18 Theorem (Gelfand-Naimark) Suppose A is a commutative B*-algebra,
with maximal ideal space A. The Gelfand transform is then an isometric isomorphism
of A onto C(A), which has the additional property that
О) *(**) = AOO ОсеЛ, А б Д),
or, equivalently, that
B) (x*r = l (xeA).
In particular, x is hermitian if and only if $ is a real-valued function.

COMMUTATIVE BANACH ALGEBRAS 277
>
The interpretation of B) is that the Gelfand transform carries the given involu-
involution on A to the natural involution on C(A), which is conjugation. Isomorphisms
that preserve involutions in this manner are often called *-isomorphisms.
proof Assume first that ue A, u = u*, he A. We have to prove that h(u) is
real. Put z = и + ite, for real t. If h{u) = oc + ifi, with oc and ft real, then
h(z) = a + i(j8 + 0, *z* = w2 + f2e,
so that
a2 + 05 + 02 = \h(z)\2 < \\z\\2 = ||zz*|| < ||и||2 + f2,
or
C) ., a2 + p2 + 2pt < \\u\\2 (-00 < t < oo).
By C), P = 0; Kdnce А(м) is real.
If л: е Л, then x; = w + fy, with w = w*, г^ = i>*. Hence x* = и — iv. Since
w and v are real, B) is proved. ^ s4
Thus A is closed under complex conjugation. By the Stone-Weierstrass
. theorem, Я is therefore dense in С(Л).
If x e A and у = xx*, then j> = j* so that \\y2\\ = ||j||2. It follows, by
induction on /i, that |bm|| = IMP for m = 2\ Hence Ш*, = IMI, ЬУ tne
spectral radius formula and (c) of Theorem 11.9. Since у = хх*, B) implies
that j) = |jc|2. Hence
№=Ш1со = Ы1 = 11^*11 = IWI2,
or H-^ll oo = ||jc||. Thus x-+% is an isometry. Hence A is closed in С (A). Since
A is also dense in С(Л), we conclude that Я =С(А). This completes the proof.
////
The next theorem is a special case of the one just proved. We shall state it in
a form that involves the inverse of the Gelfand transform, in order to make contact
with the symbolic calculus.
11.19 Theorem If A is a commutative B*-algebra which contains an element x
such that the polynomials in x and х* are dense in A, then the formula
a)
defines an isometric isomorphism *F of C(a(x)) onto A which satisfies
B) ¥/=(ЧУГ
for every fe C(a(x)). Moreover, if /(Я) — I on a(x), then lF/= x.

278 BANACH ALGEBRAS AND SPECTRAL THEORY
proof Let A be the maximal ideal space of A. Then x is a continuous function
on A whose range is a(x). Suppose h^ e A, h2 e A, and xQi^ = x(h2), that is,
h1(x) = h2(x). Theorem 11.18 implies then that hx(x*) = h2(x*). If P is any
polynomial in two variables, it follows that
htfix, **)) = h2(P(x, x*)),
since h1 and h2 are homomorphisms. By hypothesis, elements of the form
P(x, x*) are dense in A. The continuity of ht and h2 implies therefore that
hx(y) = h2(y) for every у е A. Hence h1=^h2. We have proved that x is one-
to-one. Since A is compact, it follows that x is a homeomorphism of A onto o-(x).
The mapping f-+f°x is therefore an isometric isomorphism of C(a(x))
onto C(A) which also preserves complex conjugation.
Each/о x is thus (by Theorem 11.18) the Gelfand transform of a unique
element of A which we denote by ^/and which satisfies ||*F/|| = ll/lloo • Asser-
Assertion B) comes from B) of Theorem 11.18. If /(Я) = Я, then/° x = x, so that
A) gives ¥/= x. ////
Remark In the situation described by Theorem 11.19, it makes perfectly good
sense to write f(x) for the element of A whose Gelfand transform is/о St. This
notation is indeed frequently used. It extends the symbolic calculus (for these
particular algebras) to arbitrary continuous functions on the spectrum of x,
whether they are holomorphic or not.
The existence of square roots is often of special interest, and in algebras with
involution one may ask under what conditions hermitian elements have hermitian
square roots.
11.20 Theorem Suppose A is a commutative Banach algebra with an involution,
x e A, x = x*, and a(x) contains no real X with X < 0. Then there exists у е A with
у = j* and y2 = x.
Note that the given involution is not assumed to be continuous. We shall see
later (Theorem 11.26) that commutativity can be dropped from the hypothesis/
proof Let D be the complement (in €) of the set of all nonpositive real numbers.
There exists/e H(U) such that/2(A) = Я, and/(l) = 1. Since a(x) с п, we can
define у е A by
0) y=J(xl
as in Definition ,10.26. Then y2 = x, by Theorem 10.27. We will prove that
У* =У-

COMMUTATIVE BANACH ALGEBRAS 279
Since Q. is simply connected, Runge's theorem furnishes polynomials Pn
that converge to/, uniformly on compact subsets of п. Define Qn by
Since/(I) =/(A), the polynomials Qn converge to/in the same manner. Define
C) Jn = £„(*) («=1,2,3,...).
By B), the polynomials Qn have real coefficients. Since x = x*, it follows that
yn = y* . By Theorem 10.27,
D) ' y = \imyn,
л-* oo
since Qn ->/, so that £„(*) -*/(•*)• If the involution were assumed to be contin-
continuous, the set of hermitian elements would be closed, and j* = у would follow
directly from D).
Let R be the radical of A. Let n: A -» AjR be the quotient map. Define
an involution in AjR by
E) [п(а)Г = n(a*) (a e A).
. If a e A is hermitian, so is n(a). Since n is continuous, n(yn) -> л^у). Since Л/Я
is isomorphic to A (Theorem 11.9), A/R is semisimple, and therefore every in-
involution in A/R is continuous (Theorem 11.16). It follows that n(y) is hermitian.
Hence n(y) = n(y*).
We conclude that у* — у lies in the radical of A.
By Theorem 11.15, у = и + iv, where и = w* and v = i>*. We just proved
that г e Я. Since л: — j2, we have
F) x = u2 -v2 + 2iuv.
Let h be any complex homomorphism of A. Since i; e R, h(v) = 0. Hence Л(х) =
[//(w)]2- By hypothesis, 0 ф o(x). Thus Л(дс) ф 0; hence Л(и) / 0. By Theorem
11.5, и is invertible in A. Since x = x*, F) implies that uv = 0. Since г; =
u~x{uv), we conclude that f = 0. This completes the proof. ////
Remark If <j(x) с @, oo), then also a(y) с @, oo). This follows from A)
(the definition of y) and the spectral mapping theorem.
Applications to Noncommutative Algebras
Noncommutative algebras always contain commutative ones. Their presence can
sometimes be exploited to extend certain theorems from the commutative situation
to the noncommutative one. On a trivial level, we have already done this: In the ele-
elementary discussion of spectra, our attention was usually fixed on one element x e A;

280 BANACH ALGEBRAS AND SPECTRAL THEORY
the (closed) subalgebra Ao of A that x generates is commutative, and much of the dis-
discussion took place within Ao . One possible difficulty was that x might have different
spectra with respect to A and Ao. There is a simple construction (Theorem 11.22)
that circumvents this. Another device (Theorem 11.25) can be used when A has an
involution.
11.21 Centralizers If S is a subset of a Banach algebra A, the centralizer of S is
the set
T(S) = {x e A: xs — sx for every s e S}.
We say that S commutes if any two elements of 5 commute with each other. We shall
use the following simple properties of centralizers.
(a) T(S) is a closed subalgebra of A.
(с) IfS commutes, then Г(ГE)) commutes.
Indeed, if л: and у commute with every s ё S, so do Ix, x + y, and xy; since multi-
multiplication is continuous in A,T(S) is closed. This proves (л). Since every s e S commutes
with every x e T(S), (b) holds. If S commutes, then 5 с T(S), hence T(S) => Г(ГE)),
which proves (c), since Г(£) obviously commutes whenever T(E) с Е.
11.22 Theorem Suppose A is a Banach algebra, S cz A, S commutes, and B =
Г(ГE)). Then В is a commutative Banach algebra, S a B, and oB{x) = oA{x) for every
xeB.
proof Since e e B, Section 11.21 shows that В is a commutative Banach
algebra that contains S. Suppose xeB and x is invertible in A. We have to
show that x'1 e B. Since x e B, xy = yx for every у е ГE); hence у = x~*yx,
yx~1 = x~ xy. This says that x~x e Г(ГE)) = В. ////
11.23 Theorem Suppose A is a Banach algebra, x e А, у e A, and xy = yx. Then
<т(х + у) с a(x) + o(y) and e(xy) <=■ o(x)o(y).
proof Put S = {x, y}; put В = Г(ГE)). Then x + у е В, xy e В, and Theorem
11.22 shows that we have to prove that
°b(x + у) с сгв(*) + oB(y) and o-B(x.y) cz ав(х)ав(у).
Since ^ is commutative, aB{z) is the range of the Gelfand transform £,
for every ze5. (The Gelfand transforms are now functions on the maximal
ideal space of B.) Since
, (x + yT - * + ^ and (xjr = *Л
we have the desired conclusion. ////

COMMUTATIVE BANACH ALGEBRAS 281
- Corollary IfCx = Rx-Lx,as in Section 10.37, then o(Cx) a o(x) - o{x).
proof If the theorem is applied to the commuting elements Rx and — Lx of
the algebra &(A), the conclusion is
c(Cx) c= a(Rx) - c(Lx).
But a(Rx) = a(x) = o(Lx). mi
11.24 Definition Let A be an algebra with an involution. If x e A and xx* = x*x,
then л: is said to be normal. A set S с Л is said to be normal if S commutes and if
x* e S whenever x e S.
11.25 Theorem Suppose A is a Banach algebra with an involution, and В is a normal
subset of A that is maximal with respect to being normal. Then
(a) В is a closed commutative subalgebra of A, and
(b) °в(х) — ° а(х)/ог ^ery x e B.
•Note that the involution is not assumed to be continuous but that В neverthe-
nevertheless turns out to be closed.
proof We begin with a simple criterion for membership in B: If x e A, if
xx* = x*x, and if xy = yxfor every у e B, then x e B.
For if x satisfies these conditions, we also have xy* = y*x for all у е В,
since В is normal, and therefore x*y = yx*. It follows that В u {x, x*} is normal.
Hence x e B, since В is maximal.
This criterion makes it clear that sums and products of members of В are
in B. Thus В is a commutative algebra.
Suppose xne В and xn -> x. Since xn у = yxn for all у е В, and multiplica-
multiplication is continuous, we have xy = yx and therefore also
Х*У = (y*x)* = (xy*)* = yx*.
In particular, x*xn = xnx* for all n, which leads to x*x = xx*. Hence x e B,
by the above criterion. This proves that В is closed and completes (a).
Note also that e e B. To prove (b), assume xe В, х'1 е A. Since x is
normal, so is x ~l, and since x commutes with every у е В, so does x~x. Hence
x-'eB. ЦП
Our first application of this is a generalization of Theorem 11.20:
11.26 Theorem The word "commutative" may be dropped from the hypothesis
of Theorem 11.20.

282 BANACH ALGEBRAS AND SPECTRAL THEORY
proof By Hausdorff's maximality theorem, the given hermitian (hence normal)
x e A lies in some maximal normal set B. By Theorem 11.25 we can apply
Theorem 11.20 with В in place of A. ////
Our next application of Theorem 11.25 will extend some consequences of
Theorem 11.18 to arbitrary (not necessarily commutative) 2?*-algebras.
11.27 Definition In a Banach algebra with involution, the statement "x>0"
means that x = x* and that a(x) с [0, oo).
11.28 Theorem Every B*-algebra A has the following properties:
(a) Hermitian elements have real spectra.
(b) IfxeAisnormal,thenp(x) = \\x\\.
(c) IfyeA,thenp(yy*)=\\y\\2.
(d) IfueA,veA,u>0, andv>0, then u + v>0.
(e) If у е A, then yy* > 0.
(/) VУ E A> tnen e + УУ* is invertible in A.
proof Every normal x e A lies in a maximal normal set В <z A. By Theorems
11.18 and 11.25, В is a commutative Z?*-algebra which is isometrically isomor-
phic to its Gelfand transform В — C(A) and which has the property that
0) a(z)=z(A) (zeB).
Here o{z) is the spectrum of z relative to A, A is the maximal ideal space of B,
and z(A) is the range of the Gelfand transform of z, regarded as an element of B.
If x = x*, Theorem 11.18 shows that x is a real-valued function on A.
Hence A) implies (a).
For any normal x, A) implies p(x) = Щ]^ . Also, рЦ^ = ||x||, since В
and В are isometric. This proves (b).
If у e A, then yy* is hermitian. Hence (c) follows from (b), since p(yy*) =
\\yy*\\ = \\y\\2.
Suppose now that и and v are as in (d). Put a = ||w||, P = \\v\\, w = и + v,
у = a + p. Then a(u) с [0, a], so that
B) a(oce - и) а [0, a]
and (b) implies therefore that \\cce — u\\ < a. For the same reason, ||/te — v\\ < p.
- Hence
C) \\ye-w\\<y.
Since w = w*, (a) implies that o(ye — w) is real. Thus
D) , o(ye- w)c [-7, y],
because of C). But D) implies that a(w) с [0, 2y]. Thus w > 0, and (d) is proved.

COMMUTATIVE BANACH ALGEBRAS 283
We turn to the proof of (e). Put x = yy*. Then x is hermitian, and if В
is chosen as in the first paragraph of this proof, then Jc is a real-valued function
on A. By A), we have to show that x > 0 on A.
Since В = C(A), there exists z e В such that
E) z = \3t\-x on A.
Then z = z*, because z is real (Theorem 11.18). Put
F) ' zy = w = и + iv,
where и and v are hermitian elements of A. Then
(?)
and
(8)
therefore
w*
w
= 2w2 +
= zyy
2v2 -
*z* =
ww*
= zxz =
= 2u2
Since и = м*, <t(w) is real, by (a), hence w2 > 0, by the spectral mapping theorem.
Likewise, v2 > 0. By E), z2x < 0 on A. Since z2x e B, it follows from A) that
. —z2x > 0. Now (8) and (d) imply that w*w > 0.
But a(ww*) с g(w*w) u {0} (Exercise 2, Chapter 10). Hence ww* > 0.
By G), this means that z2x > 0 on A. By E), this last inequality holds only
when £ = |x|. Thus x > 0, and (e) is proved.
Finally, (/) is a corollary of (e). ////
Equality of spectra can now be proved in yet another situation, in which com-
mutativity plays no role.
11.29 Theorem Suppose A is a B*-algebra, В is a closed sub algebra of A, e e B,
and x* e В for every x e B. Then aA(x) = gb(x) for every x e B.
proof Suppose x e В and x has an inverse in A. We have to show that x~1 e B.
Since x is invertible in A, so is jc*, hence also xx*. Thus <ja(xx*) с @, oo),
by (e) of Theorem 11.28. Since oA(xx*) has connected complement in €,
Theorem 10.18 shows thatcB(xx*) = oA(xx*). Hence (jcjc*) e B, and finally
x'1 =x*(xx*y1eB. ЦП
Positive Functionals
11.30 Definition A positive functional is a linear functional F on a Banach alge-
algebra A with an involution, that satisfies
F(xx*) > 0

284 BANACH ALGEBRAS AND SPECTRAL THEORY
for every x e A. Note that A is not assumed to be commutative and that continuity of
Fis not postulated. (The meaning of the term "positive" depends of course on the
particular involution that is under consideration.)
11.31 Theorem Every positive functional F on a Banach algebra A with involution
has the following properties:
(a) F(x*) = F(x).
(b) \F(xy*)\2 < F(xx*)F(yy*).
(c) | F(x) |2 < F(e)F(xx*) < F(eJp(xx*).
(d) | F(x) | < F(e)p(x)for every normal x e A.
(e) F is a bounded linear functional on A. Moreover, \\F\\ = F(e) if A is commutative,
and \\F\\ < pl/2F(e) if the involution satisfies \\x*\\ < fi\\x\\ for every xeA.
proof If x e A and у е A, put
A) p = F(xx*), q = F(yy*), r = F(xy*), s = F(yx*).
Since F[(x + ocy)(x* + осу*)] > О for every осе С,
B) p + ocr + ocs + \a\2q>0.
With oc = 1 and oc = i, B) shows that s + r and i(s — r) are real. Hence s = r.
With у = е, this gives (a).
If r = 0, (b) is obvious. If г Ф 0, take a = rr/1 r | in B), where t is real.
Then B) becomes
C) p + 2\r\t + qt2 >0 (-oo</<oo),
so that |r|2 </?^. This proves (b).
Since ее* = <?, the first half of (c) is a special case of (b). For the second
half, pick t > p(xx*). Then o(te — xx*) lies in the open right half-plane. By
Theorem 11.26, there exists и e A, with и = w*, such that u2 = te — xx*. Hence
D) tF(e) - F(xx*) = F(u2) > 0.
It follows that
E) F(xx*) < F(e)p(xx*).
This completes part (c).
If x is normal, i.e., if xx* = x*x, Theorem 11.23 implies that a(xx*) c=
<r(X)ff(A;*), so that
F) P(xx*)<p(x)p(x*) = p(xJ-
Clearly, (d) follows from F) and (c).

COMMUTATIVE BANACH ALGEBRAS 285
If A is commutative, then (d) holds for every xe A, so that \\F\\ = F(e).
' If 11**11 < P\\x\U (c) implies \F(x)\ < F(e)Plf2\\x\\, since p(xx*) < \\x\\\\x*\\. This
disposes of the special cases of part (e).
Before turning to the general case, we observe that F(e) > 0 and that
F(x) = 0 for every x e A if F(e) = 0; this follows from (c). In the remainder of
this proof we shall therefore assume, without loss of generality, that
G) ' F(e)=\.
Let H be the closure of H, the set of all hermitian elements of A. Note that
H and iH are real vector spaces and that A = H + ///, by Theorem 11.15.
By (d), the restriction of Fto И is a real-linear functional of norm 1, which there-
therefore extends to a real-linear functional Ф on H, also of norm 1. We claim that
(8) ФОО = 0 ifyeHniH,
for if у = lim un = lim (ivn), where un e H and vn e H, then u\ -> y2, v2n ~> — j2,
so that (c) and (*/) imply
(9) |F(t/n)|2 < F(u2n) < F(u2n + »;) < ||t/2 + 172Ц ->0.
Since Ф(у) = lim jF(wn), (8) is proved.
By Theorem 5.20, there is a constant у < oo such that every л: е Л has a
representation
A0) . * = *! +ix2,xleH,x2eH, \\Xl\\ + ||x2|| <y|W|.
If jc = м + /f, with tie H, v e H, then jq — м and x2 — v lie in Я п /Я. Hence
(8) yields
A 0 F(x) = F(u) + iF(v) = Ф(х,) + /Ф(х2),
so that
A2) ^(х)\^\Ф(х1)\+\Ф(х2)\<\\х1\\ + ||x2|| <y||x||.
This completes the proof. ////
Exercise 13 contains further information about part (e).
Examples of positive functionals—and a relation between them and positive
measures— are furnished by the next theorem. It contains Bochner's classical theorem
about positive-definite functions as a very special case. The identifications that lead
from one to the other are indicated in Exercise 14.
11.32 Theorem Suppose A is a commutative Banach algebra, with maximal ideal
space A, and with an involution that is symmetric in the sense that
A) A(jc*) = A(jc) (xeA,heA).

286 BANACH ALGEBRAS AND SPECTRAL THEORY
Let К be the set of all positive functionah F on A that satisfy F(e) < 1. Let M be the
set of all positive regular Borel measures ц on A that satisfy fj.(A) < 1. Then the formula
B) F(x)=\xdn (xeA)
establishes a one-to-one correspondence between the convex sets К and M, which carries
extreme points to extreme points.
Consequently, the multiplicative linear functionals on A are precisely the extreme
points ofK.
proof If fi e M and F is defined by B), then F is obviously linear, and F(xx*)
= J \*\2dfi>0, because A) implies that (***)* = |*|2. Since F(e) - ц(А),
FeK.
If FeK, then F vanishes on the radical on A, by (d) of Theorem 11.31.
Hence there is a functional F on A that satisfies F(x) = F(x) for all x e A.
In fact,
C) | F(x)\ = \F(x)\ < F(e)p(x) = F(*)||*IL (x e A),
by (d) of Theorem 11.31. It follows that F is a linear functional of norm F(e)
on the subspace A of С(Л). This extends to a functional on C(A), with the same
norm, and now the Riesz representation theorem furnishes a regular Borel
measure ц, with ||/i|| = F(e), that satisfies B). Since
D)
we see that ц > 0. Thus jie M.
By A), A satisfies the hypotheses of the Stone-Weierstrass theorem and is
therefore dense in С(Л). This implies that ц is uniquely determined by F.
One extreme point of M is 0; the others are unit masses concentrated at
points he A. Since every complex homomorphism of A has the form x:-» $(h),
for some he A, the proof is complete. ////
We conclude by showing that the extreme points of К are multiplicative even if
A) is not satisfied.
11.33 Theorem Let К be the set of all positive functionals F on a commutative
Banach algebra A with an involution, that satisfy F(e) < 1. If Fe K, then each of the
following three properties implies the other two:
(a) F(xy) = F(x)F(y)for all x and у е A.
(b) F(xx*) = F(x)F(x*) for every xeA.
(c) F is an extreme point of K.

COMMUTATIVE BANACH ALGEBRAS 287
proof It is trivial that (a) implies (b). Suppose (b) holds. With x = e, (b)
shows that F(e) = F(eJ, and so F(e) - 0 or F(e) = 1. When F(e) = 0, then
F = 0, by (c) of Theorem 11.31, and so F is an extreme point of K. Assume
F(e) = 1, and 2F =Ft + F2, Ft e K, F2 e K. We have to show that Fx = F.
Clearly, FjO) - 1 - F{e). If * e A is such that F(x) = 0, then
A) | FjOI2 < Ft(xx*) < 2F(xx*) = 2F(x)F(x*) - 0,
by (&) and Theorem 11.31. Thus Ft coincides with Fon the null space of Fand at
e. It follows that Fx = F. Hence (&) implies (c).
To show that (c) implies (a), let F be an extreme point of K. Either
F(e) = 0, in which case there is nothing to prove, or F(e) = 1. We shall first
prove a special case of (a), namely,
B) F(xx*y) = F(xx*)F(y) (xeA,ye A).
Choose x so that ||xjc*||< 1. By Theorem 11.20, there exists z e A, z = z*,
such that z2 = e — xx*. Define
C) Q)(y) = F(xx*y) (ye A).
Then
D) Ф(^*) = F(xx*yy*) = F[(xy)(xy)*] > 0,
and also
E) (F - Ф)(уу*) = F[(e - xx*)jj*] = F(z2^*) = %)W*] > 0.
Since
F) 0 < Ф(е) = F(xx*) < F(e)\\xx*\\ < 1,
D) and E) show that both Ф and F - Ф are in K. If Ф(е) = 0, then Ф = 0. If
Ф0) > 0, F) shows that
a convex combination of members of K. Since F is extreme, we conclude that
(8) * Ф = Ф(е)Г.
Now B) follows from (8) and C).
Finally, the passage from B) to (a) is accomplished by any of the following
identities, which are satisfied by every involution:
Ifn = 3, 4, 5, ..., if со = exp Bni/n), if x e A, and ifzp = e + co~px, then
(9) jc = - Eco%z*.

288 BANACH ALGEBRAS AND SPECTRAL THEORY
The proof of (9) is a straightforward computation which uses the fact
that
A0) £ (Dp = J]CD2P = 0. ////
P=l p=l
Exercises
/ Prove Proposition 11.2.
2 State and prove an analogue of Wiener's lemma 11.6 for power series that converge
absolutely on the closed unit disc.
3 If X is a compact Hausdorff space, show that there is a natural one-to-one correspondence
between closed subsets of X and closed ideals of C{X).
4 Prove that the polynomials are dense in the polydisc algebra A(Un). (See Theorem 11.7.)
Suggestion: If /e A(Un\ 0 < r < 1, and /r is defined by /r(z) =f(rz), then /r is the sum
of an absolutely (hence uniformly) convergent multiple power series on Un.
5 Suppose A is a commutative Banach algebra, x e A, and / is holomorphic in some
open set О с ^ that contains the range of x. Prove that there exists ye A such that
у — /о x, that is, such that h(y) =f(h(x)) for every complex homomorphism h of A.
Prove that у is uniquely determined by x and/if A is semisimple.
6 Suppose A and В are commutative Banach algebras, В is semisimple, ф : A -> В is a
homomorphism whose range is dense in B, and a: AB -> A^ is defined by
(oA)(jc) - /#(*)) (jc e Л, Л е A*).
Prove that a is a homeomorphism of AB onto a compact subset of Ал. [The fact that
ф(А) is dense in В implies that a is one-to-one and that the topology of AB is the weak
topology induced by the Gelfand transforms of the elements ф(х), for x e A.]
Let A be the disc algebra, let В = C(K), where К is an arc in the unit disc, and
let ф be the restriction mapping of A into B. This example shows that a(AB) may be a
proper subset of Ал, even if ф is one-to-one.
Find an example in which ф(А) = В but a(AB) Ф AA.
7 In Example 11.13F) it was asserted that А Ф C(A). Find several proofs of this.
8 Which properties of Lebesgue measure are used in Example 11.13(/)? Can Lebesgue
measure be replaced by any positive measure, without changing any of the results ?
Supply the details for the last paragraph in Example 11.13(/).
Using the notation in Example 11.13(/), prove that m(S) = m(S) for every
Borel set S с A. Hence boundaries of Borel sets have measure 0. (In the text, this
was proved for open sets.)
9 Let С be the algebra of all continuously differentiable complex functions on the unit
interval [0, 1], with pointwise multiplication, normed by
11/11= II/IU+ H/'lloo.
(a) Show that С is a semisimple commutative Banach algebra. Find its maximal ideal
space.

COMMUTATIVE BANACH ALGEBRAS 289
(b) Fix /?, 0 <p < 1; let J be the set of all fe С for which f(p) =f'(p) = 0. Show that
/ is a closed ideal in С and that C'/J is a two-dimensional algebra which has a
one-dimensional radical. (This gives an example of a semisimple algebra with a
quotient algebra that is not semisimple.) To which of the two algebras described
in Exercise 14 of Chapter 10 is CV/isomorphic?
10 Let A be the disc algebra. Associate to each/e A a function/* e A by the formula
/*(г)=/Ш.
Then/->/* is an involution on Л.
(я) Does this involution turn A into a i?*-algebra?
(b) Does o(ff*) always lie in the real axis?
(c) Which complex homomorphisms of A are positive functional, with respect to this
involution ?
(d) If /x is a positive finite Borel measure on [—1, 1], then
is a positive functional on A Are there any others?
11 Show that commuting idempotents have distance >1. Explicitly, if x2 = x, y2 — y,
xy — yx for some x and у in a Banach algebra, then either x = у or ||jc — y\\ > 1. Show
that this may fail if xy ф ух.
12 If xy = yx for some x and у in a Banach algebra, prove that
p(xy) < p(x)P(y) and />(x + y) < p(x) -
/5 Let r be a large positive number, and define a norm on (£2 by
Ml = | Wi | + t\ w21 if w = (wi, w2).
Let ^ be the algebra of all complex 2-by-2 matrices, with the corresponding operator
norm:
IMI = max{|LKw)||: llw|| = 1} (yeA).
For у e A, let y* be the conjugate transpose of y. Consider a fixed x e A, namely,
x=\l
Prove the following statements.
(a) ||jc(w)|| = /||w||; hence ||x|| = i
(b) cr(x) = {t, — /} = a(x*).
(c)
id)
(e) Therefore commutativity is required in Theorem 11.23 and in Exercise 12.
(/) If F(y) is the sum of the four entries in y, for у е A, then F is a positive functional
on A.
(g) The equality ||F|| = F(e) [see (e) of Theorem 11.31] does not hold, because F(e) = 2
and F(x) = 1 + t2y so that ||F|| > /.

290 BANACH ALGEBRAS AND SPECTRAL THEORY
(h) If К is the set of all positive functional/on A that satisfy f(e) < 1 (as in Theorem
11.33), then К has many extreme points, although 0 is the only multiplicative linear
functional on A. Commutativity is therefore required in the implication (с) ->(я) of
Theorem 11.33.
14 A complex function ф, defined on Rn, is said to he positive-definite if
2 с1с]ф(х1-х^>0
for every choice of jcb ..., xr in Rn and for every choice of complex numbers ci,..., cr
(a) Show that | ф(х) | < ф@) for every x e Rn.
(b) Show that the Fourier transform of every finite positive Borel measure on Rn is
positive-definite.
(c) Complete the following outline of the converse of (b) (Bochner's theorem): If ф is
continuous and positive-definite, then ф is the Fourier transform of a finite positive
Borel measure.
Let A be the convolution algebra ^(R"), with a unit attached, as described in
(d) of Section 10.3 and (e) of Section 11.13. Define/(jc) =f(-x). Show that
is an involution on A and that
/-f aS -> J /0 fifr72n -f a<£@)
Rn
is a positive functional on A. By Theorem 11.32 and (e) of Section 11.13, there is a
positive measure p on the one-point compactification A of Rn, such that
f fф dmn + осф{0) = f (/+ a) dp.
If a is the restriction of p to Rn, it follows that
f fфdmп^ \ /da
JRn JRn
for every fe ViR"). Hence ф = a. (Actually, p is already concentrated on Rn, so
that a = p.)
(d) Let P be the set of all continuous positive-definite functions ф on Rn that satisfy
<£@) < 1. Find all extreme points of this convex set.
15 Let A be the maximal ideal space of a commutative Banach algebra A. Call a closed set
P с: Д an A-boundary if the maximum of | x \ on A equals its maximum on jS, for every
x e A. (Trivially, A is an ^-boundary.)
Prove that the intersection dA of all ^-boundaries is an ^-boundary.
dA is called the Shilov boundary of A. The terminology is suggested by the maximum
modulus property of holomorphic functions. For instance, when A is the disc algebra,
then dA is the unit circle, which is the topological boundary of A, the closed unit disc.
Outline of proof: Show first that there is an ^-boundary ft о which is minimal in the
sense that no proper subset of jS0 is an ^-boundary. (Partially order the collection of A-

COMMUTATIVE BANACH ALGEBRAS 291
boundaries by set inclusion, etc.) Then pick hoepo, pick xly ..., xneA with Xi(h0) = О,
and put
V = {heA: \xt(h)\ < 1 for 1 <i<n}.
Since fi0 is minimal, there exists xeA with \\x\U = 1 and \x(h)\ < 1 on p0 — V. If
y=-xm and m is sufficiently large, then \x-ty\ < 1 on j80, for all /. Hence \\xiy\U < I.
Conclude from this first that \y(h)\= \\y\\,» only in У, hence that V intersects every
Л-boundary p, and finally that h0 e p. Thus p0 с jg, and p0 = dA.
16 Suppose A is a Banach algebra, m is an integer, m > 2, /f < oo, and
for every x e A. Show that there exist constants Kn < oo, for n = 1, 2, 3, ..., such that
(This extends Theorem 11.12.)
17 Suppose {ал,} (— oo < n < oo) are positive numbers such that oj0 = 1 and
for all integers m and я. Let A = A{a>n} be the set of all complex functions/on the integers
for which the norm
is finite. Define multiplication in A by
n)= I f(n~k)g(k).
*= -00
{a) Show that each A{a>n} is a commutative Banach algebra.
(b) Show that R+ = limn-*co (con)l/n exists and is finite, by showing that R+ —
тЬьоСаО1'".
(c) Show similarly that R- = lim,,-» (co-n)l/n exists and that R- < R+ .
(d) Put A ={Ae (Z1: jR_ <|A| <R+}. Show that A can be identified with the maximal
ideal space of А{шп} and that the Gelfand transforms are absolutely convergent
Laurent series on A.
(e) Consider the following choices for {а>„}:
(О "л=1.
(//) шп = 2".
(///) шп = 2п if n > 0, а>„ = 1 jf п < 0.
(ш) сол = 1 + 2л2.
(у) а>„ = 1 + 2п2 if л > 0, <оп = 1 if л < 0.
For which of these is A a circle? For which choices is А{а>„} self-adjoint, in the sense
that A is closed under complex conjugation?
(/) Is A{cxjn} always semisimple ?
(g) Is there an А{шп), with A the unit circle, such that A consists entirely of infinitely
differentiable functions?

12
BOUNDED OPERATORS ON A HILBERT SPACE
Basic Facts
12.1 Definitions A complex vector space H is called an inner product space (or
unitary space) if to each ordered pair of vectors x and у in H is associated a complex
number (x, y), called the inner product or scalar product of x and y, such that the
following rules hold:
(a) (y, x) = (x, y). (The bar denotes complex conjugation.)
(b) (x + y,z) = (x, z) + (y, z).
(c) (ocx, y) = oc(x, y) if x e H, у e H, a e 0.
(d) (jc, x) > 0 for all x e H.
(e) (x, x) = 0 only if x = 0.
For fixed j, (x, y) is therefore a linear function of jc. For fixed x, it is a conjugate-
linear function of y. Such functions of two variables are sometimes called sesquilinear.
If (x, y) = 0, x is said to be orthogonal to j, and the notation x Lyis sometimes
used. Since (x, y) = 0 implies (>>, jc) = 0, the relation JL is symmetric. If E a H and
F с H, the notation ELF means that x Ly whenever x e E and у e F. Also, E1 is
the set of all у e H that are orthogonal to every xe E.

BOUNDED OPERATORS ON A HILBERT SPACE 293
Every inner product space can be normed by defining
||*|| «(JC,*I'2.
Theorem 12.2 implies this. If the resulting normed space is complete, it is called a
Hilbert space.
12.2 Theorem If x e H and yeH, where H is an inner product space, then
(!) - \{x9y)\<Z\\x\\\\y\\
and
B), \\х + У\\^\\х\\ + \\у\\.
Moreover
C) ' \\y\\ < \\kx 4- y\\ for every Xe<Z
if and only ifxLy.
proof Put a = (x, y). A simple computation gives
D) 0 < \\Лх + y\\2 = \X\2\\x\\2 + 2 Re (ой) 4- Ы|2.
Hence C) holds if a = 0. If x = 0, A) and C) are obvious. If x Ф 0, take
X = -a/||jc||2. With this A, D) becomes
E)
This proves A) and shows that C) is false when а ф 0. By squaring both sides
of B), one sees that B) is a consequence of A). ////
Note: Unless the contrary is explicitly stated, the letter H will from now on
denote a Hilbert space.
12.3 Theorem Every nonempty closed convex set E a H contains a unique x
of minimal norm.
proof The parallelogram law
A) \\x + y\\2+\\x-y\\2 = 2\\x\\2 + 2\\y\\2 (xeH,yeH)
follows directly from the definition ||x||2 = (x, x). Put
B) d=mf{\\x\\:xeE}.
Choose xn e E so that ||xj -> d Since i(xn + jcJ e E, \\xn + xm\\2 > Ad2. If x
and у are replaced by xn and xm in A), the right side of A) tends to 4d2. Hence

294 BANACH ALGEBRAS AND SPECTRAL THEORY
A) implies that {xn} is a Cauchy sequence in H, which therefore converges to
some xe E, with ||jc|| = d.
If yeE and \\y\\ = d, the sequence {x, y, x, y, ...} must converge, as we
just saw. Hence у = х. ////
12.4 Theorem If M is a closed subspace of H, then
The conclusion is, more explicitly, that M and ML are closed subspaces of H
whose intersection is {0} and whose sum is H. The space M1 is called the orthogonal
complement of M.
proof If E с H, the linearity of (x, y) as a function of x shows that EL is a
subspace of #, and the Schwarz inequality A) of Theorem 12.2 implies then that
E1 is closed.
If x e M and x e Af\ then (x, jc) = 0; hence jc = 0. Thus M n M1 = {0}.
If jc e Я, apply Theorem 12.3 to the set jc — Mto conclude that there exists
хге M that minimizes \\x — xx\\. Put x2 = x — jq. Then ||jc2|| < \\x2 + y\\ for
all у e M. Hence jc2 e M1, by Theorem 12.2. Since jc = jcx + jc2 , we have shown
that M + ML = H. ЦП
Corollary If M is a closed subspace of H, then
(M1I = M.
proof The inclusion M с (М1I is obvious. Since
м ®ml = н = м1 е (м1I,
M cannot be a proper subspace of (Af1I. ////
We now describe the dual space Я* of H.
12.5 Theorem There is a conjugate-linear isometry y-^AofH onto #*, given by
A) Ax = (x,y) (xeH).
- proof ifyeH and Л is defined by A), the Schwarz inequality A) of Theorem
12.2 shows that Л e #* and that ||Л|| < ||.y||. Since
B) \\у\\2 = (у,у) = Лу<\\Щу1
it follows that \\A\\ =\\y\\.
It remains to be shown that every Л е Я* has the form A).

BOUNDED OPERATORS ON A HILBERT SPACE 295
If Л = 0, take у = 0. If Л Ф 0, let jV(A) be the null space of Л. By
Theorem 12.4 there exists z e Ж(ЛI, z Ф 0. Since
C) (Ax)z - (Az)jc e Ж(Л) (jc e Я),
it follows that (Ax)(z, z) - (Az)(x, z) = 0. Hence A) holds with j; = (z, zy1(Kz)z.
I/I/
12.6 Theorem If {xn} is a sequence ofpairwise orthogonal vectors in Я, then each
of the following three statements implies the other two.
oo
(a) £ xn converges, in the norm topology of H.
n = l
(b) i\\xn\\2<co.
n= 1
oo
(c) Yj (xn -> У) converges, for every yeH.
Thus strong convergence (a) and weak convergence (c) are equivalent for series
of orthogonal vectors.
proof Since (xt, Xj) = 0 if i Ф}, the equality
(\\ || v a- • • • + x II2 = IIjc II2 + • • • 4- IIjc II2
K1/ W-^n ~ ' лт\\ \\лп\\ ' ' \\лт\\
holds whenever n <m. Hence (b) implies that the partial sums of £ xn form a
Cauchy sequence in H. Since H is complete, (b) implies (a). The Schwarz
inequality shows that (a) implies (c). Finally, assume that (c) holds. Define
Л„ е Я* by
B) Any = £(>>, xt) (y e H, n = 1, 2, 3, ...).
By (с), {Л„.у} converges for every yeH; hence {||AJ|} is bounded, by the Banach-
Steinhaus theorem. But
C) ||AJ = ||^ + ••• + xn\\ = {ll^ll2 + .-• + \\xn\\2}l/2.
Hence (c) implies (b).
Bounded Operators
In conformity with notations used earlier, &(H) will now denote the Banach algebra
of all bounded linear operators T on a Hilbert space H Ф {0}, normed by
, \\х\\<\\

296 BANACH ALGEBRAS AND SPECTRAL THEORY
We shall see that &(H) has an involution which makes it into a i?*-algebra.
We begin with a simple but useful uniqueness theorem.
12.7 Theorem If Те &(H) and if (Tx, x) = 0 for every xeH, then T = 0.
proof Since (T(x + y), x + y) = 0, we see that
A) (Tx,y) + (Ту, x) = 0 (xeH,yeH).
If у is replaced by /y in A), the result is
B) - i(Tx, y) + i(Ty, x) = 0 (xeH,ye Я).
Multiply B) by / and add to A), to obtain
C) (ZX>0 = O (xeH,yeH).
With j = Г*, C) gives ||7х||2 = 0. Hence Tx = 0. ////'
Corollary If S e @(H),
(Sjc, л) = (Tx, x)
for every x e H, then S —T.
proof Apply the theorem to S — T. Note that Theorem 12.7 would fail if the
scalar field were R. To see this, consider rotations in jR2. ////
12.8 Theorem Iff: H x Я-> <£ is sesquilinear and bounded, in the sense that
A) M = sup{|/(x, y)\: ||x|| = Ы| = 1} < oo,
then there exists a unique S e @{H) that satisfies
B) f(x,y) = (x,Sy) (xeH,yeH).
Moreover, \\S\\ = M.
proof Since | /O, y) | < ikr||jc|| ||j||, the mapping
is, for each у е Н, a bounded linear functional on Я, of norm at most Af || j||. It
now follows from Theorem 12.5 that to each yeH corresponds a unique element
SyeH such that B) holds; also, ||5>|| <.M\\y\\. It is clear tbat S: Я->Я is
additive. If осе 0, then
(x, S(ocy)) =f(x, осу) = aftx, у) = oc(x, Sy) = (x, ocSy)
for all x and у in Я. If follows that S is linear. Hence S e ЩН), and ||£|| < M.

BOUNDED OPERATORS ON A HILBERT SPACE 297
But Ave also have
\f(x,y)\ = \(x,Sy)\ < \\x\\\\Sy\\ < ||х||||5Ш,
which gives the opposite inequality M < \\S\\. IIII
12.9 Adjoints If Te^(H), then (Tx, y) is linear in x, conjugate-linear in y, and
bounded. Theorem 12.8 shows therefore that there exists a unique T* e M(H) for
which
•0) (Tx, y) = (x, T*y) (xeH,yeH)
and also that
B) ЦП - \\T\\.
We claim that Г-* Г* is an involution on Щ7/), that is, that the following four
properties hold:
C)
D) ,
E)
F) r** = r
Of these, C) is obvious. The computations
(аГх, у) = а(Гх, у) - а(х, Г*
(STx, у) = (Гх, 5*^) = (jc,
(Гх, у) = (Т*у, х) = (у, Г**х) = (Г**х, у)
give D), E), and F). Since
|| Гх||2 = (Tx, Tx) - (T*Tx, x) < ||T*T|| ||x||2
for every x e Я, we have ||Г||2 < ||Г*Г||. On the other hand, B) gives
Hence the equality
G) \\T*T\\ - ||ГЦ2
holds for every T
We have thus proved that @(H) is a B*-algebra, relative to the involution T-+T*
defined by A).

298 BANACH ALGEBRAS AND SPECTRAL THEORY
)
Note: In the preceding setting, jT* is sometimes called the Hilbert space
adjoint of Г, to distinguish it from the Banach space adjoint that was discussed in
Chapter 4. The only difference between the two is that in the Hilbert space setting
Г-> Т* is conjugate-linear instead of linear. This is due to the conjugate-linear nature
of the isometry described in Theorem 12.5. If T* were regarded as an operator on H*
rather than on //, we would be exactly in the situation of Chapter 4.
12.10 Theorem If Те ЩН), then
JT(T*) = ЩТI and JV(T) = «(Г*I.
We recall that JT(JT) and M(T) denote the null space and range of T, respectively.
proof Each of the following four statements is clearly equivalent to the one
that follows and/or precedes it.
A) T*y = 0.
B) (x, T*y) = 0 for every x e H.
C) (Tx, y) = 0 for every xeH.
D) у e M(T)L.
Thus jV(T*) = ЩТI. Since T** = T, the second assertion follows from
the first if T is replaced by T*. II11
12.11 Definition An operator Те ЩН) is said to be
(a) normal if ГГ* = ГТ,
(b) self-adjoint (or hermitian) if Г* = T,
(c) unitary if T*T= I = TT*, where / is the identity operator on H,
(d) a projection if T2 — T.
It is clear that self-adjoint operators and unitary operators are normal. Most of
the theorems obtained in this chapter will be about normal operators.
12.12 Theorem Suppose T e ЩЯ).
(a) T is normal if and only if || Tx\\ = || T*x\\ for every xeH.
(b) IfTis normal, then JT(T) = JT(T*) = ®(T)L.
(c) If T is normal and Tx = ax for some xeH and осе G, then T*x = ocx.
(d) IfTis normal and if a and /? are distinct eigenvalues ofT, then the corresponding
eigenspaces are orthogonal to each other.
proof To see (a), combine the equalities
||7х||2 = (Tx, Tx) = (T*Tx, x)
|| r*jc||2 = (T*x, Г*х) = (TT*x, x)

BOUNDED OPERATORS ON A HILBERT SPACE 299
with the corollary to Theorem 12.7. Obviously, (b) follows from (a) and Theorem
12.10. If (b) is applied to T - a/in place of T, (c) is obtained. Finally, if Tx = ax
and Ту = py, an application of (c) gives
a(x, y) = {ax, y) = (Tx, y) = (x, T*y) = (x, jffy) = /*(*, y).
Since a^ftwe conclude that xlj. ////
12.13 Theorem /jf Ue &(H), the following three statements are equivalent.
(a) U is unitary.
(b) <#(£/) = H and (Ux, Uy) = (x, .y) for allxeH,ye H.
(c) 0t(U) = Я оийГ || Ox|| = ||x|| for every xeH.
proof If U is unitary, then &(U) = H because Ш7* - /. Also, U*U=I, so
that
Thus (я) implies (&). It is obvious that (b) implies (c). If (c) holds, then
(U*Ux, x) = (Ux, Ux) = || Uxf = |M|2 - (x, x)
for every x e Я, so that U*U = I. But (c) implies also that (/ is a linear isometry
of H onto Я, so that £/ is invertible in #(#). Since U*U = I, U'1 = «7*, and
therefore £/ is unitary. ////
The equivalence of (я) and (b) shows that the unitary operators are
precisely those linear isomorphisms of Я that also preserve the inner product. They
are therefore the Hilbert space automorphisms.
The equivalence of (b) and (c) is also a corollary of Exercise 2.
12.14 Theorem Each of the following four properties of a projection P e &(H)
implies the other three:
(a) P is self-adjoint.
(b) P is normal.
(c) 0l(P) = JT(P)L.
(d) (Px, x) = \\Px\\2 for every x e H.
Property (c) is usually expressed by saying that P is an orthogonal projection.
proof It is trivial that (a) implies (b). Statement (b) of Theorem 12.12 shows
that Ж(Р) = M(P)L if P is normal; since P is a projection, 0t(P) = Jf(I - P), so
that 0t(P) is closed. It now follows from the corollary to Theorem 12.4 that (b)
implies (c).

300 BANACH ALGEBRAS AND SPECTRAL THEORY
If (c) holds, every xeH has the form x = у + z, with у 1 z, Py = 0,
Pz = z. Hence Px = z, and (Px, x) = (z, z). This proves (*/).
Finally, assume (d) holds. Then
||Px||2 = (Px, x) = (x, P*x) = (P*x, x).
The last equality holds because ||Px||2 is real and (x, P*x) = ||Px||2. Thus
(Px, x) = (P*x, x), for every x e #, so that P = P*, by Theorem 12.7. Hence
id) implies (fl). ////
12.15 Theorem SupposeSe@(H), and Sis self-adjoint. Then ST = 0 ifand only if
JL
proof (Sx, Ту) = (x, S7». ////
This result will be most frequently used when both S and T are orthogonal
projections.
A Commutativity Theorem
Let x and у be commuting elements in some Banach algebra with an involution. It is
then obvious that x* and y* commute, simply because x*j* = (yx)*. Does it follow
that x commutes with y* ? Of course, the answer is negative whenever x is not normal
and у = x. But it can be negative even when both x and у are normal (Exercise 28).
It is therefore an interesting fact that the answer is affirmative (if x is normal) in ЩН),
relative to the involution furnished by the Hilbert space adjoint:
IfNe &(H) is normal, if Те ®{H\ and if NT = TN, then N*T = 77V*.
In fact, a more general result is true:
12.16 Theorem (Fuglede-Putnam-Rosenblum) Assume that M, N, ТеЩН),
M and N are normal, and
A) MT=TN.
Then M*T=TN*.
proof Suppose first that S e ЩН). Put V = S - S*> and define
' B) Q = exp{V)= f
„ =
Then V* = -V, and therefore
C) Q* = exp (F*) = exp (- V) = Q~x.
Hence Q is unitary. The consequence we need is that
D) ||exp (S - S*)\\ = 1 for every S e &(H).

BOUNDED OPERATORS ON A HILBERT SPACE 301
If A) holds, then Mk T = TNk for к = 1, 2, 3, ..., by induction. Hence
E) Qxp(M)T=TQxp(N\
or
F) r=exp(-Af)Fexp(TV).
Put Ui = exp (M* — M), l/ = exp (TV - TV*). Since Af and TV are normal, it
follows from F) that
G) ' exp (M*)Texp ( - TV*) = U{TU2 .
By D), ||t/y = \\U2\\ = 1, so that G) implies
(8) ||
We now define
(9) /(A) = exp(AM*)rexp(-ATV*) (A e С).
The hypotheses of the theorem hold with iM and IN in place of M and TV.
Therefore (8) implies that \\f(X)\\ < \\T\\ for every let Thus /is a bounded
entire J>(#)-valued function. By Liouville's theorem 3.32,/(A) =/@) = T, for
■every A e С Hence (9) becomes
A0) exp (ЛМ*)Т= Гехр (ATV*) (A e €).
If we equate the coefficients of A in A0), we obtain М*Г = TTV*. ////
Remark Inspection of this proof shows that it uses no properties of
which are not shared by every i?*-algebra. This observation does not lead to a
generalization of the theorem, however, because of Theorem 12.41.
Resolutions of the Identity
12.17 Definition Let Ш be a бг-algebra in a set £1, and let H be a Hilbert space.
In this setting, a resolution of the identity (on Ш) is a mapping
with the following properties:
(а)
(b) Each E(co) is a self-adjoint projection.
(c) E(co' n a/) = E(co')E(co").
(d) If a/ n со" = 0, then До' и со") = £(ю') + Е(со").
(e) For every х е Н and у е Н, the set function £x y defined by
is a complex measure on Ш.

302 BANACH ALGEBRAS AND SPECTRAL THEORY
\ )_
When Ш is the ^-algebra of all Borel sets on a compact or locally compact
Hausdorff space, it is customary to add another requirement to (<?): Each Exy should
be a regular Borel measure. (This is automatically satisfied on compact metric spaces,
for instance. See [23].)
Here are some immediate consequences of these properties.
Since each E(co) is a self-adjoint projection, we have
A) Ex,x(p)^{E(co)x,x) = \\E(a,)xf (xeH)
so that each Exx is a positive measure on Ш whose total variation is
B) \\Ех,х\\=Ех,х(П)=\\х\\2.
By (c), any two of the projections E(co) commute with each other.
If со' n со" = 0, (a) and (c) show that the ranges of E(oj') and £(со")аге orthog-
orthogonal to each other (Theorem 12.15).
By (d), E is finitely additive. The question arises whether E is countably additive,
i.e., whether the series
C) f E(con)
converges, in the norm topology of &{H), to E(oj), whenever со is the union of the
disjoint sets con еШ. Since the norm of any projection is either 0 or at least 1, the
partial sums of the series C) cannot form a Cauchy sequence, unless all but finitely
many of the E(con) are 0. Thus E is not countably additive, except in some trivial
situations.
However, let {con} be as above, and fix x e H. Since E(con)E(com) = 0 when
пФ т, the vectors E(con)x and E(com)x are orthogonal to each other (Theorem 12.15).
By (e\
D) £ (%)xj) = №)xj)
for every у е Н. It now follows from Theorem 12.6 that
E) ^ф> = ф)х.
n=l
The series E) converges in the norm topology of H. We summarize the result just
proved:
12.18 Proposition IfE is a resolution of the identity, and if x e Я, then
со -> E(co)x
is a countably additive H-valued measure on Ш.

BOUNDED OPERATORS ON A HILBERT SPACE 303
. Moreover, sets of measure zero can be handled in the usual way:
12.19 Proposition Suppose E is a resolution of the identity. If сопеШ and
E(con) = Ofor n = 1, 2, 3, ..., and if со = \J?=i(on9 then E{co) = 0.
proof Since E(con) — 0, Ex x(a>n) = 0 for every x e H. Since дх> x is countably
additive, it follows that EXtX(co) = 0. But ||£(g>);c||2 = Exx(co). Hence E(ca) = 0.
12.20 The algebra L°°(£) Let E be a resolution of the identity on Ш, as above. Let
/be a complex 9Jl-measurable function on Q. There is a countable collection {Dt} of
open discs which forms a base for the topology of €. Let V be the union of those Dt
forwhich£(/(A-)) = 0. By Proposition 12.19, E{f-\V)) = 0. Also, Kis the largest
open subset of С with this property.
The essential range of/is, by definition, the complement of V. It is the smallest
closed subset of € that contains/(p) for almost all ^ e Q, that is, for all p e Q except
those that lie in some set со е ffi with E(co) — 0.
We say that / is essentially bounded if its essential range is bounded, hence
compact. In that case, the largest value of | X |, as I runs through the essential range
of/ is called the essential supremum Ц/Ц^ of/
Let В be the algebra of all bounded complex s«W-measurable functions on Q;
with the norm
one sees easily that В is a Banach algebra and that
# = {/е2М|/||ю = 0}
is an ideal of В which is closed, by Proposition 12.19. Hence B/N is a Banach algebra,
which we denote (in the usual manner) by U°(E).
The norm of any coset [/] =/+ N of U°(E) is then equal to ||/|L, and its
spectrum o(\f\) is the essential range of/ As is usually done in measure theory, the
distinction between/and its equivalence class [/] will be ignored.
12.21 Theorem IfE is a resolution of the identity, as above, then the formula
A) W(f)x,y)= \fdEx,y (xeH,yeH)
defines an isometric isomorphism *¥ of the Banach algebra L°°(iT) onto a closed normal
subalgebra A of${H). This isomorphism also satisfies
B)

304 BANACH ALGEBRAS AND SPECTRAL THEORY
and
C) II ЧЧ/)*||2 = f I /12 dEx, x (xeH,f€
Moreover, an operator Q e &(H) commutes with every E(a>) if and only if Q
commutes with every
Formula A) will sometimes be abbreviated to
D) П/) = f fdE.
We recall that a normal subalgebra A of ЩН) is a commutative one which has
the property that T* e /1 whenever Те A; see Definition 11.24.
proof To begin with, let {wj, ..., coj be a partition of D, with cot- e 901, and
let j be a simple function, such that s = oct on cot. Define ^(s) e <%(H) by
E)
Since each E(coi) is self-adjoint,
F) П*)*
»= i
If {wj, ..., oo'J is another partition of this kind, and if t = /?, on со}, then
Since st is the simple function that equals otipj on со,- п со}, it follows that
G)
An entirely analogous argument shows that
(8) 4>(as + jSO = oW(s
If x e // and ^ e H, E) leads to
(9) {4{s)x, y)=£ oLWcodx, y)=t «i £*,,(<».■) = f ^ £/£,,,.
By F) and G),
A0) УС*)*У0) = ^(s)^(s) = 4>(ss) = ¥( | j | 2).
Hence (9) yields

BOUNDED OPERATORS ON A HILBERT SPACE 305
so that "
02) l№)*i! < IMUW!,
by formula B) of Section 12.17. On the other hand, if x e &(E(coj))9 then
A3) 4*(s)x = a,- £(a;> = a, x,
since the projections ^(cOj) have mutually orthogonal ranges. If у is chosen so
that | a,-1 = I^IL , it follows from A2) and A3) that
04) . ITOIIHHL.
Now suppose / e U°(E). There is a sequence of simple measurable func-
functions sk that converges to /in the norm of U°(E). By A4), the corresponding
operators ^{sk) form a Cauchy sequence in <%(H) which is therefore norm-
convergent to an operator that we call 4*(/); it is easy to see that *F(/) does not
depend on the particular choice of {sk}. Obviously A4) leads to
A5) Iin/)II = II/IL lfeL">(E)]-
Now A) follows from (9) (with sk in place of s), since each Ex y is a finite
'.measure; B) and C) follow from F) and A1); and if bounded measurable func-
functions / and g are approximated, in the norm of LCC(E), by simple measurable
functions s and t, we see that G) and (8) hold with/and g in place of s and /.
Thus ¥ is an isometric isomorphism of Ь°°{Е) into #(#). Since L°°(£)
is complete, its image Л = ^(L00^)) is closed in ЩН), because of A5).
Finally, if Q commutes with every E(co), then Q commutes with 4?(s)
whenever s is simple, and therefore the approximation process used above shows
that Q commutes with every member of A. ////
The Spectral Theorem
The principal assertion of the spectral theorem is that every bounded normal operator
T on a Hilbert space induces (in a canonical way) a resolution E of the identity on the
Borel subsets of its spectrum o(T) and that Jean be reconstructed from E by an integral
of the type discussed in Theorem 12.21. A large part of the theory of normal operators
depends on this fact.
It should perhaps be stated explicitly that the spectrum a(T) of an operator
Те @{H) will always refer to the full algebra &(H). In other words, X e o{T) if and
only if T — XI has no inverse in ffl(H). Sometimes we shall also be concerned with
closed subalgebras A of ^(Я) which have the additional property that / e A and Г* е А
whenever Те A. (Such algebras are sometimes called *-algebras.) Since &(H) is a
£*-algebra, Theorem 11.29 tells us, in this situation, that <r(T) = aA(T) for every Те А.

306 BANACH ALGEBRAS AND SPECTRAL THEORY
Thus T has the same spectrum relative to all closed *-algebras in &(H) that
contain T.
Theorem 12.23 will be obtained as a special case of the following result, which
deals with normal algebras of operators rather than with individual ones.
12.22 Theorem If A is a closed normal subalgebra of @(H) which contains the
identity operator /, and if A is the maximal ideal space of A, then the following assertions
are true:
(a) There exists a unique resolution of the identity E on the Borel subsets of A, that
satisfies
A) T= [tdE
J A
for every Те A, where f is the G elf and transform ofT.
(b) E{cS) Ф 0 for every nonempty open set со a A.
(c) An operator S e M(H) commutes with every Те A if and only if S commutes
with every projection E(co).
As in Theorem 12.21, formula A) means that
B) (Tx, y) = j TdEx,y (хеН,уеН,ТеАУ
proof Since @{H) is a i?*-algebra (Section 12.9), our given algebra A is a
commutative #*-algebra. The Gelfand-Naimark theorem 11.18 asserts therefore
that T-+ Tis an isometric *-isomorphism of A onto С(Л).
This leads to an easy proof of the uniqueness of E. Suppose E satisfies
B). Since T ranges over all of C(A), the assumed regularity of the complex
Borel measures ЕХг y shows that each EXj y is uniquely determined by B); this
follows from the uniqueness assertion that is part of the Riesz representation
theorem ([23], Th. 6.19). Since, by definition
C) (E(co)x,y) = Ex,y(co\
each projection E(a>) is also uniquely determined by B).
This uniqueness proof motivates the following proof of the existence of
E. If x e H and у е Я, Theorem 11.18 shows that
D) t^{Tx9y)
is a bounded linear functional on C(A), of norm <||x|||M|, since H'TiL = Ц7Ц-

BOUNDED OPERATORS ON A HILBERT SPACE 307
The Riesz representation theorem supplies us therefore with unique regular
complex Borel measures цхty on A, such that
E) (Tx, y) = f tdiiXty (xeH,yeH, Те А).
JA
When t is real, T is self-adjoint, so that (Tx, y) and (Ту, х) are complex
conjugates of each other. Hence
F) цХ}У = рУгХ (хеН,уеН).
For fixed Те A, the left side of E) is linear in x and conjugate-linear in y.
The uniqueness of the measures ^x> y implies therefore that fiX} y(co) is, for every
Borel set со cz A, a sesquilinear functional. Since \\1лХгУ\\ < \\x\\\\y\\, it follows that
is a bounded sesquilinear functional on #, for every bounded Borel function/on
Л. By Theorem 12.8 there corresponds to every such /an operator Ф(/) e &(H)
(8) (Ф(/)х, у) = f fdnx<, (xeH,ye H).
JA
Comparison with E) shows that
(9) Ф(Т) = Т (Те А).
Thus Ф is an extension of the mapping T-> Г that takes C(A) onto A.
If / is real, then F) shows that (Ф(/)х, у) is the complex conjugate of
(Ф(/)у, х). This implies that Ф(/) is self-adjoint.
Our next objective is the equality
A0) Ф(/<7)=Ф(УЖ<7),
for bounded Borel functions/and g.
If S e A and Те A, then (STp = Sf, and E) implies
A1) f §fdvx,y = (STx,y)= f 5фгх,;.
Since A = C(AJ, it follows that
A2) , f*XiJ = *rXi,
for every choice of x, _y, and T. The integrals A1) remain therefore equal if S is
replaced by/. Hence
A3) f
JA A
= (Ф(/Oх, у) = (Tx, z) = f f rfjuXi ж,

308 BANACH ALGEBRAS AND SPECTRAL THEORY
where z = Ф(/)*у. The same reasoning as above shows that the first and Jast
integrals in A3) remain equal when Tis replaced by any bounded Borel function
g. Consequently,
A4) (ФШК y) = f fg dnx,, = f gdpXtI
JA JA
which proves A0).
We are ready to define E. If со is a Borel subset of A, let/be the character-
characteristic function of со, and put E(co) = Ф (/).
By A0), E{co n со') = E(co)E(cof). With со' = со, this shows that each E{co)
is a projection. Since Ф(/) is self-adjoint when /is real, each E(co) is self-adjoint.
It is clear that E@) = Ф@) = 0. That £(A) = / follows from (9). The finite
additivity of E is a consequence of (8), as is the relation
A5) (E(a>)x,y) = iiXt,(a>).
Hence E is a resolution of the identity.
The proof of part (a) is now complete, since B) follows from E) and A5).
Suppose next that со is open and E(co) = 0. If T e A and T has its support
in со, A) implies that T= 0; hence !f = 0. Since A = C(A), Urysohn's lemma
implies now that со = 0. This proves F).
To prove (c), choose S e ^(Я), xe H, ye H, and put z = S*y. For any
7" e Л and any Borel set ш с A we then have
A6) (STx,y) = {Tx,z)= f TdEXiZ,
A7) (ra*,jO= f
A8) 0S£(co)x, J,) = (E(co)x, z) = ^х>»,
A9) (£(co)^x,j) = £Sjc>/a)).
If 5Г= TS for every fe Л, the measures in A6) and A7) are equal, so
that SE(co) = E(co)S. The same argument establishes the converse. This
completes the proof. . ////
We now specialize this theorem to a single operator.
12.23 Theorem If T e 3$(H) and T is normal, then there exists a unique resolution
of the identity E on the Borel subsets of o(T) which satisfies
A) . T=( XdE{X).

BOUNDED OPERATORS ON A HILBERT SPACE 309
Furthermore, every projection E(co) commutes with every S e $(H) which com-
commutes with T.
We shall refer to this E as the spectral decomposition ofT.
Sometimes, it is convenient to think of £as being defined for all Borel sets in C;
to achieve this, put E(co) = 0 if со n a(T) = 0.
proof Let A be the smallest closed subalgebra of 08(H) that contains /, T, and
T*. Since Г is normal, Theorem 12.22 applies to A. By Theorem 11.19, the
maximal ideal space of A can be identified with o(T) in such a way that t(X) = X
for every X e a(T). The existence of E follows now from Theorem 12.22.
On the other hand, if Sexists so that A) holds, Theorem 12.21 shows that
B) p(T, Г*) = f p(A, X) dE{X\
Jc(T)
where p is any polynomial in two variables (with complex coefficients). By the
Stone-Weierstrass theorem, these polynomials are dense in C(a(T)). The pro-
projections E(co) are therefore uniquely determined by the integrals B), hence by
T, just as in the uniqueness proof in Theorem 12.22.
If ST= TS, then also ST* - T*S, by Theorem 12.16; hence S commutes
with every member of A. By (c) of Theorem 12.22, SE(oo) = E(a))S for every
Borei set со с a(T). /111
12.24 The symbolic calculus for normal operators If E is the spectral decom-
decomposition of a normal operator ТеЩН), and if/ is a bounded Borel function on
g(T), it is customary to denote the operator
@ 44/)= f fdE
by ДП
Using this notation, part of the content of Theorems 12.21 to 12.23 can be sum-
summarized as follows:
The mapping f-+f{T) is a homomorphism of the algebra of all bounded Borel
functions on o(T) into &(H), which carries the function 1 to I, which carries the iden-
identity function on o(X) to T, and which satisfies
B) RT)=f(T)*
and
C) ||/(Г)|]<5ир
/// e C(a(T)), then equality holds in C).

310 BANACH ALGEBRAS AND SPECTRAL THEORY
i \
Iffn ~+f uniformly, then \\fn(T) -f(T)\\ -* 0, as n -> oo.
IfS e M(H) and ST = TS, then Sf(T) = f(T)S for every bounded Borel function f
Since the identity function can be uniformly approximated, on <x(T), by simple
Borel functions, it follows that T is a limit, in the norm topology of &(H), of finite
linear combinations of projections E(co).
The following proof contains our first application of this symbolic calculus.
12.25 Theorem // T e @(H) is normal, then
\\T\\=sup{\(Tx,x)\:xeH,\\x\\<\}.
proof Choose с > 0. It is clearly enough to show that
A) \(Txo,xo)\>\\T\\-s
for some x0 e H with ||xo|| = 1.
Since || ГЦ = || f || ю = p(T) (Theorem 11.18), there exists Ao e o{T) such that
|Я0| = ||Г||. Let со be the set of all A e <т(Г) for which |A-A0| < e- If £ is the
spectral decomposition of Г, then (b) of Theorem 12.22 implies that E(oj) Ф 0.
Therefore there exists x0 e H with ||xo|| = 1 and E(co)x0 = x0 .
Define f{X) = X - Xo for X e со; put /(A) = 0 for all other A e <х(Г). Then
/(Г) = (Г-А0/)£(а>),
so that
Hence
\(Txo,xo) - Xo\ = |(/G>0, xo)\ < \\f(T)\\ < s,
since |/(A)| <e for all Xea(T). This implies A), because |A0| = ||Г||. ////
12.26 Theorem A normal Те ЩН) is
(a) self-adjoint if and only if g(T) lies in the real axis,
(b) unitary if and only ifa(T) lies on the unit circle.
proof Choose A as in the proof of Theorem 12.23. Then f(A) = A and
(T*)*(X) = I on <j(T). Hence T = Г* if and only if A = I on a(T), and ГГ* = / if
and only if Xl = 1 on а(Г). • ////■
12.27 Invariant subspaces A closed subspace M of H is an invariant subspace of
a set Л c= Щ7/) if every Ге^ maps M into Л/. For example, every eigenspace of T
is an invariant subspace of T. When dim Ж oo, the spectral theorem implies that

BOUNDED OPERATORS ON A HILBERT SPACE 311
the eigenspaces of every normal operator T span H. [Sketch of proof: The character-
characteristic function of each point in o(T) corresponds to a projection in H. The sum of these
projections is E(g{T)) = /.] If dim H = oo, it can happen that T has no eigenvalues
(Exercise 20). But normal operators still have invariant subspaces that are nontrivial
(that is, /{0}and ФН).
In fact, let A be a normal algebra, as in Theorem 12.22, and let E be its resolution
of the identity, on the Borel subsets of A. If A consists of a single point, then A
consists of the scalar multiples of /, and every subspace of H is invariant under A.
Suppose that A = со u со', where со and со' are nonempty disjoint Borel sets. Let
lM and M' be the ranges of E{co) and E{co'). Then TE{co) = E(co)T for every Те A.
IfxeM, it follows that
Tx = TE(co)x = E(co)Tx,
so that Tx e M. The same, holds for M'.
Hence M and M' are invariant subspaces of A.
Moreover, M' = M1, and H = M © M'.
Decompositions of A into finitely many (or even countably many) disjoint Borel
sets induce, in the same manner, decompositions of H into pairwise orthogonal
invariant subspaces of A.
It is an open problem whether every (nonnormal) T e &{H) has a nontrivial
invariant subspace if H is an infinite-dimensional separable Hilbert space.
Eigenvalues of Normal Operators
If Те &{Н) is normal, its eigenvalues bear a simple relation to its spectral decom-
decomposition (Theorem 12.29). This will be derived from the following application of the
symbolic calculus:
12.28 Theorem Suppose Те &(Н) is normal and E is its spectral decomposition.
Iffe C(a(T)) and ifco0 =/-1@), then
A)
proof Put g(k) = 1 on co0 , g(X) = 0 at all other points of a(T). Then fg = 0,
so that f(T)g(T) = 0. Since g(T) = E(co0), it follows that
B) ВДюо» с= ЛЧ/(Г».
If со is the complement of co0, relative to cr(T), then со is the union of dis-
disjoint Borel sets con (n — 1, 2, 3, ...), each of which has positive distance from the
compact set co0. Define
l1^ on con,
C) mbl
K } JA } @ elsewhere on a(T).

312 BANACH ALGEBRAS AND SPECTRAL THEORY
]
Each/,, is a bounded Borel function on cr(T), and
D) fn(T)f(T) = E(con) (/i = 1, 2, 3, ...).
If f(T)x = 0, it follows that E(con)x = 0. The countable additivity of the
mapping со -> E(co)x (Proposition 12.18) shows therefore that E(co)x = 0. But
£(с5) + E(co0) = /. Hence £"(соо)л; = х. We have now proved that
and A) follows from B) and E). ////
12.29 Theorem Suppose E is the spectral decomposition of a normal
Xo e a(T), and Eo = E({X0}). Then
(а)
(b) Xo is an eigenvalue of T if and only if Eo Ф 0, and
(c) every isolated point of a(T) is an eigenvalue of T.
(d) Moreover, if a(T) = {Xu X2 Дз •> • • •} ^ a countable set, then every x e H has a
unique expansion of the form
X / J Xi,
1=1
where Txt = Х{х{. Л/^б>, xf ± ;cy whenever i Ф j.
Statements (b) and (c) explain the term point spectrum of TTor the set of all eigen-
eigenvalues of T
proof Part (a) is an immediate corollary of Theorem 12.28, with/(A) = Я — Xo .
It is clear that (b) follows from (a). If Xo is an isolated point of v(T), then {Xo} is
a nonempty open subset of сг(Г); hence Eo ф 0, by (b) of Theorem 12.22. There-
Therefore (c) follows from (b).
To prove (d), put Ex = ^({AJ), i = 1, 2, 3, At limit points ^ of a(T),
Et may or may not be 0. In any case, the projections Et have pairwise orthogonal
ranges. The countable additivity of со -» E(co)x (Proposition 12.18) shows that
^Е(х = Е(а(Т))х = х (хеН).
i=l
The series converges, in the norm of H. This gives the desired representation of,
x, if Xi = Ei x. The uniqueness follows from the orthogonality of the vectors
xt, and Txi = XtXi follows from (a). ////
12.30 Theorem A normal operator Te&(H) is compact if and only if it satis-
satisfies the following two conditions:

BOUNDED OPERATORS ON A HILBERT SPACE 313
(a} o(T) has no limit point except possibly 0.
. (b) IfX^O, then dim J\T(T -XI)<oo.
proof For the necessity, see (d) of Theorem 4.18, and Theorem 4.25.
To prove the sufficiency, assume (a) and (b) hold, let {Xt} be an enumeration
of the nonzero points of a(T) such that \Xt\ > \X2 \ > •• *, define fn(X) = X if
X = Xt and / < w, and put fn(X) = 0 at the other points of о-(Г). If £,. = Д^/}),
as in Theorem 12.29, then
Since dim &(Et) = dim yV(T — XtI) < oo, each/„(Г) is a compact operator.
Since |X - fn(X)\ <\Xn\ for all Я е a(T), we have
K|->0 as /i-*oo/
It now follows from (c) of Theorem 4.18 that T is compact. ////
12.31 Theorem Suppose Те &(Н) is normal and compact. Then
(a) Thas an eigenvalue X with \X\ = || T\\, and
(b) f(T) is compact iff e C(a(T)) andf@) = 0.
proof Since T is normal, Theorem 11.18 shows that there exists X e a(T) with
|A| = ||ГЦ. If ||Г||>0, this X is an isolated point of a(T) (Theorem 12.30),
hence an eigenvalue of Г (Theorem 12.29). If \\T\\ = 0, (a) is obvious.
Since a(T) is an at most countable compact set in <£, its complement is
connected. Mergelyan's theorem (see [23]) shows therefore that there are
polynomials pn, with pn@) = 0, which converge to /, uniformly on a{T). The
operators pn(T) converge therefore, in the norm of&(H), to f(T). Since pn@) = 0,
(/) of Theorem 4.18 shows that each pn(T) is compact. Hence f(T) is compact,
by (c) of Theorem 4.18. . ////
This proof of (b) could also have been based on the classical approximation
theorem of Runge rather than on the more difficult one of Mergelyan.
Positive Operators and Square Roots
12.32 Theorem Suppose Te ^{H). Then
(a) (Tx, x) > 0 for every xe H if and only if
(b) T=T* and a(T) с [0, oo).
If Te&{H) satisfies (a), we call T a positive operator and write T > 0.
The theorem asserts that this terminology agrees with Definition 11.27.

314 BANACH ALGEBRAS AND SPECTRAL THEORY
1
proof In general, (Tx, x) and (x, Tx) are complex conjugates of each other.
But if (a) holds, then (Tx, x) is real, so that
(x, Г*х) = (Tx, x) = (x, Tx)
for every x e H. By Theorem 12.7, T= T*, and thus a(T) lies in the real axis
(Theorem 12.26). If Я > 0, (a) implies that
A||jc||2 = (Xx, x) < ((T + XI)x, x) <\\(T+ XI)x\\ \\x\\,
so that
\\(T+XI)x\\>X\\x\\.
Hence T+ Л is invertible in @(H), and —A is not in o(T). It follows that (a)
implies (b).
Assume now that (b) holds, and let E be the spectral decomposition of T,
so that
f XdEx,x(X) (xeH).
)
Since each Ex x is a positive measure, and since A > 0 on сг(Г), we have
(Tx,x)>0. Thus (ft) implies (a). ////
12.33 Theorem Every positive T e &(H) has a unique positive square root S e &(H).
IfTis invertible, so is S.
proof Let A be any closed normal subalgebra of 08(H) that contains / and T,
and let A be the maximal ideal space-of A. By Theorem 11.18, Л = С(А). Since
T satisfies condition (b) of Theorem 12.32, and since a(T) = f(A), we see that
f > 0. Since every nonnegative continuous function has a unique nonnegative
continuous square root, it follows that there is a unique SeA that satisfies
S2 = T and S > 0; by Theorem 12.32, S > 0 is equivalent to S > 0.
In particular, let Ao be the smallest of these algebras A. Then there
exists So e -<4o such that S% = T and So > 0. If S e ЩН) is any positive square
root of T, let Л be the smallest closed subalgebra of @(H) that contains / and S.
Then Те A, since T= S2. Hence v40 с A, so that »S06i. The conclusion of the
preceding paragraph shows now that S = So.
Finally, if T is invertible, then S'1 = Г"^, since 5 and Г commute.
12.34 Theorem //Те @(Н), then the positive square root ofT*T is the only posi-
positive operator P e &(H) that satisfies \\Px\\ = \\Tx\\ for every xe H.

BOUNDED OPERATORS ON A HILBERT SPACE 315
proof Note first that
A) (T*Tx9 x) = (Tx, Tx) = ||Гх||2 > 0 (x e H),
so that T*T > 0. (In the more abstract setting of Theorem 11.28 this was much
harder to prove!)
Next, if P e ЩН) and P = P*, then
B) (P2x, x) = (Px, Px) = HjPjcH 2 (jc e H).
By Theorem-12.7, it follows that ||P*|| = ||7jc|| for every xeH if and only if
P2 = 7*7
This completes the proof. ////
The fact that every complex number A can be factored in the form Л = а | Л |,
where |a| = 1, suggests the problem of trying to factor Те ЩН) in the form T= UP,
with C/ unitary and P > 0. When this is possible, we call UP a /?o/ar decomposition
ofT.
Note that 6/, being unitary, is an isometry. Theorem 12.34 shows therefore that
P is uniquely determined by 7.
12.35 Theorem
(a) If Те @{H) is invertible, then T has a unique polar decomposition T = UP.
(b) If Те ЩН) is normal, then T has a polar decomposition T = UP in which U and P
commute with each other and with T.
proof (a) If Tis invertible, so are T* and T*T, and Theorem 12.33 shows that
the positive square root P of T*T is also invertible. Put U=TP~1. Then U is
invertible, and
U*U = p-lT*TP~x - p-lP2P~l = I,
so that U is unitary. Since P is invertible, it is obvious that TP'1 is the only
possible choice for U.
(b) Put p(X) = \k\, u(k) = A/|A| if А Ф 0, и@) = 1. Then p and и are
bounded Borel functions on a(T). Put P = p(T), U = u{T). Sincep > 0, Theorem
12.32 shows that P > 0. Since ип = 1, £Л7* = C/*f/ - /. Since A = w(A)/?(A), the
relation T = UP follows from the symbolic calculus. ////
Remark It is not true that every Те @t(H) has a polar decomposition. (See
Exercise 19.) However, if P is the positive square root of T*T, then \\Px\\ =
||71*11 for every xeH, so that the formula
VPx = Tx

316 BANACH ALGEBRAS AND SPECTRAL THEORY
defines a linear isometry V of 0t{P) onto ЩТ), which has a continuous extension
to a linear isometry of the closure of ЩР) onto the closure of M(T).
If there is a linear isometry of 0t{P)L onto ^(TI, ihen К can be extended
to a unitary operator on H, and then T has a polar decomposition. This always
happens when dim H < oo, since &(P) and &(T) have then the same codimen-
sion.
If V is extended to a member of ЩН) by defining Kj = 0 for all у е ЩРI,
then V is called a partial isometry.
Every Те &(H) thus has a factorization T = VP in which P is positive and
V is a partial isometry.
In combination with Theorem 12.16, the polar decomposition leads to an
interesting result concerning similarity of normal operators.
12.36 Theorem Suppose M, N,Te @{H), M and N are normal, T is invertible,
and
A) M = TNT~1.
IfT— UP is the polar decomposition of T, then
B) M=UNU~K
Two operators M and TV that satisfy A) are usually called similar. If t/ is unitary
and B) holds, M and N are said to be unitarily equivalent. The theorem thus asserts
that similar normal operators are actually unitarily equivalent.
proof By A), MT=TN. Hence M*T=TN*, by Theorem 12.16. Conse-
Consequently,
Г*М = (М*Г)* = GW*)* = NT*,
so that
NP2 = NT*T = T*MT = T*TN = P2N,
since P2 = Г*Г. Hence N commutes with /(P2), for every fe C(a(P2)). (See
Section 12.24.) Since P > 0, <j(P2) с [0, oo). If /(A) = A1/2 > 0 on a(P2), it
follows that NP = PN. Hence A) yields
M - (UP)N(UPy1 - UPNP-iU'1 = UNU~l. ////
The Group of Invertible Operators
Some features of the group of all invertible elements in a Banach algebra A were
described at the end of Chapter 10. The following two theorems contain further
information about this group, in the special case A =

BOUNDED OPERATORS ON A HILBERT SPACE 317
12.37 Theorem The group G of all invertible operators Те.ЩН) is connected,
and every Те G is the product of two exponentials.
Here an exponential is, of course, any operator of the form exp (S) with
S E
proof Let Г= UP be the polar decomposition of some Те G. Recall that U
is unitary and that P is positive and invertible. Since a(P) с @, oo), log is a
continuous real function on cr(P). It follows from the symbolic calculus that
there is a self-adjoint Se$(H) such that P = exp E). Since U is unitary,
a(U) lies on the unit circle, so that there is a real bounded Borel function/on
a(U) that satisfies
exp {//(*)} = A [Xea(U)l
(Note that there may not exist any continuous f with this property!) Put Q =
/(£/). Then Q e ЩН) is self-adjoint, and U = exp (i0. Thus
T=UP = exp (iQ) exp E).
From this it follows easily that G is connected, for if Tr is defined, for 0 < r < 1,
by
Tr = exp (i>0 exp (rS)
then r -> Гг is a continuous mapping of the unit interval [0,1] into G, To = I, and
Ti = Г. This completes the proof. ////
It is now natural to ask whether every TeG is an exponential, rather than
merely the product of two exponentials. In other words, is every product of two
exponentials an exponential ? The answer is affirmative if dim H < oo; in fact, it is
affirmative in every finite-dimensional Banach algebra, as a consequence of Theorem
10.30. But in general the answer is negative, as we shall now see.
12.38 Theorem Let D be a bounded open set in <£ such that the set
A) Q = {ae0:a2ei)}
is connected and such that 0 is not,in the closure of D. Let H be the space of all holo-
morphic functions f in D that satisfy
B) \\f\2dm2<«>
(where m2 is Lebesgue measure in the plane), with inner product
C) (f,ff)

318 BANACH ALGEBRAS AND SPECTRAL THEORY
)
Then H is a Hilbert space. Define the multiplication operator M e &(H) by
D) (Mf)(z) = zf{z) (fe H, z e D).
Then M is invertible, but M has no square root in
Since every exponential has roots of all orders, it follows that M is not an
exponential.
proof It is clear that C) defines an inner product that makes H a unitary
space. We show now that H is complete. Let К be a compact subset of D,
whose distance from the complement of Dis д. If z e K, if A is the open circular
disc with radius 5 and center z, and if /(£) = £ ««(C - z)n for ( е Л, a simple
computation shows that
E)
f (л + 1)-1|я„|2<52" + 2 = - f \f\2dm2.
n = 0 71 JA
Since/(z) = a0, it follows that
F) imi^n-'^S-'WfW (zeKJeHX
where ||/|| = (/,/I/2. Every Cauchy sequence in H converges therefore
uniformly on compact subsets of D. From this it follows easily that H is com-
complete. Hence Я is a Hilbert space.
Since D is bounded, M e &(H). Since 1/z is bounded in D, M~l
Assume now, to reach a contradiction, that M — Q1 for some Q e
Fix a e Q. Put X = a2. Then Я е D. Define
G) MX = M-M, S=Q-aI, T=Q + oJ,
so that
(8) ST=MX = TS.
Since we are dealing with holomorphic functions, the formula
(9) (M,g)(z) = (z - X)g(z) (z e D, g e H)
shows that Mx is one-to-one and that its range &(MX) consists of exactly those
fe H that satisfy /(A) = 0. Hence F) shows-that ffl(Mx) is a closed subspace of
H, of codimension 1.
Since Mx is one-to-one, the first equation (8) shows that T is one-to-one;
the second shows that S is one-to-one. Since @(MX) Ф H, Mx is not invertible
in ЩН). Hence at least one of S and T is not invertible. Suppose S is not
invertible. Since Mx = ST, M{MX) с 0t(S), so that 0t(S) is either ^(MA) or Н.
In the latter case, the open mapping theorem would imply that S is invertible.

BOUNDED OPERATORS ON A HILBERT SPACE 319
Hence S is a one-to-one mapping of Я onto ^(Мя). But the equation Mx — ST
shows that S maps &(T) onto ЩМЛ). Hence 0t(T) = Я, and another applica-
application of the open mapping theorem shows that T~l e 0&(H).
We have now proved that one and only one of the operators S and T is
invertible in ^(Я). Therefore exactly one of the numbers a and —a lies in
o(Q\ if oleQ. It follows that п is the union of two disjoint congruent sets,
a(Q) n п and -a(Q) n Q, both of which are closed (relative to Q) since a(Q) is
compact. The assumption that M = Q2 leads thus to the conclusion that Q is
not connected, which contradicts the hypothesis.
This completes the proof. ////
- The simplest example of a region D that satisfies the hypothesis of Theorem
12.38 is a circular annulus with center at 0.
A Characterization of 2?*-algebras
The fact that every <M(H) is a i?*-algebra has been exploited throughout this chapter.
We shall now establish a converse (Theorem 12.41) which asserts that every B*-
algebra (commutative or not) is isometrically *-isomorphic to some closed subalgebra
of some &{H). The proof depends on the existence of a sufficiently large supply of
positive functionals.
12.39 Theorem If A is a B*-algebra and ifzeA, then there exists a positive func-
functional F on A such that
A) F(e)=\ and F(zz*)=\\z\\2.
proof Let Ar (the " real part" of A) be the real vector space that consists of the
hermitian elements of A, and let P be the set of all x e Ar with a(x) a [0, oo).
In the terminology of Definition 11.27, x e P if and only if x > 0. By Theorem
11.28, P is a cone: if xeP, yeP, and с is a positive scalar, then cxeP and
x + yeP. Also, P contains all elements of the form xx*, for x e A. To prove the
theorem, it is therefore enough to find a real-linear functional / on Ar that
satisfies A) and
B) f(x) > 0 for every xeP,
for we can then define F(x) =f(u) + if(v) if x = и + iv and и е Ar, veAr.
Since this definition gives F(ix) = iF(x), F is complex-linear, and B) shows that
F is positive.
Let Mo be the subspace of Ar generated by e and zz*, and define/0 on Mo
by
C) fo(ae + pzz*) = a + p\\zz*|| (a, p e R).

320 BANACH ALGEBRAS AND SPECTRAL THEORY
Note that /0 is well defined on Mo, even if e and zz* are linearly dependent.
By (a) of Theorem 11.28 ||zz*|| e a(zz*). Hence a + P\\zz*\\ lies in фе + fizz*).
In other words, fo(x) e a(x) if x e Mo, so thatfo(x) > 0 for every xeP n Mo.
Also,/satisfies A).
Assume that /0 has been extended to a real-linear functional /j on a
subspace M1 of ^4r, such that fi(x) > 0 for all x eP n Mi9 and assume that
уеА^уфМ^ Put
D) £'=7^ n(y-P), E" = M1 n(
If x'eE' and x" eE'\ then j-х'бР and x" — yeP; hence so is their
sum, x" — x\ and therefore/i(V) < f\{x"). It follows that there is a real number
с that satisfies
E) /i(x') < с </(*") (^ e ^', *"e E").
Define
F) /2(x + осу) =f,(x) + ac (xeMuoce R).
If x + j e P, then -x e £', /K-x) < c, /^x) > -c; hence /2(x + y) > 0, If
°, then x e E\ fx(x) > c, and f2(x - у) > с - с = 0. It follows from
these two cases that/2 > 0 on P n M2 .
The proof can now be completed by transfinite induction, just as in the
Hahn-Banach theorem. ////
12.40 Theorem If A is a B*-algebra and if и е А, и ф 0, there exists a Hilbert
space Hu and there exists a homomorphism Tu of A into M(HU) that satisfies Tu(e) — /,
A) rtt(x*) = Ги(х)* (х е A),
B) \\Tu(x)\\<\\x\\ (xeA),
proof We regard и as fixed and omit the subscripts u. Fix a positive functional
F on A that satisfies
C) F(e) = l and F(u*u) = ||w||2.
Such an F exists, by Theorem 12.39. Define
D) Y={yeA: F(xy) = 0 for every xeA}. '
Since F is continuous (Theorem 11.31), Y is a closed subspace of A. Denote
cosets of Y, that is, elements of A/ Y, by xf:
E) - x' = x + Y (xeA).

BOUNDED OPERATORS ON A HILBERT SPACE 321
(
We claim that
F) (a\b') = F(b*a)
defines an inner product on A/ Y.
To see that (a\ b') is well defined by F), i.e., that it is independent of the
choice of representatives a and b, it is enough to show that F(b*a) = 0 if at
least one of a or b lies in Y. If a e Y, F(b*a) = 0 follows from D). ifbeY, then
G) F(b*a) = F(a*b) = 0,
by (a) of Theorem 11.31 and another application of D). Thus (a, br) is well
defined, it is linear in a\ and conjugate-linear in b\ and
(8) (a', a') = F(a*a) > 0,
since F is a positive functional. If (a', d) = 0, then F(a*a) = 0; hence F(xa) = 0
for every x e A, by F) of Theorem 11.31, so that aeY and a' = 0.
A/Y is thus an inner product space, with norm ||д'|| = F(a*aI/2. Its
completion H is the Hilbert space that we are looking for. We define linear
operators T(x) on A/Y by
*(9) 7\x)a' = (xay.
Again, one checks easily that this definition is independent of the choice of
a e a\ for if у е 7, D) implies that xyeY. (Y is a left ideal in A.) It is obvious
that x -> T(x) is linear and that
A0) T(xl)T{x2) = T(Xlx2) (xx eA,x2e A);
in particular, (9) shows that T(e) is the identity operator on A/Y. We now claim
that
(И) ||Г(*)|| < 11*11 (xeA).
Once this is shown, the uniform continuity of the operators T(x) enables us to
extend them to bounded linear operators on H. Note that
A2) ||T(x)a'\\2 = ((**)', (xa)f) = F(a*x*xa).
For fixed ae A, define G(x) = F(a*xa). Then G is a positive functional on A, so
that
A3) G(x*x)<G(e)\\x\\\
by (d) of Theorem 11.31. Thus
A4) ЦГ(х)а'||2 = G(x*x) < F(a*a)\\x\\2 = |И|2||х||2,
which proves A1).

322 BANACH ALGEBRAS AND SPECTRAL THEORY
Next, the computation
. (T(x*)a\ b') = ((x*a)', b') = F(b*x*a) = F((xb)*a)
= (a\ (xb)') = (a', Г(*L') = (T(x)*a'9 b')
shows that T(x*)a' = T(x)*rf, for all a' e Л/У. Since Л/У is dense in Я, this
proves A).
Finally, C) and A2) show that
A5) \\u\\2 = F(u*u) = |№)б>'||2 < ||Г(и)||2
since ||б>'||2 =F(e*e) = F(e) = 1. In conjunction with A1), A5) gives \\T(u)\\ =
||m||, and the proof is complete. ////
12.41 Theorem If A is a B*-algebra, there exists an isometric ^-isomorphism of
A onto a closed subalgebra of &(H), where H is a suitably chosen Hilbert space.
proof Let H be the " direct sum " of the Hilbert spaces Hu constructed in
Theorem 12.40. Here is a precise description of H: Let njv) be the in-
incoordinate of an element v of the cartesian product of the spaces Hu. Then,
by definition, v e H if and only if
О) £1К(»I12<оо,
и
where ||яи(г;)|| denotes the 7/u-norm of nu(v). The convergence of A) implies that
at most countably many nu(v) are different from 0. The inner product in H is
given by
B) (»',»•) = I (я„(р'),я.(О) (V,v"eH),
и
so that ||r||2 = (v, v) is the left side of A). We leave it as an exercise to verify
that all Hilbert space axioms are now satisfied by H.
If Su e &(HU), if IISJI < M for all u, and if Sv is defined to be the vector
whose coordinate in Hu is
C) nu(Sv) = Sunu(v),
one verifies easily that Sv e H if v e H, that S e ЩН), and that
D) ||5||=sup|5J.
We now associate with each x e A an operator T(x) e &(H), by requiring
that
E) ' nJT(x)v) = Tu(x)(nu(v)l

BOUNDED OPERATORS ON A HILBERT SPACE 323
where Tu is as in Theorem 12.40. Since
F) пади < и = цг,(*)ц,
by Theorem 12.40, it follows from D) that
That the mapping x -> T(x) of A into @(H) has the other required proper-
properties follows from a coordinatewise application of Theorem 12.40. ////
Exercises
Throughout these exercises, the letter Я denotes a Hilbert space.
1 The completion of an inner product space is a Hilbert space. Make this statement more
precise, and prove it. (See the proof of Theorem 12.40 for an application.)
2 Suppose N is a positive integer, a e (?, a" = 1, and а2 Ф 1. Prove that every Hilbert
space inner product satisfies the identities
(x,y) = T 2 \\x + *ny\\2oc»
N n=i
and
(*,JO = ^- Г \\x + euy\\2eude.
2lT J-n
Generalize this: Which functions/and measures fj, on a set П give rise to the identity
(x,y)= f \\х+Др)у\\2с11л(рI
3 (a) Assume х„ and yn are in the closed unit ball of H, and (xn, yn) -> 1 as n -> со. Prove
that then ||х„ —jn||->0.
F) Assume xne H, xn->x weakly, and ||jc|| -> ||jc||. Prove that then \\xn — x\\ ->0.
4 Let H* be the dual space of H; define 0: #* -> H by
^*(jc) = (x, 0^*) (jc eH,y*e Я*).
(See Theorem 12.5.) Prove that Я* is a Hilbert space, relative to the inner product
If ф: Я** ->Я* satisfies z**G>*) = [у*, <£z**] for all у* е Я* and z** g Я**, prove that
фф is an isomorphism of Я** onto Я whose existence implies that Я is reflexive.
5 Suppose {и„} is a sequence of unit vectors in Я (that is \\un\\ = 1), and assume that
Г2=2|A1„ю)|а<оо.

324 BANACH ALGEBRAS AND SPECTRAL THEORY
If {oti} is any sequence of scalars, prove that
and deduce that the following three properties of {aj are equivalent to each other:
(b) 2 a< ui converges, in the norm of H.
OO
(c) У oci(ui, v) converges, for every у е Н.
This generalizes Theorem 12.6.
6 Suppose E is a resolution of the identity, as in Section 12.17, and prove that
I Ex, УИ|2< Ex, x (co)Ey, у (со)
for all x g H, у e Н, and со e M.
7 Suppose Ue &(H) is unitary, and e > 0. Prove that scalars a0,..., an can be chosen
so that
WU-1 - ccol - a,U - • • • - anUn\\ < e,
if a(U) is a proper subset of the unit circle, but that this norm is never less than 1 if a(U)
covers the whole circle.
Note: That o(U) lies on the unit circle is contained in Theorem 12.26 but can be
proved in a much more elementary way. Find such a proof.
8 Prove Theorem 12.35 with PU in place of UP.
9 Suppose T= UP is the polar decomposition of an invertible Те &(Н). Prove that Tis
normal if and only if UP = PU.
10 Prove that every normal invertible T e &(H) is the exponential of some normal S e <M(H).
11 Suppose Ne 0l{H) is normal, and Те @(Н) is invertible. Prove that TNT'1 is normal if
and only if N commutes with T* T
12 (a) Suppose S e ЩН), T e @(H\ S and T are normal, and ST = TS. Prove that S + T
and ST are normal.
(b) If, in addition, S>0 and T>0 (see Theorem 12.32), prove that 5+ T>0 and
ST>0.
(c) Show, however, that there exist S > 0 and T > 0 such that SJis not even normal (of
course, then ST ф TS). In fact, such examples exist if dim H = 2.
13 If Te @(H) is normal, show that T* = UT, for some unitary U. When is U unique?
14 Assume Те &(Н) and T*Tis a compact operator. Show that T is then compact.
75 Find a noncompact Те &(H) such that T2 = 0. Can such an operator be normal?
16 Suppose T e &(H) is normal, and a(T) is a finite set. Deduce as much information about
T from this as you can.

BOUNDED OPERATORS ON A HILBERT SPACE 325
17 Show, under the hypotheses of (d) of Theorem 12.29, that the equation Ту = x has a
solution у e Н if and only if
(If A, = 0 for one i, then xt must be 0, for this /.)
18 The spectrum a(T) of Те &(Н) can be divided into three disjoint pieces:
The point spectrum <jp(T) consists of all Л e С for which T— Mis not one-to-one.
The continuous spectrum oc(T) consists of all Ae <Z such that T— A/ is a one-to-one
mapping of H onto a dense proper subspace of H.
The residual spectrum or(T) consists of all other A e o(T).
(a) Prove that every normal Те &(Н) has empty residual spectrum.
(b) Prove that the point spectrum of a normal Те ЩН) is at most countable, if H is
separable.
(c) Let SR and SL be the right and left shifts (as defined in Exercise 1 of Chapter 10),
acting on the Hilbert space £2.
Prove that (SR)* = SL and that
°c(Sl) = Vc(Sr)={\:\\\ = 1}9
or(SL) = <jp(SR) = 0.
19 Let SR and SL be as above. Prove that neither SR nor SL has polar decompositions UP,
with U unitary and P > 0.
20 Let ju, be a positive measure on a measure space Q, let /7 = L2(/x), with the usual inner
product
(/, 9) = J /# Ф-
For 0 gL°°(ju,), define the multiplication operator M* by Мф(/) = ф/. Then M0 e 0&{H).
Under what conditions on ф does Мф have eigenvalues ? Give an example in which
о{Мф) = GС(МФ). Show that every Мф is normal. What is the relation between а(Мф)
and the essential range of ф ? Show that ф -> Мф is an isometric *-isomorphism of L°°(jLt)
onto a closed subalgebra A of ЩН). (Certain pathological measures [x have to be ex-
excluded in order to make this last statement correct.) Is A a maximal commutative sub-
subalgebra of ЩН) ? Hint: If Те 0&{H) and ТМФ = Мф Jfor all ф е £°°(/х), andif ц(П) < oo,
show that Г is a multiplication by ГA) and hence that Те A.
21 Suppose Те £%(H) is normal, A is the closed subalgebra of ЩН) generated by /, T, and
T*9 and Г can be approximated, in the norm topology of ^(#), by finite linear combina-
combinations of projections that belong to A.
Under what (necessary and sufficient) conditions on o{T) does this happen?
22 Does every normal Те &(Н) have a square root in &(H)? What can you say about the
cardinality of the set of all square roots of Г? Can it happen that two square roots of
the same Г do not commute? Can this happen when T—Il

326 BANACH ALGEBRAS AND SPECTRAL THEORY N
j )
23 Show that the Fourier transform /->/ is a unitary operator on L\Rn). What is its
spectrum? Suggestion: When n = 1, compute the Fourier transforms of
(НШ"
exp I - x2JI— I exp (-x2) (m = 0, 1, 2,.. .).
24 Show that any two infinite-dimensional separable Hilbert spaces are isometrically iso-
morphic (via countable orthonormal bases; see [23]). Show that the space H in Theorem
12.38 is separable. Show that the answer to the question that precedes Theorem 12.38
is therefore negative for every infinite-dimensional H, separable or not.
25 Suppose Те &(H) is normal, /is a bounded Borel function on a(T), and S =f(T). If
ET and Es are the spectral decompositions of T and S, respectively, prove that
Es(co) = ET(f~'(oj))
for every Borel set ш с a(S).
26 If Se ЩН) and Те @(H), the notation S > T means that S - T> 0, that is, that
(Sx, x) > (Tx, x)
for all xe H. Prove the equivalence of the following four properties of a pair of self-
adjoint projections P and Q:
(a)P>Q.
(b) Я(Р)=>Я@).
(c) PQ = Q.
(d) QP=Q.
If E is a resolution of the identity, it follows that Е{ш') > E(ui") if and only if
27 Suppose * is an involution in a complex algebra A, q is an invertible element of A such
that q* =q and x* is defined by
for every xe A. Show that # is an involution in A.
28 Let A be the algebra of all complex 4-by-4 matrices. If M = (ти) е A, let M* be the
conjugate transpose of M: mfj =^тп. Put
/о о о o\ /о о о
1 0 0 0l |0 0 0 0
о о ol 1° ooo
/ \o о о o/ \o о о iy
As in Exercise 27, define
M# = e~1M*G (Me A).
(a) Show that 5 and Гаге normal, with respect to the involution #, that ST= TS, but
that ST# Ф T*S.

BOUNDED OPERATORS ON A HILBERT SPACE 327
(b) Show that S + Г is not #-normal.
(c) Compare \\SS* || with ||S||2.
(*/) Compute the spectral radius p(S + S#); show that it is different from \\S + S# ||.
(^) Define K= (y,j)G^ so that vi2= v2a = i, v3i = y43 = —/, vtJ = 0 otherwise.
Compute a{yv*)\ it does not lie in [0, oo).
Part (a) shows that Theorem 12.16 fails for some involutions. Part (b) does the
same for part (a) of Exercise 12; (с), (яГ), and (e) show that various parts of Theorem
11.28 fail for the involution *.
29 Let X be the,vector space of all trigonometric polynomials on the real line: these are
functions of the form
r/.\ istt . . isnt
f{t) = Cie * H + cne \
where ske R and ck e (?, for 1 < к < п. Show that
(f9g)= lim-i- Г
Л->оо 2/1 J-
is an inner product on X, that
and that the completion of X is a nonseparable Hilbert space H. Show that Я contains all
uniform limits of trigonometric polynomials; these are the so-called "almost-periodic"
functions on R.
30 Let Hw be an infinite-dimensional Hilbert space, with its weak topology. Prove that the
inner product is a separately continuous function on Hw x Hw which is not jointly
continuous.
31 Assume Tn e &(H) for n = 1, 2, 3,..., and
lim ||7;*[| =0
for every x e H. Does it follow that
lim ||Г**|| = 0
Л-*оО
for every x e HI
32 Let X be a uniformly convex Banach space. This means, by definition, that the assump-
assumptions
imply that ||хл - yn\\ ->0.
For example, every Hilbert space is uniformly convex.
(a) Prove that Theorem 12.3 holds in X.
(b) Assume ||jc|| = 1, AeX*, \\A\\ = 1, and Лхл->1. Prove that {xn} is a Cauchy
sequence (in the norm-topology of X). Hint: Consider &(xn + xm).

328 BANACH ALGEBRAS AND SPECTRAL THEORY
(c) Prove that every Ael* attains its maximum on the closed unit ball of X.
(d) Assume that xn ~> x weakly and ||лгя|| -> ||jc||. Prove that \\xH — x\\ -» 0. Hint: Reduce
to the case ||jcJ| = 1. Consider A(xn + x), for a suitable Л.
(e) Show that the preceding four properties fail in certain Banach spaces (for instance,
in L1, or in C). These are therefore not uniformly convex.
33. Prove the assertion about the case dim H < oo made in the remark that follows
Theorem 12.35.

13
UNBOUNDED OPERATORS
Introduction
13.1 Definitions Let Я be a Hilbert space. By an operator in H we shall now mean
a linear mapping T whose domain @(T) is a subspace of H and whose range ^?(T)
lies in H.
It is not assumed that T is bounded or continuous. Of course, if Tis continuous
[relative to the norm topology that <2){T) inherits from H] then T has a continuous
extension to the closure of @(T), hence to H, since £^(T) is complemented in H. In
that case, Г is the restriction to $){T) of some member of &(H).
The graph &(T) of an operator T in H is the subspace of H x H that consists
of the ordered pairs {x, Tx}, where x ranges over @(T). Obviously, S is an extension
of T [that is, 9{T) a @(S) and Sx = Tx for x e @(T)] if and only if &(T) a &(S).
This inclusion will often be written in the simpler form
A) TaS.
A closed operator in H is one whose graph is a closed subspace of H x H. By
the closed graph theorem, Те @(H) if and only if @(T) = H and Tis closed.

330 BANACH ALGEBRAS AND SPECTRAL THEORY
We wish to associate a Hilbert space adjoint T* to T. Its domain ££(T*) is to
consist of all у е Н for which the linear functional
B) x-+(Tx,y)
is continuous on ^(T). If у е @(T*)9 then the Hahn-Banach theorem extends the
functional B) to a continuous linear functional on H, and therefore there exists an
element T*y e H that satisfies
C) (Tx9y) = (x9T*y) [хеЗД
Obviously, T*y will be uniquely determined by C) if and only if *F(T) is dense in H,
that is, if and only if T is densely defined. The only operators T that will be given an
adjoint T* are therefore the densely defined ones. Routine verifications show then
that T* is also an operator in H, that is, that *F(T*) is a subspace of Я and that T* is
linear.
Ordinary algebraic operations with unbounded operators must be handled with
care, because the domains have to be watched. Here are the natural definitions for the
domains of sums and products:
D) @(S + Г) = @(S) n ®(T)9
E) @(ST) = {xe@(T):Txe
13.2 Theorem Suppose S, T, and ST are densely defined operators in H. Then
A) T*S* с (ST)*.
If, in addition, S e &(H), then
B) T*S* = (ST)*.
Note that A) asserts that (ST)* is an extension of T*S*. The equality B) implies
that r*S* and (ST)* actually have the same domains.
proof Suppose x e @(ST) and у е @(T*S*). Then
C) (Tx9 S*y) = (x, T*S*y),
because x e @(T) and S*y e @(T*)9 and
. D) (STx, y) = (Tx, S*y),
because Tx e @(S) and у е ®(S*). Hence
E) (STx, y) = (x, T*S*y).
This proves A). .

UNBOUNDED OPERATORS 331
Assume now that S e &(H) and у e ^((ST)*). Then S* e @(H), so that
@(S*) = H, and
F) (Tx, S*y) = (STx, y) = (x, (ST)*JO
for every xe^(ST). Hence S*ye@(T*), and therefore je^(T*S*). Now
B) follows from A). ////
13.3 Definition An operator Tin Я is said to be symmetric if
A) (Tx, y) = (x, Ту)
whenever x e Q)(T) and у е Sf(T). The densely defined symmetric operators are thus
exactly those that satisfy
B) . ГсГ*.
If T = T*9 then T is said to be self-adjoint.
These two properties evidently coincide when Te^(H). In general, they
do not.
13.4 • Example Let H =L2 =L2([0, 1]), relative to Lebesgue measure. We define
operators Tl9 T2, and T3 in L2. Their domains are as follows:
Tj) consists of all absolutely continuous functions/on [0, 1] with derivative
> n {f: ДО) =/A)}.
®(Г3) = ЭД) п {/:/@) =/A) = 0}.
These are dense in L2. Define
A) Tkf= if for/e ®(Tk), * = 1, 2, 3.
We claim that
B) Tf=T3, T2*=T2, 7t=7V
Since Г3 <= Т2 с Tj, it follows that T2 is a self-adjoint extension of the symmetric
(but not self-adjoint) operator T3 and that the extension Tt ofT2 is not symmetric.
Let us prove B). Note first that
C) (Tk f,g) = j \if')9 = ?№) = (/> Tm g)
when /e ЩТк), д е ЩТт), and от + к = 4, since then /A)^A) = /@)g@). It follows
that TmcT*, or
D) rlC:7i, T2e:T;, Г3с]*.

332 BANACH ALGEBRAS AND SPECTRAL THEORY
Suppose now that ge@(T?) and ф = Tt* g. Put Ф(х) = JS <£. Then, for
E) /o if'g = (Ttf,g)=(f,4>)= /AЖ1) - If Ф-
When к — 1 or 2, then @(Tk) contains nonzero constants, so that E) implies ФA) = О.
When k = 3, then/(l) = 0. It follows, in all cases, that
F) ig
Since M{TX) = L2, ig = Ф if к = 1, and since ФA) = 0 in that case,
Thus T? cz T3.
If к = 2 or 3, then ЩТк) consists of all ueL2 such that JJ w = 0. Thus
G) ад) = ^(г3) = у-\
where Г is the one-dimensional subspace of L2 that contains the constants. Hence F)
implies that ig — Ф is constant. Thus g is absolutely continuous and g' e L2, that is,
де9(Тг). Thus Г3* с ^ Ti
If A: = 2, then ФA) = 0, hence g@) = g(l), and g e @{T2). Thus Г2* с Г2 .
This completes the proof.
Before we turn to a more detailed study of the relations between symmetric
operators and self-adjoint ones, we insert another example.
13.5 Example Let H = L2, as in Example 13.4, define Df= f for fe @(T2), say
(the exact domain is now not very important), and define (Mf)(t) = tf(i). Then
(DM - MD)f = f, or
A) DM-MD = I,
where / denotes the identity operator on the domain of D.
The identity operator appears thus as a commutator of two operators, of which
only one is bounded. The question whether the identity is the commutator of two
bounded operators on H arose in quantum mechanics. The answer is negative, not
just in &(H), but in every Banach algebra:
13.6 Theorem If A is a Banach algebra with unit element e, if x e A and ye A,
then
xy - ухФ e.
The following proof, due to Wielandt, does not even use the completeness of A.
proof Assume xy — yx = e. Make the induction hypothesis
A) ' xny-yxn = nxn~1^0,

UNBOUNDED OPERATORS 333
which is assumed to hold for n = 1. If A) holds for some positive integer n,
then x" Ф 0 and
xn + iy - yxn + i = x"(xy - yx) + (xny - yx")x
= x"e + nxn~1x = (л
so that A) holds with n + 1 in place of n. It follows that
п\\хп~'\\ = Wx-y-yx-W <2||х"||Ы| <2||x"-1|||
or n < 21| x || || у ||, for every positive integers. This is obviously impossible. ////
Graphs and Symmetric Operators
13.7 Graphs If H is a Hilbert space, then H x H can be made into a Hilbert
space by defining the inner product of two elements {a, b) and {c, d} of H x H to be
A) . ({я, b), {c, d}) = (a, c) + (A, rf),
where (a, c) denotes the inner product in H. We leave it as an exercise to verify that
this satisfies all the properties listed in Section 12.1. In particular, the norm in H x H
is given by
B) ' \\{a,b}\\2 = \\a\\2 + \\b\\\
Define
C) V{a, b} = {-b,a} (aeH,be H).
Then V is a unitary operator on H x H, which satisfies V2 = —I. Thus V2M = M
if M is any subspace of H x H.
This operator yields a remarkable description of T* in terms of T:
13.8 Theorem IfTis a densely defined operator in H, then
A) &(T*) = [ИЗГ(Г)]\
the orthogonal complement of V&(T) in H x H.
Note that once ЩТ*) is known, so are $}(T*) and Г*.
proof Each of the following four statements is clearly equivalent to the one
that follows and/or precedes it.
B) {y,z}e9(T*).
C) (Tx, y) = (x, z) for every x e 3>(T).
D) ({- Tx, x), {y, z}) = 0 for every x e @{T).
E) {y,z}e[VnT)]L. Illl

334 BANACH ALGEBRAS AND SPECTRAL THEORY
13.9 Theorem // T is a densely defined operator in H, then Г* is a closed operator.
In particular, self-adjoint operators are closed.
proof ML is closed, for every M с H x H. Hence &(T*) is closed in H x H,
by Theorem 13.8. ////
13.10 Theorem IfTis a densely defined closed operator in H, then
A) H x H = V&(T)®&(T*)9
a direct sum of two orthogonal subspaces.
proof If ЩТ) is closed, so is V&(T), since К is unitary, and therefore Theorem
13.8 implies that V<$(T) = ЩТ*)]1; see Theorem 12.4. ////
Corollary IfaeH and b e H, the system of equations
- Jx + у = а
x + T*y = b
has a unique solution with x e @(T) and у е
Our next theorem states some conditions under which a symmetric operator is
self-adjoint.
13.11 Theorem Suppose T is a densely defined operator in H, and T is symmetric.
(a) If@(T) = H, then T is self-adjoint and Te ЩН).
(b) If T is self-adjoint and one-to-one, then 3%{T) is dense in H, and T~l is self-
adjoint.
(c) If ffl(T) is dense in H, then T is one-to-one.
(d) If m(T) = Я, then T is self-adjoint, and T~l e ЩН).
proof (a) By assumption, TaT*. If 9(T) = H, it is thus obvious that
T = T*. Hence T is closed (Theorem 13.9) and therefore continuous, by the
closed graph theorem. (We could also refer to Theorem 5.1.)
(b) Suppose у 1 ЩТ). Then x -> (Tx, у) = О is continuous in 9{T), hence
у e ®{T*) = Q)(T), and (jc, Ту) = (Tx, y) = 0 for all x e &(T). Thus Ту = О.
Since T is assumed to be one-to-one, it follows that у = 0.. This proves that
01{J) is dense in H.
T'1 is therefore densely defined, with Q)(T~X) = ЩТ), and (T)*
exists. The relations
A) %(T-1) = V^(-T) and VViT-1) = &(-T)

UNBOUNDED OPERATORS 335
are easily verified. Since T is self-adjoint, so is —T. Hence Theorem 13.10,
applied to T ~ * and to — T, yields the orthogonal decompositions
B) HxH =
and
C) н х
Consequently,
D) '
Avhich shows that (Г)* = T~K
(c) Suppose Tx = 0. Then (x, Ту) = (Tx, y)=0 for every у е Q){T).
ThiTs x 1 ^G), and therefore x = 0.
(d) Since ^G) = Я, (c) implies that Г is one-to-one, and 9(T~X) = H.
IfxeHandyeH, then x = Tz and у = TV, for some z e^G) and w e$(T),
so that
(T-'x, y) = (z, TV) = Gz, w) - (x, T~ V).
Hence 7 is symmetric, (я) implies that Г is self-adjoint (and bounded),
and now it follows from (b) that T = (T) is also self-adjoint. ////
13.12 Theorem // T is a densely defined closed operator in H, then <3{T*) is dense
and T** = T.
proof Since V is unitary, and V2 = — /, Theorem 13.10 gives the orthogonal
decomposition
A) HxH = &(T)®V&(T*).
Suppose z 1 @(T*). Then (z, у) = О and therefore
B) ({05z},{-T*j;,j;}) = 0
for all ye@(T*). Thus {0, z} e [V<g(T*)]L = ЩТ), which implies that
z = T@) = 0. Consequently, ^(Г*) is dense in Я, and Г** is defined.
Another application of Theorem 13.10 gives therefore
C) Я x H = V$(T*) ©
By A) and C),
D)
so that Г** - Г. ////
We shall now see that operators of the form T*T have interesting properties.
In particular, @(T*T) cannot be very small.

336 BANACH ALGEBRAS AND SPECTRAL THEORY
13.13 Theorem Suppose T is a densely defined closed operator in H, and
Q = I + T*T.
(a) Under these assumptions, Q is a one-to-one mapping of
0@ = $(T*T) = {xe 0(T): Tx e ^(T*)}
onto H, and there are operators В e &(Н), Ce@(H) that satisfy \\B\\ < 1,
\\C\\<l,C = TB,and \
A) B(l + T*T) с (I + T*T)B = I.
Also, B>0, and T*T is self adjoint.
(b) IfV is the restriction ofT to @(T*T), then У(Т) is dense in &(T).
Here, and in the sequel, the letter / denotes the identity operator with domain H.
proof If x e Q){Q) then Tx e &{J*\ so that
B) (x, jc) + (Tx, Tx) = (jc, x) + (x, T*Tx) = (x, Qx).
Therefore ||x||2 < ||x||||6x||, which shows that Q is one-to-one.
By Theorem 13.10 there corresponds to every h e H a unique vector
BheQ}(T) and a unique Che®(T*) such that
C) {0, h} = {-TBh, Bh} + {Ch, T*Ch}.
It is clear that В and С are linear operators in H, with domain H. The two
vectors on the right of C) are orthogonal to each other (Theorem 13.10). The
definition of the norm in H x H implies therefore that
D) ||A||2>№||2 + ||C/z||2 (heH),
so that ||J5|| < 1 and ||C|| < 1.
Consideration of the components in C) shows that С — ТВ and that
E) ?, h = Bh + T*Ch = Bh + T*TBh = QBh
for every heH. Hence QB = I. In particular, В is a one-to-one mapping of
H onto 0@. If у e 0@, then у = Bh for some heH, hence Qy = QBh = h,
and BQy = Bh=y. Thus BQ cz J, and A) is proved.
И heH, then he Qx for some x e Q)(Q), so that
F) (Bh, h) - (BQx, Qx) = (jc, Qx) > 0,
by B). Thus В > 0, В is self-adjoint (Theorem 12.32), and now (b) of Theorem
13.11 shows that Q is self-adjoint, hence so is 7*Г = Q-I.
This completes the proof of part (a).

UNBOUNDED OPERATORS 337
Since T is a closed operator, $(T) is a closed subspace of Я х Я; hence
is a Hilbert space. Assume {z, Tz} e У(Т) is orthogonal to <&(T). Then,
for every x e @(T*T) =
0 = ({z, 7z}, {x, Tx}) = (z, x) + (Tz, Tx) = (z, x) + (z, Г*Гх) - (z, Qx).
= #. Hence z = 0. This proves (b). ////
13.14 Definition A symmetric operator Tin Я is said to be maximally symmetric
if Thas no proper symmetric extension, i.e., if the assumptions
A) T с S, S symmetric
imply that S = T.
13.15 Theorem Self-adjoint operators are maximally symmetric.
proof Suppose T is self-adjoint, S is symmetric (that is, S с S*)9 and T cz S.
This inclusion implies obviously (by the very definition of the adjoint) that
S* с 7*. Hence
5 с S* с Г* = Г с 5,
which proves that S = T. ////
13.16 Theorem If T is a symmetric operator in H (not necessarily densely defined),
the following statements are true.
(a) ||Г.х + /х||2 = |Н|2 + ||Г.х||2 [х
(b) Tis a closed operator if and only if 0t{J 4- //) is closed.
(c) T H- // is one-to-one.
(d) If ЩТ + il) = H, then T is maximally symmetric.
(e) The preceding statements are also true if i is replaced by — i.
proof Statement (a) follows from the identity
\\Tx + ix\\2 = \\x\\2 + ||7x||2 + (ix, Tx) + (Tx, ix\
combined with the symmetry of T. By (a),
)x*+{x, Tx}
is an isometric one-to-one correspondence between the range of T -f // and the
graph of T. This proves (b). Next, (c) is also an immediate consequence of (a).
If M(T + //) = H and 7\ is a proper extension of T [that is, Q)(T) is a proper
subset of ^(Tt)], then 7\ + // is a proper extension of T + il which cannot be
one-to-one. By (c), Tt is not symmetric. This proves (d).
It is clear that this proof is equally valid with — / in place of /. ////

338 BANACH ALGEBRAS AND SPECTRAL THEORY
The Cayley Transform
13.17 Definition The mapping
sets up a one-to-one correspondence between the real line and the unit circle (minus
the point 1). The symbolic calculus studied in Chapter 12 shows therefore that every
self-adjoint Те &(H) gives rise to a unitary operator
B) U = (T-iI)(T+iI)
and that every unitary U whose spectrum does not contain the point 1 is obtained
in this way.
This relation T<--> U will now be extended to a one-to-one correspondence between
symmetric operators, on the one hand, and isometries, on the other.
Let T be a symmetric operator in H. Theorem 13.16 shows that
C) \\Tx + «||2 = И2 + ||Гдс||2 = ||Г* - «||2 (xe®(T)).
Hence there is an isometry U, with
D) @(U) = M(T + il), M(U) = m(T - /i),
defined by
E) U(Tx + ix) = Tx- ix (x e ®{J)).
Since (T + iiy1 maps <3}(U) onto @(T), U can also be written in the form
F) U = {T
This operator U is called the Cayley transforWi of T. Its main features are
summarized in Theorem 13.19. It will lead to an eas(y proof of the spectral theorem
for self-adjoint (not necessarily bounded) operators.
13.18 Lemma Suppose U is an operator in H which is an isometry; \\Ux\\ — ||jc||
for every x e
(a) Ifxe @(U) and у e @(U), then (Ux, Uy) = (jc, y).
(b) If 01A — U) is dense in H, then I — U is one-to-one.
(c) If any one of the three spaces @(U), &(U), and &(U) is closed, so are the other
two.
proof Any of the identities listed in Exercise 2 of Chapter 12 proves (a).
To prove (b), suppose x e 9(U) and (/ - U)x = 0, that is, x = Ux. Then
(x, (/- U)y) = (x, y) - (x, Uy) = (Ux, Uy) - (x, Uy) = 0

UNBOUNDED OPERATORS 339
for every ye@(U). Thus x 1 0t{I - U), so that x = 0 if 0t(l - U) is dense in
H. The proof of (c) is left as an exercise. ////
13.19 Theorem Suppose U is the Cay ley transform of a symmetric operator T in
H. Then the following statements are true.
(a) U is closed if and only if T is closed.
(b) &(I — U)= @(T), I — U is one-to-one, and T can be reconstructed from U by
the formula ,.
. T=i(I+ U)(I- I/).
(The Cayley transforms of distinct symmetric operators are therefore distinct.)
(c) U is unitary if and only if T is self-adjoint.
Conversely, if V is an operator in H which is an isometry, and if I — V is one-to-
one, then V is the Cayley transform of a symmetric operator in H.
proof By Theorem 13.16, T is closed if and only if ЩТ + H) is closed. By
Lemma 13.18, U is closed if and only if &(U) is closed. Since 0A/) = ^(Г+ П),
by the definition of the Cayley transform, (a) is proved.
The one-to-one correspondence x<->z between @(T) and @(U) =
M(T + /7), given by
A) z = Tx + ix, Uz = Tx- ix
can be rewritten in the form
B) (/ - U)z = 2ix, (I + U)z = 2Tx.
This shows that / - U is one-to-one, that 01A - U) = @(T), so that (/ - C/)
maps 9(T) onto 0A/), and that
C) 27* = (J + U)z = (I + l/)(/ - Uy'ilix) [x e 0(T)].
This proves (b).
Assume now that Г is self-adjoint. Then
D) 01A + T2) = H
by Theorem 13.13. Since
E) (Г + H)(T - il) =I+T2 = (T- П)(Т + П)
[the three operators E) have domain ^(T2)], it follows from D) that
F) @(U) = &(T+iI) = H
and
G) Щи) = ЩТ-П) = Н.

340 BANACH ALGEBRAS AND SPECTRAL THEORY
Since U is an isometry, F) and G) imply that U is unitary (Theorem 12.13).
To complete the proof of (c), assume that U is unitary. Then
(8) \®A - I/)]1 = JTif - U) = {0},
by (b) and the normality of / - U (Theorem 12.12), so that 0(T) = 01A - U)
is dense in H. Thus Г* is defined, and Та Г*.
Fix у e ^(T*). Since ^(Г + il) = @(U) = Я, there exists j0 6 9(T) such
that
(9) (Г* + И)У = (T+ il)yo = (T* + il)yo.
The last equality holds because Та Г*. If yi = у — yOi then yy e @(T*) and,
for every x e
A0) (G - il)x, yt) = (x, (Г* + /ЛЛ) = fe 0) = 0.
Thus yx 1 ^(Г - il) = ^(U) = Я, and so ^ - 0, and у = y0 e ®(T).
Hence Г* с= Г, and (c) is proved.
Finally, let V be as in the statement of the converse. Then there is a one-
to-one correspondence z<~+x between <3(V) and M(I — V), given by
A1) x = z-Vz.
Define 5 on 0(S) = ^(/ - 7) by
A2) 5x = i(z +Vz) if x = z - Vz.
If x e @(S) and j; e 0(S), then jc = z - Fz and у = и - Vu for some z e
and w e^(F). Since Fis an isometry, it now follows from (a) of Lemma 13.18
that
A3) (Sx, y) = i(z +Vz,u- Vu) = i(Vz, u) - i(z9 Vu)
= (z - Vz, iu + iVu) = (jc, Sy). v
Hence 5 is symmetric. Since A2) can be written in the form
A4) 2/Fz - Sx - ix, 2iz = Sx + ix [z e @(V)]9
we see that
A5) V(Sx + ix) = Sx- ix [xe&(S)\
and that @(V) = 0l(S + il). Therefore F is the Cayley transform of S. ////
13.20 The deficiency indices If JJ1 and U2 are Cayley transforms of symmetric
operators Tx and T2, it is clear that Tt cz T2 if and only if (/jC U2. Problems about
symmetric extensions of symmetric operators reduce therefore to (usually easier)
problems about extensions of isometries.

UNBOUNDED OPERATORS 341
. For example, every isometry U extends (uniquely) to an isometry whose domain
is the closure of @(U). Therefore it follows from (a) of Theorem 13.19 that every
symmetric operator in H has a closed symmetric extension.
Let us now consider a closed and densely defined symmetric operator T in H,
with Cayley transform U. Then M(T 4- il) and ЩТ - il) are closed, and U is an
isometry carrying the first onto the second. The dimensions of the orthogonal com-
complements of these two spaces are called the deficiency indices of T. (The dimension of a
Hilbert space is, by, definition, the cardinality of any one of its orthonormal bases.)
Since M(I — U) = <2t(T) is now assumed to be dense in Я, every isometric
extension Ux of U has M{I — UJ dense in Я, so that / - Ux is one-to-one (Lemma
13.18) and Ut is the Cayley transform of a symmetric extension Tj of T.
The following three statements are easy consequences of Theorem 13.19 and
the preceding discussion; we still assume that Г is closed, symmetric, and densely
defined.
(a) Tis self-adjoint if and only if both its deficiency indices are 0.
(b) T is maximally symmetric if and only if at least one of its deficiency indices is 0.
(c) Т has a self-adjoint extension if and only if its two deficiency indices are equal.
The proofs of (a) and (b) are obvious. To see (c), use (c) of Theorem 13.19
and note that every unitary extension of U must be an isometry of [M(T + /J)]1
onto [ЩТ-И)]\
13.21 Example Let V be the right shift on ^2. Then V is an isometry and I - V
is one-to-one (Chapter 12, Exercise 18), and so Fis the Cayley transform of a sym-
symmetric operator T. Since @(V) = ^2 and 8%(V) has codimension 1, the deficiency
indices of T are 0 and 1.
This provides us with an example of a densely defined, maximally symmetric,
closed operator T which is not self-adjoint.
Resolutions of the Identity
13.22 Notation Ш will now be a cr-algebra in a set £2, H will be a Hilbert space, and
Е:Ш-*В(Н) will be a resolution of the identity, with all the properties listed in
Definition 12.17. Theorem 12.21 describes a symbolic calculus which associates to
every fe L°°(£) an operator ¥(/) e ЩН\ by the formula
A) (¥(/>, У) = f fdEXt y (xeH,ye H).
This will now be extended to unbounded measurable functions / (Theorem 13.24).
We shall use the same notations as in Definition 12.17.

342 BANACH ALGEBRAS AND SPECTRAL THEORY
13.23 Lemma Letf: u->Cbe measurable. Put
A)
Then Q}f is a dense subspace ofH. IfxeH andyeH, then
B)
Iff is bounded and v = 4?(f)z, then
C) dEx>v=fdEx>z (xeH,zeH).
proof If z = x + y, and со e Ш, then
\\E(co)z\\2 < (\\E(co)x\\ + \\E(co)y\\J < 2\\E(co)x\\2 + 2\\E(co)y\\2
or
D) £z> z(w) < 2EX< x(oj) + 2£,, ,(o).
It follows that @f is closed under addition. Scalar multiplication is even easier.
Thus Я)f is a subspace of H.
For n = 1, 2, 3, ..., let ш„ be the subset of Q in which \f\ <n. If
x e ЩЕ(соп)) then
E) £(to)x = E(co)E((on)x = E(co n соп)х
so that
F) EXf» = £х>х(ш n шп) (со е 9И),
and therefore
G) \\f\2dEx>x = \ \f\2dEXfX<n2\\x\\2<<x>.
Thus ЩЕ(соп)) a $)s. Since п = (J^L х шл, the countable additivity of со -> £(co)j
implies that j> = lim £(сол)^ for every у еЯ, so that j/ lies in the closure of @f.
Hence Q)f is dense.
If л: е Я, j; g Я, and/is a bounded measurable function on Q, the Radon-
Nikodym theorem [23] shows that there is a measurable function и on Q, with
|w| = 1, such that
(8)
Hence
(9) f \f\d\EXi,\ - №/)i, j) < || V(uf)x\\\\y\\.

UNBOUNDED OPERATORS 343
By Theorem 12.21,
A0) \muf)x\\2=j \uf\2dEx,x = j \f\*dEx,x.
Now (9) and A0) give B) for bounded/. The general case follows from this.
Finally C) holds because
f g dEx,, = Q¥(g)x, v) = QV(g)x,
= (?(jyP(g)x9 z) = m/^)x, z) = f gfdEx> z
for every bounded measurable g, by Theorem 12.21. ////
13.24 Theorem Let E be a resolution of the identity, on a set п.
(a) To every measurable f: Q. -> С corresponds a densely defined closed operator ¥(/)
in H, with domain @Q¥(f)) = Q)f, which is characterized by
A) (П/>, У) = f fdEXt y (xe$f,yeH)
and which satisfies
B) №(/М2= f \f\2dEXf
(b) The multiplication theorem holds in the following form: Iff and g are measurable,
then
C) Ч(/У¥(д) <= 4*(fg) and
Hence Ч?(/У*(9) = TO) if and only if&fg
(c) For every measurable f: Q -> C,
D) Y(/)*
and
E) ПЛУ(Л* = ч(\ /I2) =
proof If xe@f then у -*lnfdEx y is a bounded conjugate-linear functional
on #, whose norm is at most (f|/|2 dEXfXI/2, by B) of Lemma 13.23. Jit follows
that there is a unique element *¥(f)x e H that satisfies A) for every yeH and
that
F) 1Ш/>1|2< f \f\2dEXtX
The linearity of ^F(/) on Q)f follows from A), since EXf y is linear in x.

344 BANACH ALGEBRAS AND SPECTRAL THEORY
Associate with each/its truncations fn = /фп, where фп(р) = 1 if \f(p) I < n,
фп(р) = 0 if \f(p)\>n.
Then @f_fn = @f, since each/, is bounded, and therefore F) shows, by
the dominated convergence theorem, that
G) \mf)x-4>(fn)x\\2< f \f-f.\2dEXti-+0 as/i-*oo,
for every xe@f. Since/, is bounded, B) holds with/, in place of /(Theorem
12.21). Hence G) implies that B) holds as stated.
This proves (a), except for the assertion that 4?(f) is closed. The latter
follows from Theorem 13.9 if D) (to be proved presently) is applied to /in place
of/
We turn to the proof of (&). 0
Assume first that/is bounded. Then ®f~<£- @g. If z e H and v = *¥(J)z,
Equation C) of Lemma 13.23 and Theorem 12.21 show that
, z) = (ВДх, W(f)z) = (Пд)х, v)
= f g dEx,v = f fgEx>z = Q¥{fg)*> z)-
Hence
(8) 4(fmg)x = V<f9)x (x e$g,fe L").
If у = vf(g)x, it follows from (8) and B) that
(9) f \f\2dEy,y=\ \fg\2dEx,x (xeD.JeL").
Now let / be arbitrary (possibly unbounded). Since (9) holds for all
/eL00, it holds for all measurable/ Since ®Q¥(jy¥(g)) consists of all xe@g
such that ye@f, and since (9) shows that у e $) s if and only if x e $bSg, we see
that
A0) ®Q¥(jy¥(g)) = ®,n®f
If x e <3g n Q)Sq , if у = Ч/(д)х, and if the truncations fn are defined as
above, then/, ->/in L?(Eyy),fng-+fg in L2(EXfX), and now (8) (with/, in place
of/) and B) imply
4(fy¥(g)x = ^(/)j; = lim ^(/n)y = lim 4(fng)x = 4<(fg)x.
n~* oo n-*oo
This proves C) and hence (b).
Suppose now that xe@f and уe$)j = <2)s. It follows from G) and
Theorem 12.21 that
, y) = lim (ЧЧ/л)х, у) = lim (x,

UNBOUNDED OPERATORS 345
Thus ye@D*(f)*)9 and
A1) ¥(/) с У(/)*.
To pass from A1) to D) we have to show that every ze^(T(/)*) lies in
^y. Fix z; put у = *F(/)*z. Since/, = /фя, the multiplication theorem gives
A2) Ш) = У(/тФп).
Since Ч*(фп) is self-adjoint, we conclude from Theorems 13.2 and 12.21 that
Hence
A3) У (</>> = У(/> A1=1,2 3,...).
Since |<£„| < 1, A3) and B) imply
A4) f \fn\2dEZtZ=\ \фп
for n = 1,2,3, Hence ze^/5 and D) is proved.
Finally, E) follows from D) by another application of the multiplication
theorem, because <3fJ cz <%f. ////
Remark If g is bounded, then @fg cz Q)Q (simply because <3g — H) so that
y¥(fLf(g) = 4J(fg). This was used in A2). It also shows that
A5) %)ПЛ^(/)%),
because ^(дУ^{/) с *¥(gf) = ^{fg). If g is the characteristic function of a
measurable set со a Q, (\5) becomes
A6) E{tiy¥{f)
If x e @f n &(E(co)), it follows that
A7) E(<oL(f)x = 4^(/)
Thus 4*(f) maps ^ п ЭДсо)) into &(E(co)).
This should be compared with the discussion of invariant subspaces in
Section 12.27.
13.25 Theorem In the situation of Theorem 13.24, $f = H if and only Iffe L»(E).
proof Assume Qls = H. Since ¥(/) is a closed operator, the closed graph
theorem implies that ¥(/) e ^(//). If/n =/ф„ is a truncation of/ it follows
from the multiplication theorem, combined with Theorem 12.21, that
или. =
since \тф„Ц = \\ф„\\х < 1. Thus H/IL ^ ЦВД11, and/eL00^). The converse
is contained in Theorem 12.21. ////

346 BANACH ALGEBRAS AND SPECTRAL THEORY
13.26 Definition The resolvent set of a linear operator 7 in Я is the set of all
X e С such that 7 — XI is a one-to-one mapping of Q)(T) onto H whose inverse
belongs to ЩН).
In other words, 7 — XI should have an inverse S e &(H), which satisfies
S(T - XI) a G - XI)S = I.
For instance, Theorem 13.13 states that - 1 lies in the resolvent set of 7*7 if 7
is densely defined and closed.
The spectrum o(T) of 7 is the complement of the resolvent set of 7, just as for
bounded operators.
Some properties of o(T), for unbounded 7, are described in Exercises 17 to 20.
For the next theorem, we refer to Section 12.20 for the definition of the essential
range of a function, with respect to a given resolution of the identity.
13.27 Theorem Suppose E is a resolution of the identity on a set Q,f:£l-+C is
measurable, and V
a} (a e C).
(a) If a is in the essential range of f and E(coa) Ф 0, then *¥(f) — ocl is not one-to-
one.
(b) If ol is in the essential range off but E(oja) = 0, then ^(f) — al is a one-to-one
mapping of Q)f onto a dense proper subspace of H, and there exist vectors xn e H,
with \\xn\\ — 1, such that
Km mf)xn - ах„] = О.
П~* 00
(c) aC¥(f)) is the essential range off
In the terminology used earlier for bounded operators, we may say that a lies
in the point spectrum of Ч^/) in case (a) and in the continuous spectrum of ¥(/) in
case (b). The conclusion of (b) is sometimes stated by saying that a is an approximate
eigenvalue of4?(f).
proof We shall assume, without loss of generality, that a = 0.
(a) lfE(co0) Ф 0, there exists x0 e @(E(co0)) with ||xo|| = 1. Let ф0 be the
characteristic function of co0. Then /ф0 = 0, hence ^(/)^((/>0) = 0, by the
multiplication theorem. Since ^(фо) — Е((о0), it follows that
¥(/>o = 4(f)E(aH)x0 = ¥<УТОо)*о = .0.
(b) The hypothesis is now that E(co0) = 0 but E(con) Ф 0 for n = 1, 2, 3,
..., where

UNBOUNDED OPERATORS 347
Choose xn бЭДо)„)), ||xj = 1; let фп be the characteristic functions of con.
The argument used in (a) leads to
— n oo — n
Thus ¥(/>„ -> 0 although ||хл|| = 1.
If *F(/)a: = 0 for some xe^, then
и
Since |/| >0 a.e. [EXfX], we must have EXX(Q) = 0. But £Xfje(fll)=
Hence *F(/) is one-to-one.
Likewise ¥(/)* = ¥(/) is one-to-one. If у 1 &tQ¥(f)), then x
= 0 is continous in Q)f, hence у е &(*¥(/)*), and
(x, 4(J)y) = (¥(/>, y) = 0 (x e 2f).
Therefore, *¥(J)y = 0, and у = 0. This proves that ^0F(/)) is dense in #.
Since ^(/) is closed, so is П/). If ^W)) filled я? the closed graPh
theorem would imply that Ч^/) е£й(Н). But this is impossible, in view of
the sequence {xn} constructed above.
Hence (b) is proved.
(c) It follows from (a) and (b) that the essential range of/is a subset of
(j^f/)). To obtain the opposite inclusion, assume 0 is not in the essential
range of/ Then g - \lfeL™(E)Jg = 1, hence У(/)¥(<7) = ¥A) = J, which
proves that M(^{f)) = H and therefore that ¥(/) ~i e ^(Я), by the closed
graph theorem.
This completes the proof. ////
The following theorem is sometimes called the change of measure principle.
13.28 Theorem Suppose
(a) 9Л and Ж are о-algebras in sets Q and Q',
(b) E: Ш -> &(H) is a resolution of the identity, and
(c) ф: Q -> Q' has the property that ф~1(сог) е $Jlfor every со' е Ш'.
IfE'(co') = Е(ф~\<о')\ then E'\ Ж -> ^(Я) is also a resolution of the identity,
and
A) ,y \
for every Ш'-measurable f: Q' -> С for which either of these integrals exists.

348 BANACH ALGEBRAS AND SPECTRAL THEORY
proof For characteristic functions /, A) is just the definition of E'. Hence
A) holds for simple functions/. The general case follows from this. The proof
that E' is a resolution of the identity is a matter of straightforward verifications
and is omitted. ////
The Spectral Theorem
13.29 Normal operators A (not necessarily bounded) linear operator T in H is
said to be normal if T is closed and densely defined and if
jr*T = TT*
Every \F(/) that arises in Theorem 13.24 is normal; this is part of the statement
of the theorem. We shall now see, just as in the bounded case discussed in Chapter 12,
that all normal operators can be represented in this way, by means of resolutions of
the identity on their spectra (Definition 13.26). For self-adjoint operators, this can be
deduced very quickly from the unitary case, via the Cayley transform (Theorem 13.30).
For normal operators in general, a different proof will be given in Theorem 13.33.
13.30 Theorem To every self-adjoint operator A in H corresponds a unique resolution
E of the identity, on the Borel subsets of the real line, such that
A) {Ax, y)=C t dEx, /0 (x e 9{A), у е Я). __
J — oo
Moreover, E is concentrated on o(A) cz (—oo, oo), in the sense that E(a(A)) = I.
As before, this E will be called the spectral decomposition of A.
proof Let U be the Cayley transform of A, let п be the unit circle with the
point 1 removed, and let E' be the spectral decomposition of U (see Theorems
12.23 and 12.26). Since / - U is one-to-one (Theorem 13.19), E'({!})'= 0, by
(b) of Theorem 12.29, and therefore
B) (Ux, y) = f Я dE'Xf y(X) (xeH,ye H).
Define
' C) ?
and define *¥(/) as in Theorem 13.24 with E in place of E:
D) • (¥(/>, y) = \ fdE'x<, (xe9,,yeH).

UNBOUNDED OPERATORS 349
Since / is real-valued, *F(/) is self-adjoint (Theorem 13.24), and since
— X) = i(l + A), the multiplication theorem gives
- U) = i(I + U).
In particular, E) implies that Щ1 - U) cz SjQV(f)). By Theorem 13.19,
F) A(I -U) = i(I + U),
and &(A) = 01A - U) cz &(¥(/)). Comparison of E) and F) shows now that
is a self-adjoint extension of the self-adjoint operator A. By Theorem
13.15, A = 4*(f). Thus
G) (Ax, y) = f /Ж;,, [x e @(A), у е Н].
By (c) of Theorem 13.27, o{A) is the essential range of/ Thus a(A) cz
( — oo, oo). Note that/is one-to-one in п. If we define
(8) E(f(co)) = E'(co)
for every Borel set со с Q, we obtain the desired resolution E which converts
G) to A).
• Just as A) was derived from B) by means of the Cayley transform, B) can
be derived from A) by using the inverse of the Cayley transform. The uniqueness
of the representation B) (Theorem 12.23) leads therefore to the uniqueness of
the resolution Z^that satisfies A).
This completes the proof. ////
The whole machinery developed in Theorem 13.24 can now be applied to self-
adjoint operators. The following theorem furnishes an example of this.
13.31 Theorem Let A be a self-adjoint operator in H.
(a) (Ax, x) > Ofor every x e @(A) (briefly: A>0) if and only if o(A) cz [0, oo).
(b) If A > 0, there exists a unique self-adjoint В > 0 such that B2 = A.
proof The proof of (a) is so similar to that of Theorem 12.32 that we omit it.
Assume A > 0, so that c(A) cz [0, oo), and
A) (Ax, y)=Ct dEXt y(t) [x e ®(A\ у е Н],
where $)(A) = {x e H: J t2 dExy(t) < oo}; the domain of integration is [0, oo).
Let ^@ be the nonnegative square root of t > 0, and put В = ^(s); explicitly,
B) (Bx, y) = fs(i) dEXt y(t) (xe®s,ye H).

350 BANACH ALGEBRAS AND SPECTRAL THEORY
The multiplication theorem (b) of Theorem 13.24, with /= g = s, shows that
B2 — A. Since s is real, В is self-adjoint [(c) of Theorem 13.24], and since
s(t) > 0, B), with x = y, shows that В > 0.
To prove uniqueness, suppose С is self-adjoint, С > 0, С2 = A, and Zsc is
its spectral decomposition:
C) (Cx, y) = f °°j <ffi£,(*) (x g 0(O, j; g Я).
Apply Theorem 13.28 with D = [0, oo), </>(» = Л Л0 = ^ and
D) Е'(ф((о)) = .Ес(со) for со с [0, oo),
to obtain
E) (Ax, y) = (C2x, y) = f*> J£^, /5) - f % d£; y@.
By A) and E), the uniqueness statement in Theorem 13.30 shows that E' = E.
By D), E determines Ec, and hence C. ////
The following properties of normal operators will be used in the proof of the
spectral theorem 13.33.
13.32 Theorem If N is a normal operator in H, then
(a) @(N)=@(N*),
(b) \\Nx\\ - ||jV*jc|| for every x e ®(N\ and
(c) N is maximally normal.
proof If у e @(N*N) = @(NN% then (Ny, Ny) = (y, N*Ny) because Ny e
®(N*)9 and (N*y,N*y) = (y9NN*y) because N*ye$(N) and N = N**
(Theorem 13.12). Since N*7V = NN*, it follows that
A) \\Ny\\=\\N*y\\ irye$(N*N).
Now pick xe$J(N). Let N' be the restriction of N to @(N*N). By
Theorem 13.13, {x, TVx} lies in the closure of the graph of N'. Hence there are
vectors уi e <3(N*N) such that
B) Hy.-^II^Oas/^oo
and
C)
By A), \N*y.t - N*yj\\ =\\Nyi-Nyj\\9 so that C) implies that {iV*y«} *s a
Cauchy sequence in H. Hence there exists z eH such that

UNBOUNDED OPERATORS 351
D)
Since jV* is a closed operator, B) and D) imply that {x, z} e <&(N*).
From this we conclude first that xe@(N*), so that <2(N) с @(N*)9 and
secondly that
E) \\N*x\\ = ||z|| = lim \\N*yi\\ = lim \\Nyt\\ = ||№с||.
This proves (b) and half of (a). For the other half, note that TV* is also normal
(since 7У** = TV), so that
F)
Finally, suppose M is normal and N a M. Then M* с TV*, so that
G) Q){M) = ^(M*) с 0(W*) = &(N) с 0(M),
which gives 0(M) = ^(^V); hence M = N. ////
13.33 Theorem jE'feA'j; normal operator N in H has a unique spectral decomposition
E, which satisfies
A) (Nx, y)=\ X c!Ex,y(X) (x e ®(N\ у е Н).
Moreover, E(co)S = SE(co) for every Borel set со с a(N) and for every S e &(H)
that commutes with N9 in the sense that SN с TVS.
It also follows from A) and Theorem 13.24 that E(co)N с NE(oo).
proof Our first objective is to find self-adjoint projections Pi9 with pairwise
orthogonal ranges, such that PtN с NPt e &(H), NPt is normal, and x = ]T Ptx
for every x e H. The spectral theorem for bounded normal operators will then
be applied to the operators NP-t, and this will lead to the desired result.
By Theorem 13.13, there exist В е ЩН) and СеЩН) such that В > О,
||J5|| < 1, C = NB, and
B) B(I + N*N) c/ = (/ + N*N)B.
Since N*N = NN*, B) implies
C) BN = BN(I + N*N)B = B(I + N*N)NB с NB = C.
Consequently, ВС = B(NB) = (BN)B a CB. Since В and С are bounded, it
follows that ВС = CB and therefore that С commutes with every bounded
Borel function of B. (See Section 12.24.)
Choose {tH so that 1 = t0 > tt > t2 > • • •, lim t{ = 0. Let />£ be the

352 BANACH ALGEBRAS AND SPECTRAL THEORY
characteristic function of (ti9 ^-J, for / = 1, 2, 3, ..., and put ft(t) = Pi(t)/t.
Each/; is bounded on a(B) с [0, 1]. Let EB be the spectral decomposition of B.
The equality B) shows that В is one-to-one, that is, 0 is not in the point spectrum
of B. Hence EB({0}) = 0, and EB is concentrated in @, 1].
Define
D) Pi=p№ (i=1,2,3,...).
Since PiPj = 0 if 1ф}9 the projections Pt have mutually orthogonal ranges.
Since YjPi 1S tne characteristic function of @, 1], we have
E) £plx = ^B(@, 1])jc = jc A6Я).
F) NPi
and PiiV =fi(B)BN cft{B)C9 by C), so that
G) Pt N с ТУР^.
By F), ^(^Pf) = Я, so that
(8) <*(Pt)c:@(N) (/=1,2,3,...)-
Hence, if P£jc = x, G) implies PtNx = NPtx = x. Thus JV carries ЩР^ into
&(Pi), or: ^(P£) w an invariant subspace of N.
Next, we wish to prove that each NPt is normal. By G) and Theorem
13.2,
(9)
But NPt e &(H)9 so that (NPt)* has domain H. Hence
A0) (NPf.)* = N*P£,
and now Theorem 13.32 shows, by (8) and A0), that
A1) ||ЛН>,*|| = ||N*P,*|| = \\(МРд*х\\ (xeH).
By Theorem 12.12, A1) implies that NPt is normal.
Hence E), F), and G) show that our first objective has now been reached.
By Theorem 12.23, each NPt has a spectral decomposition E\ defined on
the Borel subsets of С
Since N carries ЩР^) into ^(/\), P£ commutes with NPt. Therefore Pt
commutes with Е1(со)9 for every Borel set ш c C, so that
A2) Яг(о))Р,- x = Pt EXco)x e ЩР,) (х е Я, i = 1, 2, 3, ...).

UNBOUNDED OPERATORS 353
Since these ranges are pairwise orthogonal, and since E) implies
(i3) f; надран2 < f; iip,.x||2 = imi2,
i 1 i 1
f;
i = 1 i = 1
the series £ £'(Ш)Л* converges, in the norm of H, and it makes sense to
define
A4) E(w)=YE\co)Pl
for all Borel sets со с С.
It is easy to check that E is a resolution of the identity. Hence there is a
normal operator M, defined by
A5) (Mx,y)
where the domain of integration is <£, and
A6) ЩМ) = LeH: j\X\2dEXtX(X) < oo}.
Our assertion A) will now be proved by showing that M = N.
For any x e H, A4) shows that
A7) EXfX(co) = \\E(cd)x\\2 = £ ||Е'(со)Р£х||2= £ EXitXi(co),
i = 1 i = 1
where xt = P,- jc. If x e @(N), then P£ Nx = JVP,- x, so that
oo « oo oo
A8) X f |Я|2 <,„№ = E HiVP.-x.-H2 = J] ||P,.iVx||2 = ||iVx||2.
It follows from A7) and A8) that the integral in A6) is finite for every x e
Hence
A9)
If x e &(Pi)9 then x = Ptx, and so E(co)x = El(co)x\ thus £^у = J?i>y for
every у е Н. Hence
(AT*, у) = (ЛГЛ-*, >-) = /^ <Ki.,W = JA Ж»./A) = (M*, y).
Consequently
B0) P,.Nx = NP( x = MPt x [xe ®(N\ i = 1, 2, 3,...].
If Gi = Л + """+ Л-, it follows that Qt-Nx = M|2tx. Thus
B1) {Qtx, QtNx} e 9(M) [xe@(N), i = 1, 2, 3, ...].

354 BANACH ALGEBRAS AND SPECTRAL THEORY
Since ^(M) is closed, it follows from E) and B1) that {x, Nx} e ЩМ), that is,
that JVx = Mx for every xe@(N). ThusiV с М, by A9), and now the maximality
of N (Theorem 13.32) implies N = M.
This gives the representation A), with € in place of cr(N). That E is actually
concentrated on a(N) follows from (c) of Theorem 13.27.
To prove the uniqueness of E, consider the operator
B2) T /
where y/N*N is the unique positive square root of JV*iV. If A) holds, it follows
from Theorem 13.24 that
B3) T=j<l>dE,
where ф(Х) = 1/A + |Л|), so that Te ЩН), and since ф is one-to-one on C,
Theorem 13.28 implies that the spectral decomposition ET of T satisfies
B4) E(co) = Ет(ф(со))
for every Borel set со с С. The uniqueness of E follows now from that of ET
(Theorem 12.23).
Finally, assume S e @(H) and SN с NS. Put Q = Qn = E(cb), Where
(b = {A: |Л| < л}, and n is some positive integer. Then NQ e@(H) is normal
and is given by
B5) NQ = jfdE,
where f(X) = X on cb,f(X) = 0 outside (b. Theorem 13.28 implies that the spectral
decomposition E' of NQ satisfies E\w) = E{f~1{w)), or
B6) F'(co) = E((D n &) = QE(co) if ° *ш'
\£'({0}) = Ц{0} u (C - C)) = £({0}) + / - Q.
Hence
B7) E(co) = QE(co) = QE'(co) if со с й.
By Theorem 13.24, QN aNQ = QNQ, so that
B8) (QSQXNQ) = QSNQ с gN^S с (NQXQSQ).
Since (QSQ)(NQ) e &(H), the inclusions in B8) are actually equalities. Now
Theorem 12.23 implies that QSQ commutes with every E'(co).

UNBOUNDED OPERATORS 355
Consider a bounded со, and take n so large that со а со. By B7)
QSE(co) = QSQE'{w) = £'(co)<2Se = E(co)SQ
so that
B9) е„ S£(co) = Д(оMеи (л-1,2,3,...).
It now follows from Proposition 12.18 that
C0) SE(co) = E(co)S
if со is bounded [let n -> oo in B9)], and hence also if со is any Borel set in €.
Ill/
Semigroups of Operators
13.34 Definitions Let ХЫ a Banach space, and suppose that to every t e [0, oo)
is associated an operator Q(t) e ЩХ), in such a way that
(b) . Q(s + t) = Q(s)Q(t) for all s > 0 and Г > 0, and
(c) lim || Q(t)x — jc|| = 0 for every x e X.
t-*0
If (a) and (b) hold, {Q(t)} is called a semigroup (or, more precisely, a one-param-
one-parameter semigroup). Such semigroups have exponential representations, provided that
the mapping t -+ Q(t) satisfies some continuity assumption. The one that is chosen
here, namely (c), is easy to work with.
Motivated by the fact that every continuous complex function that satisfies
f(s + t) =f(s)f(t) has the form/@ = exp (At), and that/is determined by the number
A =/'@), we associate with {Q(t)} the operators Ae9 by
A) AE x = - [Q(e)x -x] (xeX,e> 0),
and define
B) Ax = lim AE x
for all x e @(A), that is, for all x for which the limit B) exists in the norm topology
of X.
It is clear that <$(A) is a subspace of X and that A is thus a linear operator
in A".
This operator, which is essentially Q'@), is called the infinitesimal generator of
the semigroup {Q(t)}.

13.35 Theorem If the semigroup {6@} satisfies the preceding hypotheses, then
(a) t -> Q(t)x is a continuous mapping of [0, oo) into X, for every x e X,
(b) A is a closed densely defined linear operator in X,
(c) for every x e @(A), Q(i)x satisfies the differential equation
d
~ Q(t)x = AQ{t)x = Q(t)Ax,
dt
and
(d) for every x e X,
Q(t)x = lim [exp (tAe)]x,
e-»O
the convergence being uniform on every compact subset of [0, oo).
It is remarkable that the conclusion (d) holds for every x e X, not just for x e i
The limit in (d), as well as the one that is implicit in the derivative used in (c), is under-
understood to refer to the norm topology of X.
proof If there were a sequence tn->0 such that ||6(Oil -* °°> tne Banach-
Steinhaus theorem would imply the existence ofanxel for which {|| Q(tn)x\\}
is unbounded. This contradicts our assumption that
A) \\Q(t)x-x\\-+0 ast->0.
Hence there exist S > 0 and y0 < oo such that
B) 116@11 <y0 ifO<f<<5.
Put у = sup {|| Q(s)\\: 0 < s < 1}. By the functional equation
C) 60+ 0 = 60N@,
B) implies that у < oo. Moreover, у > 1, and
D) \\Q(t)\\<yl+t @<f<oo),
for if n < t < n + 1, then 6@ = QW"Q(t - n).
The equality D) can be applied to
and yields
||exp (MJH < e-'/£ Y, ^ПГ = У exP \*L
If 0 < e < 1, then y* - 1 < e(y - 1). Hence
E) ||exp (M£)|| < у exp (ty) @ < г < 1, 0 ^ t < oo).
After these preparations, we turn to the main part of the proof.

UNBOUNDED OPERATORS 357
If x e X and e > 0, A) shows that there exists ц = t](x, e) > 0 such that
|| Q(t)x - x\\ < a if 0 < t < t]. If 0 < s < t < s + ц < n, it follows from D)
that
|| Q(t)x - Q(s)x\\ = || Q(s)[Q(t - s)x - x]\\
<\\Q(s)\\\\Q(t-s)x-x\\<yn + ic.
This proves (a).
The X-valued integrals
F) Mtx = - ( Q(s)xds (xeX,t> 0)
t Jo
can therefore be defined. In fact, Mt e ЩХ) and \\Mt\\ < y1 +f, by D). We claim
that the identity
G) AeMt = AtMe (e>0,t>0)
holds.
To prove G), rewrite the obvious relation
in the form
and insert the integrand Q{s)x ds. By C), the left side of (8) becomes
f [6(e + *) - Q(s)]x d* = [Q(z) - I] f Q(s)x ds = eAE tMt.
In the same way, the right side becomes tAt e ME. This gives G).
We can now prove (b). By F), Mt x -> x as t -> 0, for every x e X. For
t > 0, At e &(X). Hence G) gives
(9) lim AEMtx = At limM£i = Atx.
It follows that Mtxe @(A), for every t > 0, so that Q)(A) is dense in X, and
that
A0) AMtx = Atx (xeX,t>0).
Since Q(s) commutes with Q(i), Ae commutes with Mt. If x e @(A), G)
implies therefore, for t > 0, that
A1) MtAx = Mt lim AEx = lim Me Atx = Atx.

358 BANACH ALGEBRAS AND SPECTRAL THEORY
If now xne@(A), xn-+x, and Axn-+у as n -+ oo, then A1) shows that
Atxn = MtAxn. Hence Atx = Mty. Since Mty->y as t -» 0, it follows that
;t g @(A) and that Лх = y. Thus >4 is a closed operator, and (b) is proved.
Assume now that x e .^(Л). Then, for all t > 0,
A2) AEQ(t)x = GCK*- 6@^* as e - 0,
so that Q(t)x e Q){A) and
A3) AQ(t)x = Q(f)Ax.
If A1) is multiplied by t, we get
A4) f Q(s)Ax ds = Q(t)x - x.
The continuity of the integrand, proved in (a), shows that the derivative of this
integral is Q(t)Ax. In conjunction with A3), this proves (c).
We turn to (d). If x e 9(A) and 0 < s < t, then (c) shows that
- [exp {(t - s)AE}Q(s)x] = exp {(t - s)AE}Q(s)(Ax - AEx),
and since
exp {(t — s)AF}Q(s)x z=\^ '
f iv j esvz\ j |exp (tAE)x when s = 0,
we have
6@* - exP (tA^x = f exp {(f - j)^JG(j)(^* -Л*) ^>y-
Jo
By D) and E), the norm of this integrand is at most
yexp{(t-s)y}yl+s\\Ax-AEx\\,
so that
A5) ||S@* - exp(L4>|| < K(t)\\Ax - AEx\\,
where К is an increasing continuous function on [0, oo).
To complete the proof, fix t0 > 0, and define
A6) S(t9 e) = 6@ - exp (tAE) (t>O,O<e<l).
By D) and E), there exists Ko < oo such that
A7) ||S(f, 8)|| <KQ @<t<tOi0<e<l).

UNBOUNDED OPERATORS 359
If x0 e X, and rj > 0, there exists x e $)(A) such that
08) ll*-*oll<J-
Then A5) to A8) imply that
\\S(t, s)xo\\ < \\S(t, e)x || + ||5ft e)|| ||* - jcqII
<K(to)\\Ax-AEx\\ + n
if 0 < f < f0. Since x e ®{A), we finally get
■ A9) II Q(t)x0 - exp (tAE)x0 \\<n @<t<to)
for all sufficiently small e.
This completes the proof. ////
It is now natural to ask whether the limit can be removed from the conclusion
(d), that is, under what conditions the exponential representation Q(t) = exp (tA) is
valid. The next two theorems give answers to these questions.
13.36. Theorem //{6@} is as in Theorem 13.35, then any of the following three
conditions implies the other two:
(a)
(b)
E->0
(c) A e &(X) and Q(t) = etA @ < t < oo).
proof We shall use the same notations as in the proof of Theorem 13.35.
If (a) holds, the Banach-Steinhaus theorem implies that the norms of the
operators Ae are bounded, for all sufficiently small e > 0. Since Q(s) — I — eAE,
(b) follows from (a).
If (b) holds, then also ||Mf - /|| -*0 as t -> 0. Fix t > 0, so small that
Mt is invertible in &(X). Since MtA£ = AtME, we have
A) А. = (М,Г1А,М..
As e -> 0, A) shows first of all that AE x converges, for every x e X [since ME x -> x
and (Mt)At e ®{X)\ secondly that A = (Mt)~xAt, and thirdly that
B) \\A£ -A\\< ||(МГГ 4|| ||M£ - /|| - 0 as г -*0.
The formula Q(t) = exp (tA) follows now from (d) of Theorem 13.35, since B)
implies that
C) lim ||exp (tAE) - exp (tA)\\ =0 @ < t < oo).
e->0

60 BANACH ALGEBRAS AND SPECTRAL THEORY
Thus (c) follows from (b).
The implication (c) -> (a) is trivial. - ////
For our final theorem, we return to the Hilbert space setting.
13.37 Theorem Assume that {Q(t): 0 < t < oo} is a semigroup of normal operators
2@ 6 &(H), which satisfies the continuity condition
1) lim || Q(t)x - x\\ = 0 (xeH).
The infinitesimal generator A of {Q(t)} is then a normal operator in H, there is a
> < oo such that Re X < у for every X e cr(A), and
2) Q(t) = etA @<t< oo).
// each 2@ is unitary, then there is a self-adjoint operator S in H such that
3) 2@ = eits @<t< oo).
This representation of unitary semigroups is a classical theorem of M. H.
Stone.
Note: Although Q)(A) may be a proper subspace of H, the operators etA are
defined in all of Я and are bounded. To see this, let EA be the spectral decomposition
of A (Theorem 13.33). Since | ea\ < etyfov all X e a(A), the symbolic calculus described
in Theorem 12.21 allows us to define bounded operators etA by v
D) etA = Г ea dE\X) @ <: t < oo).
Ja(A)
The theorem has an easy converse: If A is as in the conclusion, then B) obviously
defines a semigroup of normal operators, and A) holds because
E) || 6@* - *H2 = f Wx ~ 1 I2 dEix(X) -> 0
Ja(A)
as t -> 0, by the dominated convergence theorem.
proof Since each Q(s) commutes with each Q(t), Theorem 12.16 implies that
Q(s) and 6@* commute. The smallest closed subalgebra of 0l(K) that contains
all g@ and all 2@* ^s therefore normal. Let A be its maximal ideal space, and
let Z^be the corresponding resolution of the identity, as in Theorem 12.22.
Let f and a£ be the Gelfand transforms of 2@ and Ae, respectively.
Then
F) ^@)

UNBOUNDED OPERATORS 361
and.a simple computation gives
G) ^2e - ae = - (aEJ,
since/2e = (/£J. Define
(8) b(p) = Km a2-n(p)
n~* oo
for those p e A at which this limit exists (as a complex number), and define
b(p) = 0 at all other p e A. Then b is a complex Borel function on A. Put
В = Ч*(Ь), as in Theorem 13.24, with domain
(9) ^(Б)= хеЯ: f \b\2 dEx x< oo .
I ja " )
Then В is a normal operator in H.
We will show that A = B.
If xe@(A) then \\A£x\\ is bounded, as e-^-O. Hence there exists Cx < oo
such that
A0) f \ac\2dEXtX=\\Aex\\2<Cx
f
and therefore
by G). Take e = 2~n (n = 1, 2, 3, ...) in A1) and add the resulting inequalities.
It follows that
A2) f>2-» + i-«2-l <<» a.e.[EXfJ.
n= 1
The limit (8) exists therefore a.e. [Exx], and now Fatou's lemma and A0) imply
that
A3) \\b\2dEXtX<Cx.
Consequently, 9{A) с
Formula E) in the proof of Theorem 13.35 shows that ||ехр(Ле)|| < yx < со
for 0 < e < 1, where yt depends on {6@}- Hence |exp аг(р)\ < yt for every
p e A, since the Gelfand transform is an isometry on 2?*-algebras. It now follows

362 BANACH ALGEBRAS AND SPECTRAL THEORY
from (8) that | exp b(p) \ < yt for every p e A. Hence there exists у < oo such
that
A4) Re%)<7 (peA).
For every x e $)(A) and every / > 0,
A5) ||exp (tAE)x - exp (tB)x\\2 = f |exp (taE) - exp (tb)\2 dEXfX
tends to 0 as e -> 0 through the sequence {2~n}, because the integrand is bounded
by 4yf and its limit is 0 a.e. [EXtX]. Hence (d) of Theorem 13.35 implies that
A6) Q(t)x = etBx [xe@(A)].
However, etb is a bounded function on A, hence е*веЩН), and since
A6) shows that the continuous operators Q(t) and etB coincide on the dense set
), we conclude that
A7) Q(t) = etB @<t<oo).
It follows from A7) that
A8) AEx-Bx=(e——--b\x
so that
A9) \\AEx-Bx\\2= b
- 1
dEXtX.
As e -> 0, the integrand A9) tends to 0, at every point of A. Since \(ez - l)/z|
is bounded on every half-plane {z: Re z < c}, and since the integrand A9) can
be written in the form
eb
it follows from A4) and the dominated convergence theorem that
B0) lim \\AEx-Bx\\2 =
£-»O
This proves that 9{B) a @(A) and that A = B.
That the real part of c{A) is bounded above follows now from A4) and
(c) of Theorem 13.27.
This completes the proof, except for the final statement about unitary
semigroups. If each Q(t) is unitary, then \fE\ = 1, F) shows that lim aE is pure

UNBOUNDED OPERATORS 363
imaginary at every point at which it exists, as e -> 0, hence b{p) is pure imaginary
at every p e A, and if S = — iB then A7) gives C), and (c) of Theorem 13.24
shows that S is self-adjoint. ////
Exercises
Throughout this set of exercises, the letter H denotes a Hilbert space, unless the contrary is
stated.
/ The associative law G, T2)T3 = TV(T2 T3) has been used freely throughout this chapter.
Prove it. Prove also that Tx с т2 implies STi с ST2 and TXS с Т2 S.
2 Let 7\be a densely defined operator in H. Prove that Thas a closed extension if and only
if <3(T*) is dense in H. In that case, prove that T** is an extension of T.
3 By Theorem 13.8, @{T*) = {0} for a densely defined operator T in H if and only if ЩТ)
is dense in И х Н. Show that this can actually happen.
4 Suppose T is a densely defined, closed operator in Я, and T*T с 7Т*. Does it follow that
Г is normal ?
5 Suppose T is a densely defined operator in Я, and G>, x) = 0 for every jc e @(T). Does
it follow that 7x = 0 for every x e Q)(TI
6 If T is an" operator in Я, define
= {xe 9(T): Tx = 0}.
is dense, prove that
Jf{J*) = 0t{JY n
If Г is also closed, prove that
This generalizes Theorem 12.10.
7 Consider the following three boundary value problems. The differential equation is
f"-f=g,
where g e L2([0, 1]) is given. The choices of boundary conditions are
@
(//) /40) =/4O = o.
(ш) /@)=/(l)and/'@)=/'@.
Show that each of these problems has a unique solution/such that/' is absolutely
continuous and/"e£2([0, 1]). Hint: Combine Example 13.4 with Theorem 13.13.
Do this also by solving the problems explicitly.
8 (a) Prove the self-adjointness of the operator T in L2(R\ defined by Tf= //N with
consisting of all absolutely continuous feL2 such that/' eZA
Hint: You may need to know that /(/) -> 0 as t -> ± oo for every /e

364 BANACH ALGEBRAS AND SPECTRAL THEORY
Prove this. Or prove more, namely, that every fe@(T) is the Fourier transform of
an //-function.
(b) Fixg eL2(R). Use Theorem 13.13 to prove that the equation
f"-f=9
has a unique absolutely continuous solution fe L2, which has /' eL2, f" el2, and
/' absolutely continuous.
Prove also, by direct calculation, that
1 r* 1
fix) = - - j e'-*g(O dt--j e'-*g{t) dt.
This solution can also be found by means of Fourier transforms.
9 Let H2 be the space of all holomorphic functions /(z) = 2 Cn z" *n tne °Pen umt disc that
satisfy
= 2 \cn\2<co.
Show that H2 is a Hilbert space which is isomorphic to /2 via the one-to-one corre-
correspondence / «-► {cn}.
Define Ve @{H2) by (Vf)(z) = zf(z). Show that V is the Cayley transform of the
symmetric operator Tin H2, given by
Find the ranges of Г+ //and of T— il\ show that one is H2 and one has codimension 1.
(Compare with Example 13.21.)
10 With H2 as in Exercise 9, define Know by
(Vf)(z)=zf(z2).
Show that Fis an isometry which is the Cayley transform of a closed symmetric operator
T in H2, whose deficiency indices are 0 and oo.
// Prove part (c) of Lemma 13.18. /
12 (a) In the context of Theorem 13.24, how are the operators T(/+#) and T(/) +
related?
(b) If/and g are measurable and g is bounded, prove that Щд) maps <3f into
(c) Prove that T(/) = Г(#) if and only if/ = g a.e. [£*], that is, if and only if
13 Is the operator С that occurs in the proof of Theorem 13.33 normal ?
14 Prove that every normal operator N in //, bounded or not, has a polar decomposition
N=UP=PU,
where U is unitary, P is self-adjoint, P > 0. Moreover, <${P) =

UNBOUNDED OPERATORS 365
15 Prove the following extension of Theorem 12.16: If Te @{H\ if M and N are normal
operators in H9 and if TM <=■ NT, then also TM* с N*T.
16 Suppose ris a closed operator in Я, ^(Г) = ^(Г*), and ||7*|| = ||r*x||foreveryxe @(T)
Prove that T is normal. Hint: Begin by proving that
(Tx9 Ту) = (T*x, T*y) (x e @(T\ у е 9{T)).
17 Prove that the spectrum a(T) of any operator Tin Я is a closed subset of <?. (See Defini-
Definition 13.26.) Hint: If ST 5= TS = /, and S e @{H\ then S{I - \S)~x is a bounded inverse
of 7-A/, for small |A|.
75 Put ^(/) = exp (-t2). Define S e ^(£2), where L2 = L\R\ by
E/)@=^@/a-l) (/el2),
so that (S2f)(t) = <f>(t)(f)(t - \)f{t ~ 2), etc. (Note that S is presented in its polar de-
decomposition S = PU.)
Find S*. Compute that
( (n - \)n(n + 1))
Ц5-Ц = exp I ^ ^J (n = 1,2,3,...).
Conclude that S is one-to-one, that 0t{S) is dense in L2, and that o{S) = {0}. Define T,
with domain Q)(T) =M(S\ by
TSf = f (/gL2).
Prove that a(T) is empty.
19 Let Ji, Г2, Тъ be as in Example 13.4, put
*) = {/4=0G-0: ДО) = 0},
and define J4/= i/' for all/e
Prove the following assertions.
(a) Every Л e (I is in the point spectrum of 7i.
(b) o(T2) consists of the numbers 2ттпу where n runs through the integers; each of these
is in the point spectrum of T2.
(c) 01{Тъ — A/)hascodimension 1 for every Л е <£. Hence о{Тъ) = С The point spectrum
of Тъ is empty.
(d) a(TA) is empty.
Hint: Study the differential equation if — Xf = g.
This illustrates how sensitive the spectrum of a differential operator is to its domain
(in this case, to the boundary conditions that are imposed).
20 Show that every nonempty closed subset of С is the spectrum of some normal operator
in H (if dim H = oo).
21 Define unitary operators Q(t) e ®{L2\ where L2 = L2(R) by
№t)fKs)=f(s+t).

366 BANACH ALGEBRAS AND SPECTRAL THEORY
Show that {Q(t)} satisfies the conditions imposed in Definition 13.34. Find the infinitesi-
infinitesimal generator of {Q(t)}. Include a precise description of its domain, and describe its
spectrum. (The Plancherel theorem can be used here.)
Show that the exponential representation of {G@}> given by Theorem 13.37, is
(formally) the Taylor expansion that is familiar from elementary calculus.
22 If /g Я2 (see Exercise 9) and f(z) = 2 cn z", define
lQ(t)f](z) = I (n + l)-'cz» @ < / < oo).
n = 0
Show that each Q{t) is self-adjoint (and positive). Find the infinitesimal generator A of
the semigroup {Q(t)}. Is A self-adjoint? Show that A has pure point spectrum, at the
points log 1, log A/2), log A/3),....
23 For /g L2(R\ xeR,0<y<oo, define
and put Q(O)f=f. Show that {Q{y)\ 0 <y < oo} satisfies the conditions imposed in
Definition 13.34 and that HGOOII = 1 for all y.
[The integral represents a harmonic function in the upper half-plane, with bound-
boundary values /. The semigroup property of {Q(y)} can be deduced from this, as well as
from a look at the Fourier transforms of the functions Q(y)f.]
Find the domain of the infinitesimal generator A of {GOO}» and prove that
Af= -Hf\
where H is the Hilbert transform (Chapter 7, Exercise 24).
Prove that — A is positive and self-adjoint.

A
APPENDIX
COMPACTNESS AND CONTINUITY
A1 Partially ordered sets A set & is said to be partially ordered by a binary relation
<if:
(/) a < b and b < с implies a < c,
(//) a < a for every ae&9
(in) a <b and b <a implies a = b.
A subset п of a partially ordered set ^ is said to be totally ordered if every pair a,
beй satisfies either a <b or b <a.
Hausdorff's maximality theorem states :
Every nonempty partially ordered set 0 contains a totally ordered subset й which is
maximal with respect to the property of being totally ordered.
A proof (using the axiom of choice) may be found in [23]. Explicit applications of the
theorem occur in the proofs of the Hahn-Banach theorem, of the Krein-Milman theorem,
and of the theorem that every proper ideal in a commutative ring with unit lies in a maximal
ideal. It will now be applied once more (A2) to prepare the way to an easy proof of the
Tychonoff theorem.

368 APPENDIX A
A 2 Subbases A collection £f of open subsets of a topological space X is said to be
a subbase for the topology r of X if the collection of all finite intersections of members of £?
forms a base for r. (See Section 1.5.) Any subcollection of Sf whose union is X will be called
an <9*-cover of X. By definition, Xis compact provided that every open cover of Xhas a
finite subcover. It is enough to verify this property for ^-covers:
Alexander's subbase theorem If У is a subbase for the topology of a space X, and
if every fif-cover of X has a finite subcover, then X is compact.
proof Assume Х\ъ not compact. We will deduce from this that Xhas an ^-cover Г
without finite subcover.
Let 0 be the collection of all open covers of X that have no finite subcover. By
assumption, 0* Ф 0. Partially order 0* by inclusion, let п be a maximal totally ordered
subcollection of 0, and let Г be the union of all members of П. Then
(a) Г is an open cover of X,
(b) Г has no finite subcover, but
(c) Ги{V] has a finite subcover, for every open V$ Г.
Of these, (a) is obvious. Since Q is totally ordered, any finite subfamily of Г lies
in some member of Q, hence cannot cover X; this gives (b), and (c) follows from the
maximality of Q.
Put Г = ГпУ. Since Г с: Г, (b) implies that f has no finite subcover. To
complete the proof, we show that f covers X.
If not, some x e X is not covered by f. By (a\ xe W for some WeY. Since
Sf is a subbase, there are sets Vi, ..., Vn e 9> such that x e f] Vt <= W. Since x is
not covered by f, no Vt belongs to Г. Hence (c) implies that there are sets Yt, ...,
Yn, each a finite union of members of Г, such that X = Vt и Yt for 1 < i < n. Hence
X= Fju-u Г„иA Vi с Y i u • • • и YnvW,
i=i
which contradicts (b). ////
A 3 Tychonoff's theorem If X is the cartesian product of any nonempty collection of
compact spaces Xa, then X is compact.
proof If тта(х) denotes the ^-coordinate of a point xe X, then, by definition, the
topology of Xis the weakest one that makes each ттл\ Х-> Ха continuous; see Section
3.8. Let £fa be the collection of all sets 7ra~ \V^9 where Va is any open subset of Xx.
If Sf is the union of all <9*a, it follows that £f is a subbase for the topology of X.
Suppose Г is an ^-cover of X. Put Га = Г n £f a. Assume (to get a contra-
contradiction) that no Га covers X. Then there corresponds to each a a point x* e Xa such
that Га covers no point of the set ттг 1(xa\ and if x e X is chosen so that тга(х) = xa,
then x is not covered by Г. But Г is a cover of X.
Hence at least one I\ covers X. Since Xa is compact, some finite subcollection
of Га covers X. Since Tx а г, Г has a finite subcover, and now Alexander's theorem
implies that Xis compact. ////

COMPACTNESS AND CONTINUITY 369
A 4 Theorem If К is a closed subset of a complete metric space X, then the following
three properties are equivalent:
(a) К is compact.
(b) Every infinite subset of К has a limit point in K.
(c) К is totally bounded.
Recall that (c) means that К can be covered by finitely many balls of radius e, for
every s > 0.
proof Assume (a). If E <= К is infinite and no point of К is a \\m\t point of E, there
is an open cover {Уя} of К such that each V* contains at most one point of E. There-
Therefore {Va} has no finite subcover, a contradiction. Thus (a) implies (b).
Assume (/>), fix e > 0, and let d be the metric of X. Pick xte K. Suppose
xt, ..., xn are chosen in К so that d{x<, xj) >e if / ^ j. If possible, choose xn+i e К
so that d(xi9 xn + i)>e for 1 </<«. This process must stop after a finite number
of steps, because of (b). The e-balls centered at xt,..., xn then cover K. Thus (b)
implies (c).
Assume (c), let Г be an open cover of K, and suppose (to reach a contradiction)
that no finite subcollection of Г covers K. By (с), К is a union of finitely many closed
sets of diameter < 1. One of these, say K\, cannot be covered by finitely many members
of Г. Do the same with Ki in place of K, and continue. The result is a sequence of
closed sets Kt such that
(/) K^ Кi => K2^ • • •,
(/V) diam Kn < 1 //2, and
(///) no Kn can be covered by finitely many members of Г.
Choose xn e Kn. By (/) and (/7), {х„} is a Cauchy sequence which (since X is
complete and each К„ is closed) converges to a point x e f] Kn. Hence x e V for some
Ve Г. By (//), АГЯ с К when /г is sufficiently large. This contradicts (/77). Thus (c)
implies (a). Ill I
Note that the completeness of X was used only in going from (c) to (a). In fact, (a) and (b)
are equivalent in any metric space.
A 5 Ascoli's theorem Suppose X is a compact space; C(Jf) w the sup-normed Banach
space of all continuous complex functions on X, and Ф c: C(X) is pointwise bounded and equi-
continuous. More explicitly.
(a) sup {\f(x) I :/g Ф} < 00 ybr £геги л: е Jf, яга/
(/)) // e > 0, £*;<?ry л: е X te a neighborhood V such that \f(y) — /(x) | < e /<?r я// д> е
/or tf///e Ф.
77z£7* Ф /j tota//>> bounded in C(X).
Corollary Since C(X) is complete, the closure of Ф is compact, and every sequence
in Ф contains a uniformly convergent subsequence.

370 APPENDIX A
proof Fix e > 0. Since X is compact, (b) shows that there are points xi, . .\ , xn e X,
with neighborhoods Vt, ..., Va, such that X = (J Vt and such that
A) |/«-/(л:,)| <е (/еФ,хе
If (a) is applied to Xi, ..., xn in place of x, it follows from A) that Ф is uniformly
bounded:
B) sup {|/U)|: xe Х,/еФ}=М<оо.
Put D ={Xg (?:\\\ < M}, and associate to each/ e Ф apoint/?(/)e Dn с <£", by setting
C)
Since /)" is a finite union of sets of diameter < e, there exist /i, ...,/,„ e Ф such that
every /?(/) lies within e of some /?(/*).
If/e Ф, there exists k,\ <k <m, such that
D) |/(*,)-/*(*,)|<£ (\<
Every x e X lies in some Vt, and for this /
E) \f(x) - f(xd \<e and \fk(x) - fk(x,) \ < e.
Thus | f(x) -fk(x)| < 3e for every xe X.
The Зг-balls centered at/ , ... ,/fc therefore cover Ф. Since e was arbitrary, Ф
is totally bounded. ////
A6 Sequential continuity If X and Y are HausdorfT spaces and if /maps X into У,
then /is said to be sequentially continuous provided that Итп_оо f(xn) =f(x) for every se-
sequence^} in X that satisfies limn-oo xn = x.
Theorem
(a) Iff: X-> Y is continuous, then f is sequentially continuous.
(b) Iff: X —> Y is sequentially continuous, and if every point of X has a countable local
base {in particular, if X is metrizable\ then f is continuous.
proof (a) Suppose xn -> x m X, V is a neighborhood of f(x) in Y, and U =
f~ l(V). Since/is continuous, U is a neighborhood of x, and therefore х„е U for all
but finitely many n. For these n9f(xn) e V. Thus/Or,,) ->/(*) as n -> oo.
(b) Fix л: e X, let {t/n} be a countable local base for the topology of X at x, and
assume that/is not continuous at x. Then there is a neighborhood Fof/(x) in У
such that/" !(K) is not a neighborhood of x. Hence there is a sequence xn9 such that
xne Un, хЛ -> x as л -> oo, and х„ ^/~ ЧЮ- Thus/(xn) ^ K, so that /is not sequentially
continuous. ////
A 7 Totally disconnected compact spaces A topological space X is said to be totally
disconnected if none of its connected subsets contains more than one point.

COMPACTNESS AND CONTINUITY 371
A set E с X is said to be connected if there exists no pair of open sets Vi, V2 such that
hutEnV1nV2=0.
Theorem Suppose K^ V с X, where X is a compact Hausdorff space, V is open, and
К is a component ofX. Then there is a compact open set A such that К <= A c: V.
Corollary If X is a totally disconnected compact Hausdorff space, then the com-
compact open subsets of X form a base for its topology.
proof J.et Г be the collection of all compact open subsets of X that contain K.
Since X e T, Г Ф 0. Let H be the intersection of all members of Г.
Suppose H' <^W, where W is open. The complements of the members of Г form
an open cover of the compact complement of W. Since Г is closed under finite inter-
intersections, it follows that A <=-W for some А еГ.
We claim that H is connected. To see this, assume H = Яои Hl9 where Ho
and Hi are disjoint compact sets. Since K^-H and К is connected, К lies in one of
these. Say Ic Ho. By Urysohn's lemma, there are disjoint open sets Wo, W\ such
that HQ с Wo , Hi^-Wi, and the preceding paragraph shows that some A e Г
satisfies A^Wq\jWi. Put Ao = A n Wo. Then K^ Ao, Ao is open, and Ao is
compact, because A n tV0 — A n Wo. Thus AoeT. Since H с Ао, it follows that
Thus H is connected. Since Ic ff and X is a component, we see that К = H.
The preceding argument, with К and V in place of # and W, shows now that A <= F
for some Л е Г. ////

в
APPENDIX
NOTES AND COMMENTS
The abstract tendency in analysis which developed into what is now known as functional
analysis began at the turn of the century with the work of Volterra, Fredholm, Hilbert,
Frechet, and F. Riesz, to mention only some of the principal figures. They studied integral
equations, eigenvalue problems, orthogonal expansions, and linear operations in general.
It is of course no accident that the Lebesgue integral was born in the same period.
The normed space axioms appear in F. Riesz' work on compact operators in С {[a, b])
(Acta Math., vol. 41, pp. 71-98, 1918), but the first abstract treatment of the subject is in
Banach's 1920 thesis (Fundam. Math., vol. 3, pp. 133-181, 1922). His book [2], published in
1932, was tremendously influential. It contains what is still the basic theory of Banach
spaces, but with some omissions which, from our vantage point, seem curious.
One of these is the complete absence of complex scalars, in spite of Wiener's observa-
observation (Fundam. Math., vol. 4, pp. 136-143, 1923) that the axioms can be formulated just as
well over <£, and, more importantly, that a theory of Banach-space-valued holomorphic
functions can then be developed whose basic features are very similar to the classical complex-
valued case. Very little (if anything) was done with this until 1938. (See the notes for
Chapter 3 in this appendix.)

NOTES AND COMMENTS 373
Even more puzzling, in retrospect, is Banach's treatment of weak convergence—surely
one of h:s most important contributions to the subject. In spite of the vigorous development
of topology in the twenties, and in spite of von Neumann's explicit description of weak
neighborhoods in a Hilbert space and inoperator algebras {Math. Ann., vol. 102, pp. 370-
427, 1930; see p. 379), Banach deals only with weakly convergent sequences. Since the
adjunction of all limits of weakly convergent subsequences of a set need not lead to a weakly
sequentially closed set (see Exercise 9, Chapter 3), he is forced into complicated notions such
as transfinite closures, but he never uses the much simpler and more satisfactory concept of
weak topologies.
Occasionally, unnecessary separability assumptions are made in [2]. This is also true
of von Neumann's axiomatization of Hilbert space {Math. Ann., vol. 102, pp. 49-131, 1930),
where separability is included among the defining properties. In this fundamental paper
on unbounded operators, he establishes the spectral theorem for them, thus generalizing
what Hilbert had done for the bounded ones more than 20 years earlier. Another basic
contribution to operator theory was M. H. Stone's 1932 book [28].
Although continuous functions obviously play an important role in Banach's book,
he considers only their vector space structure. They are never multiplied. But multiplication
was not neglected for very long. In his work on the tauberian theorem {Ann. Math., vol. 33,
pp. 1-100, 1932) Wiener stated and used the fact that the Banach space of absolutely con-
convergent Fourier series satisfies the multiplicative inequality \\xy\\ < \\x\\\\y\\. M. H. Stone's
generalization of the Weierstrass approximation theorem {Trans. Amer. Math. Soc, vol. 41,
pp. 375-481, 1937; especially pp. 453-481) is undoubtedly the best-known instance of the
explicit use of the ring structure of spaces of continuous functions. Von Neumann's interest
in operator theory, which stemmed from quantum mechanics, led him to a systematic study
of operator algebras. M. Nagumo {Jap. J. Math., vol. 13, pp. 61-80, 1936) initiated the
abstract study of normed rings. But what really got this subject off the ground was Gelfand's
discovery of the important role played by the maximal ideals of a commutative algebra
{Mat. Sbornik N. S., vol. 9, pp. 3-24, 1941) and his construction of what is now known as
the Gelfand transform.
Before the middle forties, the interest of functional analysts was focused almost
exclusively on normed spaces. The first major paper on the general theory of locally convex
spaces is that of J. Dieudonne and L. Schwartz in Ann. Inst. Fourier {Grenoble), vol. 1, pp.
61-101, 1949. One of its principal motivations was Schwartz' construction of the theory of
distributions [26]. (The first version of this book appeared in 1950.) Just as Banach and
Gelfand had predecessors, so did Schwartz. As Bochner points out in his review of Schwartz'
book {Bull. Amer. Math. Soc, vol. 58, pp. 78-85, 1952), the idea of "generalized functions "
goes back at least as far as Riemann. It was applied in Bochner's " Vorlesungen iiber Fourier-
sche Integrate" (Leipzig, 1932), a book that played a very important role in the develop-
development of harmonic analysis. Sobolev's work also predates Schwartz. But it was Schwartz
who built all this into a smoothly operating very general structure that turned out to have
many applications, especially to partial differential equations.
The following expository articles describe some of the history of our subject in greater
detail.

374 APPENDIX В
F. F. Bonsall: A Survey of Banach Algebra Theory, Bull. London Math. Soc, vol. 2, pp. 257-
274, 1970.
E. R. Lorch: The Structure of Normed Abelian Rings, Bull. Amer. Math. Soc, vol. 50, pp.
447-463, 1944.
Т. Н. Hildebrandt: Integration in Abstract Spaces, Bull. Amer. Math. Soc, vol. 59, pp.
111-139, 1953.
J. Horvath: An Introduction to Distributions, Amer. Math. Monthly, vol. 77, pp. 227-240,
1970.
F. Treves: Applications of Distributions to PDE Theory, Amer. Math. Monthly, vol. 77,
pp. 241-248, 1970.
A. E. Taylor: Notes on the History and Uses of Analyticity in Operator Theory, Amer.
Math. Monthly, vol. 78, pp. 331-342, 1971.
Volume 1 of the series " Studies in Mathematics " (published by the Mathematical
Association of America, 1962, edited by R. C. Buck) contains articles by
E. J. McShane: A Theory of Limits
M. H. Stone: A Generalized Weierstrass Approximation Theorem
E. R. Lorch: The Spectral Theorem
C. Goffman: Preliminaries to Functional Analysis
There are two special issues of Bull. Amer. Math. Soc: One (May 1958) is devoted to
the work of John von Neumann; the other (January 1966) to that of Norbert Wiener.
We now give detailed references to some items in the text.
Chapter 1
For the general theory of topological vector spaces, see [5], [14], [15], [31], [32].
Section 1.8 (e). In Banach's definition of an F-space, he postulated only the separate
continuity of scalar multiplication and proved that joint continuity was a consequence.
See [4], pp. 51-53, for a proof based on Baire's theorem. Another proof (due to S. Kakutani)
does not require completeness of X but uses Lebesgue measure in the scalar field; see [33],
pp. 31-32.
Theorem 1.24. This metrization theorem was first proved (in the more general context
of topological groups) by G. Birkhoff {Compositio Math., vol. 3, pp. 427-430, 1936) and by
S. Kakutani (Proc. Imp. Acad. Tokyo, vol. 12, pp. 128-142, 1936). Part (d) of the theorem is
perhaps new.
Section 1.33. The Minkowski functional of a convex set is sometimes called its support
function.
Theorem 1.39 is due to A. Kolmogoroff (Stadia Math., vol. 5, pp. 29-33, 1934). It
may well be the first theorem about locally convex spaces.
Section 1.46. The construction of the function g by repeated averaging may be found
on pp. 80-84 of S. Mandelbrojt's 1942 Rice Institute Pamphlet "Analytic Functions and
Classes of Infinitely Differentiable Functions," where it is credited to H. E. Bray.

NOTES AND COMMENTS 375
Section 1.47. Of particular interest among the F-spaces that are not locally convex
but have enough continuous linear functional to separate points are certain subspaces of
Lp, the #p-spaces (with 0 <p < 1). For a detailed study of these, see the paper by P. L.
Duren, B. W. Romberg, and A. L. Shields in /. Reine Angew. Math., vol. 238, pp. 32-60,
1969, and those by Duren and Shields in Trans. Amer. Math. Soc, vol. 141, pp. 255-262,
1969, and in Рас J. Math., vol. 32, pp. 69-78, 1970.
Chapter 2
Basically, all results of this chapter are in [2].
Exercise 11. If X, Y,Z are Banach spaces, and if В is a continuous bilinear mapping
of X x Y onto Z, it does not seem to be known whether В must be open at @, 0).
Exercise 13. A barrel is a elosed, convex, balanced, absorbing set. A space is barreled
if every barrel contains a neighborhood of 0. Exercise 13 asserts: Topological vector spaces
of the second category are barreled. There exist barreled spaces of the first category, and
certain versions of the Banach-Steinhaus theorem are valid for them. See [14], p. 104;
also [15]. Barreled spaces with the Heine-Borel property are often called Montel spaces; see
Sec. 1.45.
Chapter 3 .
Theorem 3.2 is in [2]. Its complex version, Theorem 3.3, was proved by H. F.
Bohnenblust and A. Sobczyk, Bull. Amer. Math. Soc, vol. 44, pp. 91-93, 1938, and by G. A.
Soukhomlinoff, Mat. Sbornik, vol. 3, pp. 353-358, 1938.
Theorem 3.6. For a partial converse, see J. H. Shapiro, Duke Math. J., vol. 37, pp.
639-645, 1970.
Theorem 3.15. See L. AlaogJu, Ann. Math., vol. 41, pp. 252-267, 1940. For separable
Banach spaces, the theorem is in [2], p. 123.
Theorem 3.18. A shorter proof based on seminorms, may be found on p. 223 of
[32].
Theorem 3.21 was proved, for weak *-compact convex subsets of the dual of a Banach
space, by M. Krein and D. Milman, in Studia Math., vol. 9, pp. 133-138, 1940.
The history of vector-valued integration is described by Т. Н. Hildebrandt in Bull.
Amer. Math. Soc, vol. 59, pp. 111-139, 1953. The "weak" integral of Definition 3.26 was
developed by B. J. Pettis, Trans. Amer. Math. Soc, vol. 44, pp. 277-304, 1938.
The history of vector-valued holomorphic functions is described by A. E. Taylor in
Amer. Math. Monthly, vol. 78, pp. 331-342, 1971.
Theorem 3.31. That weakly holomorphic functions (with values in a complex Banach
space) are strongly holomorphic was proved by N. Dunford in Trans. Amer. Math. Soc,
vol. 44, pp. 304-356, 1938.
Theorem 3.32 was used by A. E. Taylor to prove that the spectrum of every bounded
linear operator on a complex Banach space is nonempty {Bull. Amer. Math. Soc, vol. 44,
pp. 70-74, 1938). Since every Banach algebra A is isomorphic to a subalgebra of &{A) (see
the proof of Theorem 10.2), Taylor's result contains (a) of Theorem 10.13.

376 APPENDIX В
Exercise 9 is due to von Neumann, Math. Ann., vol. 102, pp. 370-427, 1930; see
p. 380.
Exercise 10 is patterned after a construction in the Appendix of [2].
Exercise 25. If К is also separable and metric, then suchua ^ exists even on E, rather
than on E. This is Choquet's theorem. See [20]. For a recent paper on this, see R. D.
Bourgin, Trans. Amer. Math. Soc, vol. 154, pp. 323-340, 1971.
Exercise 28 (c). This is the easy part of the Eberlein-Smulian theorem. See [4], pp.
430-433 and p. 466. Another characterization of weak compactness has been given by R.
C. James, Trans. Amer. Math. Soc, vol. 113, pp. 129-140, 1964: A weakly closed set S in a
Banach space X is weakly compact if and only if every x* e X* attains its supremum on S.
Chapter 4
A large part of this chapter is in [2].
Compact operators used to be called completely continuous. As defined by Hilbert (in €2)
this means that weakly convergent sequences are mapped to strongly convergent ones. The
presently used definition was given by F. Riesz (Acta Math., vol. 41, pp. 71-98, 1918). In
reflexive spaces, the two definitions coincide (Exercise 18).
Section 4.5. R. C. James has constructed a nonreflexive Banach space X which is
isometrically isomorphic with X** (Proc. Natl. Acad. Sci. USA, vol. 37, pp. 174-177, 1951).
Theorems 4.19 and 4.25 were proved by J. Schauder (Studia Math., vol. 2, pp. 183—
196, 1930). For generalizations to arbitrary topological vector spaces, see J. H. Williamson,
J. London Math. Soc, vol. 29, pp. 149-156, 1954; also [5], chap 9.
Exercise 15. These operators are usually called Hilbert-Schmldt operators. See [4],
chap. XI.
Exercise 17. Operators of this type are discussed by A. Brown, P. R. Halmos, and
A. L. Shields in Acta Sci. Math. Szeged., vol. 26, pp. 125-137, 1965.
Exercise 19. This "max-min duality" was exploited by W. W. Rogosinski and H. S.
Shapiro to obtain very detailed information about certain extremum problems for holomorphic
functions. See Acta Math., vol. 90, pp. 287-318, 1953.
Exercise 21. This was proved by M. Krein and V. Smulian in Ann. Math., vol 41,
pp. 556-583, 1940. See also [4], pp. 427-429.
Chapter 5
Theorem 5.1. For a more general version, see R. E. Edwards, J. London Math. Soc,
vol. 32, pp. 499-501, 1957.
Theorem 5.2 is due to A. Grothendieck, Can. J. Math., vol. 6, pp. 158-160, 1954. His
proof is less elementary than the one given here.
Theorem 5.3. For more on trigonometric series with gaps, see J. Math. Mech., vol. 9,
pp, 203-228, 1960; also, sec. 5.7 of [24], and J. P. Kahane's article in Bull. Amer. Math. Soc,
vol. 70, pp. 199-213, 1964.

NOTES AND COMMENTS 377
Theorem 5.5 was first proved by A. Liapounoff, Bull. Acad. Sci. USSR, vol. 4, pp.
465-478, 1940. The proof of the text is due to J. Lindenstrauss, /. Math. Mech., vol. 15,
pp. 971-972, 1966. J. J. Uhl (Proc. Amer. Math. Soc, vol. 23, pp. 158-163, 1969) generalized
the theorem to measures whose values lie in a reflexive Banach space or in a separable dual
space.
Theorem 5.7. The idea to use Krein-Milman to prove Stone-Weierstrass is due to L.
de Branges, Proc. Amer. Math. Soc, vol. 10, pp. 822-824, 1959. E. Bishop's generalization
is in Рас. J. Math., vol. 11, pp. 777-783, 1961. The proof given here is that of I. Glicksberg,
Trans. Amer. Math. Soc, vol. 105, pp. 415-435, 1962.
Theorem 5.9. Bishop proved this in Proc. Amer. Math. Soc, vol. 13, pp. 140-143,
1962. For the special case of the disc algebra, sec Proc. Amer. Math. Soc, vol. 7, pp. 808-811,
1956, and L. Carleson's paper in Math Z., vol. 66, pp. 447-451, 1957. Other applications
occur in chap. 6 of [25].
Theorem 5.10. The proof follows that of M. Heins, Ann. Math., vol. 52, pp. 568-573,
1950, where the same method is applied to a large class of interpolation problems.
Theorem 5.11 was proveS by S. Kakutani in Proc. Imp. Acad. Tokyo, vol. 14, pp. 242-
245, 1938.
Theorem 5.14. This simple construction of the Haar measure of a compact group is
essentially that of von Neumann {Compositio Math., vol. 1, pp. 106-114, 1934). His is even
more elementary and self-contained, though a little longer, since he uses no fixed point
theorem. (In Trans. Amer. Math. Soc, vol. 36, pp. 445-492, 1934, he uses the same method
to construct mean values of almost periodic functions.) If compactness is replaced by local
compactness, the construction of Haar measure becomes more difficult. See [18], [11], [16].
Theorem 5.18 was proved (for Banach spaces) in Proc. Amer. Math. Soc, vol. 13,
pp. 429-432, 1962. For further results on uncomplemented subspaces, see H. P. Rosenthal's
1966 AMS Memoir "Projections onto Translation-invariant Subspaces of LP(G)" and his
paper in Acta Math., vol. 124, pp. 205-248,1970. There are also positive results. For example
Co is complemented in any separable Banach space which contains it (isomorphically) as a
closed subspace. A very short proof of this theorem of A. Sobczyk was recently obtained
by W. A. Veech in Proc. Amer. Math. Soc, vol. 28, pp. 627-628, 1971.
Chapter 6
The standard reference is, of course, [26]. See also [5], [8], [27], [31]. [13] contains a
very concise introduction to the subject.
Definition 6.3. <3>{€l) is here topologized as the inductive limit of the Frechet spaces
^*(£ty. See [15], pp. 217-225, for a systematic discussion of this notion in an abstract
setting.
Chapter 7
For those aspects of Fourier analysis that are related to distributions, we refer to [26]
and [13]. The group-theoretic aspects of the subject are discussed in [11] and [24]. The
standard work on Fourier series is [34].

378 APPENDIX В
Theorem 7.4. The intimate relation between Fourier transforms and differentiation
is no accident; Fourier series were invented, in the eighteenth century, as tools to solve
differential equations.
Theorem 7.5 is sometimes called the Riemann-Lebesgue lemma.
Theorem 7.9 was originally proved by M. Plancherel in Rend. Palermo., vol. 30, pp.
289-335, 1910.
Theorems 7.22 and 7.23. TrTese proofs are as in [13] but contain more details.
Theorem 7.25 is due to S. L. Sobolev, Mat. Sbornik, vol. 4, pp. 471-497, 1938.
Exercise 16. This is taken from L. Schwartz' first counterexample to the spectral
synthesis problem (C. R. Acad. Sci. Paris, vol. 227, pp. 424-426, 1948). For further informa-
information on this problem, see C. S. Herz {Trans. Amer. Math. Soc, vol. 94, pp. 181-232, 1960)
and chap. 7 of [24].
Exercise 17. See С S. Herz, Ann. Math., vol. 68, pp. 709-712, 1958.
Chapter 8
General references: [1], [13], [27], [30].
The existence of fundamental solutions (Theorem 8.5) was established independently
by L. Ehrenpreis (Amer. J. Math., vol. 76, pp. 883-903, 1954) and by B. Malgrange in his
thesis (Ann. Inst. Fourier, vol. 6, pp. 271-355, 1955-1956). Lemma 8.3 is Malgrange's. He
proves it for Fourier transforms/of test functions. He integrates over a ball where we have
used a torus. As far as applications are concerned, this makes hardly any difference. The
point is to get some useful majorization of/by/P, that is, to have division by P under control.
Ehrenpreis solved this division problem in a different way and went on to solve more general
division problems of this type. See [13] and [30] for further references and more detailed
results.
It is essential in Theorem 8.5 that the coefficients of the differential operator under
consideration be constant. This follows from an equation constructed by H. Lewy (Ann.
Math., vol. 66, pp. 155-158, 1957), which has C00 coefficients but no solution. Hormander
([13], chap. VI) has investigated this nonexistence phenomenon very completely.
Section 8.8. Many other types of Sobolev spaces have been studied. See [13], chap. II.
Theorem 8.12. See K. O. Friedrichs, Comm. Pure Appl. Math., vol. 6, pp. 299-325,
1953, and P. D. Lax, Comm. Pure Appl. Math., vol. 8, pp. 615-633, 1955. Lax treats the
periodic case first, via Fourier series, and then uses the bootstrap proposition to obtain the
general case. He does not assume that the highest-order terms are constant. See also [4],
pp. 1703-1708.
Exercise 10. G is the so-called " Green's function " of P(D).
Exercise 16. This is a theorem about zero sets of homogeneous polynomials (with
complex coefficients) in Rn. See [1], p. 46.
Chapter 9
Section 9.1. See A. Tauber, Monatsh. Math., vol. 8, pp. 273-277, 1897, and J. E.
Littlewood, Proc. London Math. Soc, vol. 9, pp. 434-448, 1910.

NOTES AND COMMENTS 379
Theorem 9.3. The use of distributions in this proof is as in J. Korevaar's paper in
Proc. Amer. Math. Soc, vol. 16, pp. 353-355, 1965.
Theorem 9.4 to Theorem 9.7. N. Wiener, Ann. Math., vol. 33, pp. 1-100, 1932, and
H. R. Pitt, Proc. London Math. Soc, vol. 44, pp. 243-288, 1938. Later proofs gave various
generalizations; see [24], p. 159, for further references. See also A. Beurling, Ada Math.,
vol. 77, pp. 127-136, 1945.
Section 9.9. The prime number theorem was first proved, independently, by J.
Hadamard (Bull. Soc. Math. France, vol. 24, pp. 199-220, 1896) and by Ch. J. de la Vallee-
Poussin (Ann. Soc. Sci. Bruxelles, vol. 20, pp. 183-256, 1896). Both used complex variable
methods. Wiener gave the first tauberian proof, as an application of his general theorem.
" Elementary " proofs were found in 1949 by A. Selberg and by P. Erdos. For a simpler
elementary proof, see N. Levinson, Amer. Math. Monthly, vol. 76, pp. 225-245, 1969.
The complex variable proofs still give the best error estimates; see W. J. Le Veque, "Topics
in Number Theory," vol. II, p. 251, Addison-Wesley Publishing Company, Inc., Reading,
Mass., 1956.
Theorem 9.12. A. E. Ingham, /. London Math. Soc, vol. 20, pp. 171-180, 1945.
The material on the renewal equation is from S. Karlin, Рас. J. Math., vol. 5, pp. 229-
257, 1955, where references to earlier work may be found. Nonlinear versions of the renewal
equation are discussed by J. Chover and P. Ney in /. d'Analyse Math., vol. 21, pp. 381-413,
1968; see also 33. Henry, Duke Math. J., vol. 36, pp. 547-558, 1969.
Exercise 7. This approximation problem is much less delicate in L2. See [23], sec. 9.16.
Chapter 10
General references: [7], [12], [16], [19], [21]. In [16] and [21], a great deal of basic
theory is developed without assuming the presence of a unit. [21] contains some material
about real algebras.
Gelfand's paper (Mat. Sbornik, vol. 9, pp. 3-24, 1941) contains Theorems 10.2, 10.13,
and 10.14, some symbolic calculus, and Theorem 11.9. For Fourier transforms of measures,
the spectral radius formula (b) of Theorem 10.13 had been obtained earlier by A. Beurling
(Proc. IX Congres de Math. Scandinaves, Helsingfors, pp. 345-366, 1938). See also the note
to Theorem 3.32
Theorem 10.9. The commutative case was obtained independently by A. M.Gleason (/.
Anal. Math., vol. 19, pp. 171-172, 1967) and by J. P. Kahane and W. Zelazko (Studia Math.,
vol. 29, pp. 339-343, 1968). W. Zelazko (Studia Math., vol. 30, pp. 83-85, 1968) removed the
commutativity hypothesis. The proof given in the text contains some simplifications. See
also Theorem 1.4.4 of [3], and J. A. Siddiqi, Can. Math. Bull., vol. 13, pp. 219-220, 1970.
Theorem 10.19. H. A. Seid (Amer. Math. Monthly, vol. 77, pp. 282-283, 1970) obtains
the same conclusions, without assuming that A has a unit, if M = 1.
Theorem 10.20 says that a(x) is an upper semicontinuous function of x. An example
of Kakutani ([21], p. 282) shows that a(x) is not, in general, a continuous function of x. See
also Exercise 20.

380 APPENDIX В
Section 10.21. The terms operational calculus ox functional calculus are also frequently
used. [12] contains a very thorough treatment of the symbolic calculus in Banach algebras.
Theorem 10.38. These differentiation formulas may be new.
Theorem 10.42. N. M. Riviere showed me this proof, for the case of the exponential
function acting on the algebra of all complex 2-by-2 matrices.
Section 10.43. These results were found by E. Hille {Math. Ann., vol. 136, pp. 46-57,
1958), as was Exercise 12.
Theorem 10.44 (d) is due to E. R. Lorch (Trans. Amer. Math. Soc, vol. 52, pp. 238-
248, 1942).
Exercise 16. This is one of the simplest cases of the Arens-Royden theorem for com-
commutative Banach algebras. It relates the group GjGi to the topological structure of the
maximal ideal space of A. See H. L. Royden's article in Bull. Amer. Math. Soc, vol. 69,
pp. 281-298, 1963, and that by R. Arens in F. T. Birtel, ed., "Function Algebras," pp. 164-
168, Scott, Foresman and Company, Glenview, 111., 1966.
Exercise 17. For the precise structure of GjGi in this case, see J. L. Taylor, Ada Math.,
vol. 126, pp. 195-225, 1971.
Exercise 25. See C. Le Page, C. R. Acad. Scl Paris, vol. 265, pp. A235-A237, 1967.
Chapter 11
Theorem 11.7. The case n = 1 was proved in elementary fashion by P. J. Cohen in
Proc. Amer. Math. Soc, vol. 12, pp. 159-163, 1961. For n > 1, the proof of the text seems to
be the only one that is known.
Theorem 11.9. When A has no unit, then Д is locally compact (but not compact) and
^cCo(A); the origin of A* is then in the closure of A. See [16], pp. 52-53.
Example 11.13 (d) shows why there are very close relations between commutative
Banach algebras, on the one hand, and holomorphic functions of several complex variables
on the other. This topic is not at all pursued in the present book. Very good, up-to-date
accounts of it may be found in the books by Browder [3], Gamelin [6], and Stout [29]. A
symbolic calculus for functions of several Banach algebra elements can be developed. See
R. Arens and A. P. Calderon, Ann. Math., vol. 62, pp. 204-216, 1955, and J. L. Taylor, Acta
Math., vol. 125, pp. 1-38, 1970.
Example 11.13 (e) shows why certain parts of Fourier analysis may be derived easily
from the theory of Banach algebras. This is done in [16] and [24].
Theorem 11.18 was proved by Gelfand and Naimark in Mat. Sbornik, vol. 12, pp.
197-213, 1943. In the same paper they also proved that every J?*-algebra A (commutative
or not) is isometrically *-isomorphic to an algebra of bounded operators on some Hilbert
space (Theorem 12.41), if e + x*x is invertible for every x e A. That this additional hypoth-
hypothesis is redundant was proved 15 years later by I. Kaplansky [(/) of Theorem 11.28]. See
[21], p. 248, for references to the rather tangled history of this theorem. B. J. Glickfeld (///.
J. Math., vol. 10, pp. 547-556, 1966) has shown that A is a J?*-algebra if ||exp (ijc)|| = 1 for
every hermitian x e A.

NOTES AND COMMENTS 381
Theorem 11.20. The idea to pass from A to AjR, in order to prove the theorem without
assuming the involution to be continuous, is due to J. W. M. Ford (/. London Math. Soc,
vol. 42, pp. 521-522, 1967).
Theorem 11.23. See R. S. Foguel, Ark. Mat., vol. 3, pp. 449-461, 1957.
Theorem 11.25. See P. Civin and B. Yood, Рас. J. Math., vol. 9, pp. 415-436, 1959;
especially p. 420. Also [21], p. 182.
Theorem 11.28. A recent treatment of these matters was given by V. Ptak, Bull. London
Math. Soc, vol. 2, pp. 327-334, 1970. Also, see the note to Theorem 11.18.
Theorem 11.31. See [19], [21]. H. F. Bohnenblust and S. Karlin (Ann. Math., vol. 62,
pp. 217-229, 1955) have found relations between positive functionals, on the one hand,
and the geometry of the unit ball of a Banach algebra on the other.
Theorem 11.32. See [7]. Also [16], p. 97, and [21], p. 230.
Theorem 11.33 is in [20], for continuous involutions.
Exercise 13. Part (g) contradicts the second half of corollary D.5.3) in [21]. It also
affects Theorem D.8.16) of [21].
Exercise 14. This was first proved by S. Bochner (Math. Ann., vol. 108, pp. 378-410,
1933; especially p. 407), using essentially the same machinery that we used in Theorem 7.7.
See [24] for a somewhat different proof. The proof that is suggested here shows that the
presence or absence of a unit element makes a difference in studying positive functionals.
See [16], p. 96,'and [21], p. 219.
Chapter 12
General references: [4], [9], [10], [17], [22].
Theorem 12.16. B. Fuglede proved the case M = N in Proc. Natl. Acad. Sci. USA,
vol. 36, pp. 35-40, 1950, including the unbounded case (Chapter 13, Exercise 15). His proof
used the spectral theorem and was extended to the case M ф N by C. R. Putnam (Amer. J.
Math., vol. 73, pp. 357-362, 1951), who also obtained Theorem 12.36. The short proof of
the text is due to M. Rosenblum, J. London Math. Soc, vol. 33, pp. 376-377, 1958.
Theorem 12.22. The extension process that is used here to go from continuous func-
functions to bounded ones is as in [16], pp. 93-94.
Theorem 12.23. See [4], pp. 926-936, for historical remarks about the spectral theorem.
See also P. R. Halmos' article in Amer. Math. Monthly, vol. 70, pp. 241-247, 1963, for a
different description of the spectral theorem.
Section 12.27. Л Aronszajn and К. Т, Smith (Ann. Math., vol. 60, pp. 345-350,
1954) proved that every compact operator on a Banach space has a proper invariant sub-
space. A. R. Bernstein and A. Robinson (Рас. J. Math., vol. 16, pp. 421-431, 1966) obtained
the same conclusion for bounded operators Г on a Hilbert space that have p(T) compact
for some polynomial p. Their proof uses nonstandard analysis; P. R. Halmos converted it
into one that uses only classical concepts (Рас. J. Math., vol. 16, pp. 433-437, 1966).
Theorem 12.38 was proved by P. R. Halmos, G. Lumer, and J. Schaffer, in Proc.
Amer. Math. Soc, vol. 4, pp. 142-149, 1953. D. Deckard and C. Pearcy (Ada ScL, Math.

382 APPENDIX В
Szeged., vol. 28, pp. 1-7, 1967) went further and proved that the range of the exponential
function is neither open nor closed in the group of invertible operators. Their paper contains
several references to intermediate results.
Theorem 12.39. See [21], p. 227.
Theorem 12.41. See the note to Theorem 11.18.
Exercise 2 is very familiar if TV = 4.
Exercise 18. The relation between shift operators and the invariant subspace problem
is discussed by P. R. Halmos in /. Reine Angew. Math., vol. 208, pp. 102-112, 1961.
Exercise 27. See P. Civin and B. Yood, Рас. J. Math., vol. 9, pp. 415-436, 1959, for
many results about involutions.
Exercise 32. Part (c) implies that every uniformly convex Banach space is reflexive.
See Exercise 1 of Chapter 4 and the note to Exercise 28 of Chapter 3. All //-spaces (with
1 <p < go) are uniformly convex. See J. A. Clarkson, Trans. Amer. Math. Soc, vol. 40,
pp. 396-414, 1936, or [15], pp. 355-359.
Chapter 13
General references: [4], [12], [22].
Theorem 13.6 was first proved by A. Wintner, Phys. Rev., vol. 71, pp. 738-739, 1947.
The more algebraic proof of the text is H. Wielandt's, Math. Ann., vol. 121, p. 21, 1949. It
was generalized by D. C. Kleinecke {Proc. Amer. Math. Soc, vol. 8, pp. 535-536, 1957), to
yield the following theorem about derivations: If D is a continuous linear operator in a
Banach algebra A such that D{xy) = xDy + {Dx)y for all x, у е A, then the spectral radius
of Dx is 0 for every x that commutes with Dx. See p. 20 of I. Kaplansky's article " Functional
Analysis" in "Some Aspects of Analysis and Probability," John Wiley & Sons, Inc.,New
York, 195&
A. Brown and С Pearcy (Ann. Math., vol. 82, pp. 112-127, 1965) have proved, for
separable H, that an operator Те &(H) is a commutator if and only if T is not of the form
Л/ + C, where Л Ф 0 and С is compact. See also C. Schneeberger, Proc. Amer. Math. Soc.,
vol. 28, pp. 464-472, 1971.
The Cayley transform, its relation to deficiency indices, and the proof of Theorem
13.30 are in von Neumann's paper in Math. Ann., vol. 102, pp. 49-131, 1929-1930, and
so is the spectral theorem for normal unbounded operators. The material on graphs is in his
paper in Ann. Math., vol. 33, pp. 294-310, 1932. Our proof of Theorem 13.33 is like that of
F. Riesz and E. R. Lorch, Trans. Amer. Math. Soc, vol. 39, pp. 331-340, 1936. See also
[4], chap. XII.
Definition 13.34. The continuity condition we impose can be weakened: If (a) and
(b) hold, and if Q(t)x -> x weakly, as t -> 0, for every x e X, then (c) holds. See [33], pp.
233-234. The proof uses more from the theory of vector-valued integration than the present
book contains.
Theorem 13.35 is proved in [4], [12], [22], [33].
Theorem 13.37. M. H. Stone, Ann. Math., vol. 33, pp. 643-648, 1932; B. Sz.-Nagy,
Math. Ann., vol. 112, pp. 286-296, 1936.

NOTES AND COMMENTS 383
Appendix A
Section A2. J. W. Alexander, Proc. Natl. Acad. Sci. USA, vol. 25, pp. 296-298,1939.
Section A3. A. Tychonoff proved this for cartesian products of intervals {Math. Ann.,
vol. 102, pp. 544-561, 1930) and used it to construct what is now known as the Cech (or
Stone-Cech) compact ification of a completely regular space. E. Cech {Ann. Math., vol. 38,
pp. 823-844, 1937; especially p. 830) proved the general case of the theorem and studied
properties of the compactification. Thus it appears that Cech proved the Tychonoff theorem,
whereas Tychonoff found the Cech compactification—a good illustration of the historical
reliability of mathematical nomenclature.

BIBLIOGRAPHY
1 agmon, s.: ''Lectures on Elliptic Boundary Value Problems," D. Van Nostrand
Company, Inc., Princeton, N.J., 1965.
2 banach, s.: "Theorie des Operations lineaires," Monografje Matematyczne, vol. 1,
Warsaw, 1932.
3 browder, a. : " Introduction to Function Algebras," W. A. Benjamin, Inc., New York,
1969.
4 dunford, n., and j. т. Schwartz: "Linear Operators," Interscience Publishers,
a division of John Wiley & Sons, Inc., New York, pt. I, 1958; pt. II, 1963.
5 edwards, r. e. : " Functional Analysis," Holt, Rinehart and Winston, Inc., New York,
1965.
6 gamelin, т. w.: " Uniform Algebras," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969.
7 gelfand, i. м., d. raikov, and G. e. shilov: " Commutative Normed Rings," Chelsea
'Publishing Company, New York, 1964. (Russian original, 1960.)
8 gelfand, i. м., and g. e. shilov: " Generalized Functions," Academic Press, Inc., New
York, 1964. (Russian original, 1958.)
9 halmos, p. R.: "Introduction to Hilbert Space and the Theory of Spectral Multi-
Multiplicity," Chelsea Publishing Company, New York, 1951.
10 halmos, p. r.: "A Hilbert Space Problem Book," D. Van Nostrand Company, Inc.,
Princeton, N.J., 1967.

BIBLIOGRAPHY 385
11 hewitt, e., and к. a. ross: "Abstract Harmonic Analysis," Springer-Verlag OHG,
Berlin, vol. 1, 1963; vol. 2, 1970.
12 hille, e., and r. s. Phillips, " Functional Analysis and Semigroups," Amer. Math. Soc.
Colloquium Publ. 31, Providence, R.I., 1957.
13 hormander, L.: "Linear Partial Differential Operators," Springer-Verlag OHG,
Berlin, 1963.
14 kelley, j. l., and i. namioka: "Linear Topological Spaces," D. Van Nostrand Com-
Company, Inc., Princeton, N.J., 1963.
15 kothe, g.: "Topological Vector Spaces, I," Springer-Verlag New York Inc., New
York, 1969.
16 loomis, l. н.: "An Introduction to Abstract Harmonic Analysis," D. Van Nostrand
Company, Inc., Princeton, N.J., 1953.
17 lorch, e. r.: "Spectral Theory," Oxford University Press, New York, 1962.
18 nachbin, l.: "The Haar Integral," D. Van Nostrand Company, Inc., Princeton,
N.J., 1965.
19 naimark, M. a. : "Normed Rings," Erven P. Noordhoff, Ltd., Groningen, Netherlands,
1960. (Original Russian edition, 1955.)
20 phelps, r. r.: "Lectures on Choquet's Theorem," D. Van Nostrand Company, Inc.,
Princeton, N.J., 1966.
21 rickart; c. e.: "General Theory of Banach Algebras," D, Van Nostrand Company,
Inc., Princeton, N.J., 1960.
22 riesz, f., and в. sz.-nagy, "Functional Analysis," Frederick Ungar Publishing Co.,
New York, 1955.
23 rudin, w.: " Real and Complex Analysis," McGraw-Hill Book Company, New York,
1966.
24 rudin, w.: "Fourier Analysis on Groups," Interscience Publishers, a division of
John Wiley & Sons, Inc., New York, 1962.
25 rudin, w.: "Function Theory in Polydiscs," W. A. Benjamin, Inc., New York, 1969.
26 Schwartz, l.: "Theorie des distributions," Hermann & Cie, Paris, 1966.
27 shilov, g. e.: "Generalized Functions and Partial Differential Equations," Gordon
and Breach, Science Publishers, Inc., New York, 1968. (Russian original, 1965.)
28 stone, м. н.: "Linear Transformations in Hilbert Space and Their Applications to
Analysis," Amer. Math. Soc. Colloquium Publ. 15, New York, 1932.
29 stout, e. l.: "The Theory of Uniform Algebras," Bogden and Quigley, Tarrytown,
N.Y., 1971.
30 treves, f. : " Linear Partial Differential Equations with Constant Coefficients," Gordon
and Breach, Science Publishers, Inc., New York, 1966.
31 treves, F.: "Topological Vector Spaces, Distributions, and Kernels," Academic
Press, Inc., New York, 1967.
32 wilansky, a. : " Functional Analysis," Blaisdell, New York, 1964.
33 yosida, к.: "Functional Analysis," Springer-Verlag New York Inc., New York, 1968.
34 zygmund, a. :" Trigonometric Series," 2d ed., Cambridge University Press, New York,
1959.

LIST OF SPECIAL SYMBOLS
The numbers that follow the symbols indicate the sections where their meanings are
explained.
Spaces
С(П)
С£
Rn
<Zn 1
X/N 1
U
С°°(П)
Lip a !
Л 1
t™ 1
z* :
L3
1.3
1.3
1.19
.19
1.40
1.43
1.46
1.46
Exercise 22, Chapter 1
Exercise 5, Chapter 2
Exercise 4, Chapter 3
U
xw
@{X, Y)
&{X)
X**
M1
*N
jV(T)
01{T)
Я1
Si
9{O)
Cn(Rn)
3.11
4.1
4.1
4.5
4.6, 12.4
4.6
4.11
4.11
5.19
6.1
6.2
6.7
7.3
7.5

LIST OF SPECIAL SYMBOLS 387
C(p)(Ot)
Hs
ЩАп)
A(U")
A
7.11
7.24
8.2
8.8
10.26
11.7
11.8
ra.dA
И
ПТ)
11.8
12.1
12.20
13.1
13.1
13.23
Operators
D*
T*
I
Rs
Is
Ts
sx
Af
A,
Tx
P(D)
1.46
4.10, 13.1
4.17
5.12
5.12
5.19
6.9
6.11
6.11 *
6.29
7.1
7.1
D-
A
д
"В
мх
(DF)a
Lx
Rx
cx
SL
sR
V
7.24
8.5
Exercise 8,
Exercise 8,
10.2
10.34
10.37
10.37
10.37
Exercise 1,
Exercise 1,
13.7
Chapter
Chapter
Chapter
Chapter
8
8
10
10
tx]
d\n
A(n)
9.9
9.10
9.10
9.10
Number Theoretic Functions and Symbols
ф(х)
F(x)
Us)
9.10
9.10
9.11
С
R
\\x\\ \
dim X
0
E
E° 1
f:X->Y 1
fD) 1
f~\B) 1
1.1
1.1
.2
1.4
1.4
1.5
[.5
.16
.16
.16
Other
complex field
real field
norm
dimension
empty set
closure
interior
function notation
image
inverse image
Symbols
TN
An)
Indr (z)
<x,x*>
a(T)
1.33
Minkowski
functional
1.40
1.46
1.46
Exercise
Fourier
3.30
4.2
4.17, 13
quotient topology
order of multi-index
seminorm
6, Chap. 2
coefficient
index
value of x* at x
.26 spectrum

388 LIST OF SPECIAL SYMBOLS
©
1*1
/Ie
x -у
1*1
xa
SA
й
и * v
C't
fit)
rB
Cz
E
On
/A.
Z(Y)
/*«, /*s
4.20
5.5
direct sum
total variation of
measure
5.6
6.2
6.10
6.10
6.10
6.24
6.29
6.29,
restriction
norm in^(H)
scalar product
length of vector
monomial
support
u{x) = u(—x)
6.34, 6.37, 7.1 con-
volution
7.1
Rn
7.1
7.1
7.22
7.20
8.1
8.2
jn
8.8
Hs
9.3
9.14
sition
10.1
Lebesgue measure on
character
Fourier transform
ball of radius r
exponential
fundamental solution
Haar measure on
measure related to
zero set
Lebesgue decompo-
I Of /X
unit element
G(A)
o(x)
pM
An
f
(Q/)(jc;//)
exp (x)
A
U"
X
T(S)
(х,У)
±
E
Ex.y
T с S
10.10
group of inver-
tible elements
10.10
10.10
10.26
spectrum
spectral radius
members of A
with spectrum in О
10.26
morphic
10.35
quotieni
10.37
function
11.5
space
11.7
11.8
form
11.21
12.1
12.1
relation
12.17
identity
12.17
measure
13.1
/4-valued holo-
: functions
difference
t
exponential
maximal ideal
polydisc
Gelfand trans
central izer
inner product
orthogonality
resolution of
spectral
inclusion of
operators

INDEX
Absorbing set, 24
Adjoint, 92, 298, 330
Alaoglu, L., 66, 375
Alexander, J. W., 383
Algebra, 98, 115,227
commutative, 228
self-adjoint, 115
semisimple, 268
*-Algebra, 305
Almost periodic function, 327, 377
Annihilator, 90, 115
Antisymmetric set, 115
Approximate identity, 157
Arens, Richard R, 380
Arens-Royden theorem, 380
Aronszajn, Nachman, 381
Ascoli's theorem, 369
£*-algebra, 276
Baire's theorem, 42
Balanced local base, 12
Balanced set, 6
Ball, 4
Banach, Stefan, 372, 374
Banach-Alaoglu theorem, 66
converse of, 108
Banach algebra, 228
Banach limit, 82
Banach space, 4
Banach-Steinhaus theorem, 43, 44
Barrel, 375
Base of a topology, 7
Basis of a vector space, 15
Bernstein, Allen R., 381
Beurling, Arne, 379
Bilinear mapping, 51, 54, 375
Birkhoff, George D., 374
Bishop, Errett, 377
Bishop's theorem, 115, 117
Blaschke product, 118
Bochner, Salomon, 373, 381

390 INDEX
Bochner's theorem, 285, 290
Bohnenblust, H. F., 375, 381
Bonsall, Frank F., 374
Bootstrap proposition, 202, 378
Borel measure, 74
regular, 76
Borel set, 74
Bounded linear functional, 14, 23
Bounded linear transformation, 23
Bounded set, 8, 22
Bourgin, Richard D., 376
Branges, Louis de, 377
Bray, Hubert E., 374
Browder, Andrew, 380
Brown, Arlen, 376, 382
Buck, R. Creighton, 374
Calderon, Alberto P., 380
Carleson, Lennart, 377
Cartesian product, 49
Category, 41
Category theorem, 42
Cauchy formula, 79, 205
Cauchy-Riemann equation, 204
Cauchy sequence, 20
Cauchy's theorem, 79
Cayley transform, 338
Cech, Eduard, 383
Centralizer, 280
Chain rule, 260
Change of measure, 347
Character, 166
Characteristic polynomial, 198
Choquet's theorem, 376
Chover, Joshua, 379
Civin, Paul, 381,382
Clarkson, James A., 382
Closed convex hull, 70
Closed graph theorem, 50
Closed operator, 329
Closed range theorem, 96
Closed set, 6
Closure, 6
Codimension, 38
Cohen, Paul J., 380
Commutator, 250, 332, 382
Compact operator, 97
Compact set, 7
Complete metric, 20
Completely continuous operator, 376
Complex algebra, 227
Complex homomorphism, 231
Complex-linear functional, 56
Complex vector space, 5
Component, 238
principal, 257
Cone, 319
Conjugate-linear function, 292
Continuity, 7
of scalar multiplication, 40
Continuous spectrum, 325
Continuously differentiable mapping, 249
Contour, 241
Convergent sequence, 7
of distributions, 146
Convex base, 12
Convex combination, 36
Convex hull, 36
Convex set, 6
Convolution, 155, 166
of distributions, 155, 159, 160, 178
of measures, 219
of rapidly decreasing functions, 172
Convolution algebra, 222, 230, 261, 272
Deckard, Don, 381
Deficiency index, 341
Degree of polynomial, 193
Dense set, 14
Densely defined operator, 330
Derivation, 382
Diagonal, 49
Dieudonne, Jean, 373
Diffeomorphism, 255
local, 253
Difference quotient, 164, 249
Differential operator, 33, 185, 198
elliptic, 198
order of, 33, 198
Differentiation:
in Banach algebras, 248
of distributions, 143

INDEX 391
Dimension, 6, 341
Dirac measure, 141, 150, 177
Direct sum, 100
of Hilbert spaces, 322
Disc algebra, 117,230
Distance, 4
Distribution, 136, 141
on a circle, 164
locally H \ 200
periodic, 190, 206
tempered, 174
on a torus, 190
Distribution derivative, 143
Domain, 330
Dual space, 55
of c, cOi 83, 109
of C(K), 66, 76
of C(C), 84
of a Hilbert space, 294, 323
of /p, 82
of Lp, 35,36
of a quotient space, 91
of a reflexive space, 105
second, 90, 105
of a subspace, 91
Dunford, Nelson, 375
Duren, Peter L., 375
Eberlein-Smulian theorem, 376
Edwards, Robert E., 376
Ehrenpreis, Leon, 192, 378
Eigenfunction, 107
Eigenvalue, 98, 311
Eigenvector, 98
Elliptic operator, 198
Entire function, 180
Equicontinuity, 43, 369
Equicontinuous group, 120
Erdos, Paul, 379
Essential range, 273, 303
Essential supremum, 83, 303
Essentially bounded function, 273, 303
Evaluation functional, 150
Exact degree, 193
Exponential function, 246, 256, 317
Extension of holomorphic function, 243
Extension theorem, 56, 57, 59
Extremally disconnected space, 274
Extreme point, 70, 286
Extreme set, 70
F-space, 8
First category, 41
Foguel, Shaul R., 381
Ford, J. W. M., 381
Fourier coefficient, 53
of a distribution, 191
Fourier-Plancherel transform, 172
Fourier transform, 167
of convolutions, 167
of derivatives, 167
of L ^functions, 172
of polynomials, 178
of rapidly decreasing functions, 168
of tempered distributions, 175
Frechet, Maurice, 372
Frechet derivative, 248
Frechet space,8
Fredholm, Ivar, 372
Fredholm alternative, 107
Friedrichs, Kurt O., 378
Fuglede, Bent, 300, 381
Function:
almost periodic, 327, 377
entire, 180
essentially bounded, 273, 303
exponential, 246, 256, 317
harmonic, 163, 366
Heaviside, 164
holomorphic, 32, 78
infinitely differentiable, 33
locally integrable, 136
locally L2, 185
positive-definite, 290
rapidly decreasing, 168
slowly oscillating, 211
strongly holomorphic, 78
weakly holomorphic, 78
{See also Functional; Operator)
Functional, 13
bounded, 14, 23
complex-linear, 56

392 INDEX
Functional:
continuous, 14, 55
linear, 13
multiplicative, 230
positive, 283
on quotient space, 91
real-linear, 56
sesquilinear, 292
on subspace, 91
{See also Dual space)
Functional calculus, 380
Fundamental solution, 192
Gamelin, Theodore W., 380
Gelfand, Izrail M, 237, 373, 379, 380
Gelfand-Mazur theorem, 237
Gelfand-Naimark theorem, 276, 380
Gelfand topology, 268
Gelfand transform, 268
Gleason, Andrew M., 233, 379
Glickfeld, Barnett W., 380
Glicksberg, Irving, 377
Goffman, Casper, 374
Graph, 49, 329
Green's function, 378
Grothendieck, Alexandre, 376
Group:
compact, 122
of invertible elements, 234, 257
of operators, 120, 127, 317
topological, 122
Haar measure, 123, 377
of a torus, 193
Hadamard, Jacques, 379
Hahn-Banach theorems, 55-59
Halmos, Paul R., 376, 381, 382
Hamel basis, 52
Harmonic function, 163, 366
Hausdorff separation axiom, 10, 49
Hausdorff space, ,7
Hausdorff topology, 7, 61
Hausdorff's maximality theorem, 367
Heaviside function, 164
Heine-Borel property, 9
Heins, Maurice, 377
Hellinger-Toeplitz theorem, 110
Henry, Bruce, 379
Hermitian element, 275
Hermitian operator, 298
Herz, Carl S., 378
Hilbert, David, 372, 376
Hilbert-Schmidt operator, 376
Hilbert space, 293
adjoint, 298
automorphism, 299
Hilbert transform, 191, 366
Hildebrandt, Т.Н., 374, 375
Hille, Einar, 380
Holder's inequality, 113
Holomorphic distribution, 204
Holomorphic function, 32
of several variables, 180
vector-valued, 78
Homomorphism, 167, 230, 264
Hormander, Lars, 378
Horvath, John M., 374
Hyperplane, 81
Ideal, 263
maximal, 263
proper, 263
Idempotent element, 247
Image, 13
Index, 79
Inductive limit, 377
Infinitesimal generator, 355
Ingham, Albert E., 379
Ingham's theorem, 215
Inherited topology, 7
Inner product, 292
Integral of vector function, 74, 85, 179,
236, 240
Integration by parts, 260
Interior, 6
Internal point, 81
Invariant measure, 123
Invariant metric, 18
Invariant subspace, 310, 382
Invariant topology, 8
Inverse, 231, 346

INDEX 393
Inverse function theorem, 252
Inverse image, 13
Inversion theorem, 170
Invertible element, 231
Invertible operator, 98
Involution, 275
^Isomorphism, 277
Kahane, Jean-Pierre, 233, 376, 379
Kakutani, Shizuo, 374, 377, 379
Kakutani's fixed point theorem, 120
Kaplansky, Irving, 380, 382
Karlin, Samuel, 219, 379, 381
Kleinecke, David C, 382
Kolmogorov, A., 374
Korevaar, Jacob, 379
Krein, M., 375, 376
Krein-Milman theorem, 70, 377
Laplace equation, 197
Laplacian, 189
La Vallee-Poussin, Ch.-J. de, 379
Lax, Peter D., 378
Lebesgue decomposition, 219
Lebesgue integral, 372
Lebesgue spaces, 31, 35, 111
Left continuity, 228
Left multiplication, 229
Left shift, 259
Left translate, 122
Leibniz formula, 144, 145
Le Page, Claude, 380
Le Veque, William J., 379
Levinson, Norman, 379
Lewy, Hans, 378
Liapounoff, A., 377
Limit, 7
Lindenstrauss, Joram, 377
Linear functional (see Functional)
Linear mapping, 13
Liouville's theorem, 81
Lipschitz space, 40
Uttlewood, John E., 208, 378
Littlewood's tauberian theorem, 209,
222
Local base, 7, 122
balanced, 12
convex, 12
Local compactness, 8
Local convexity, 8, 24
Local diffeomorphism, 253
Local equality of distributions, 147
Local finiteness, 147
Locally bounded space, 8
Locally convex space, 8
Locally integrable function, 136
Locally L2 function, 185
Logarithm, 246
Lorch, Edgar R., 374, 380, 382
Lumer, Gunter, 381
McShane, Edward J., 374
Malgrange, Bernard, 192, 378
Mandelbrojt, Szolem, 374
Mapping:
bilinear, 51, 54, 375
continuously differentiable, 249
open, 29, 46
(See also Operator)
Max-min duality, 376
Maximal ideal space, 268
Maximally normal operator, 350
Maximally symmetric operator, 337
Measure:
Borel, 74
Я-valued, 302
Haar, 123
nonatomic, 113
normalized Lebesgue, 166
probability, 74
projection-valued, 302
regular, 76
Mergelyan's theorem, 117
Metric, 4
compatible, 7
complete, 20
euclidean, 14
invariant, 18
Metric space, 4
Metrization theorem, 18, 61, 374
in locally convex spaces, 27, 28

394 INDEX
Milman, D., 375
Minkowski functional, 24
Monomial, 142
Montel space, 375
Multi-index, 32
Multiplication operator, 318, 325
Multiplication theorem, 343
Multiplicative functional, 230
Multiplicative inequality, 227
Nagumo, M., 373
Naimark, M. A., 380
Neighborhood, 7
Neumann, John von, 373, 374, 376, 377,
382
Ney, Peter, 379
Nonatomic measure, 113
Norm, 4
in dual space, 89
Norm topology, 4
Normable space, 8
Normal element, 281
Normal operator, 298, 348
Normal subset, 281
Normalized Lebesgue measure, 166
Normed dual, 87
Normed space, 4
Nowhere dense set, 41
Null space, 13
Open mapping, 29
Open mapping theorem, 46
Open set, 6
Operational calculus, 380
Operator;
bounded, 23
closed, 329
compact, 97
completely continuous, 376
densely defined, 330
differential, 33, 185, 198
elliptic, 198
hermitian, 298
invertible, 98
linear, 13
maximally normal, 350
Operator:
maximally symmetric, 337
normal, 298, 348
positive, 313, 349
self-adjoint, 298, 331
symmetric, 110, 331
unitary, 298
Order:
of a differential operator, 198
of a distribution, 141
of an operator on Sobolev spaces, 199
partial, 367
total, 367
Origin, 5
Original topology, 63, 64
Orthogonal complement, 294
Orthogonal vector, 292
Paley-Wiener theorems, 181, 183
Parallelogram law, 293
Parseval formula, 172
Partial isometry, 316
Partially ordered set, 367
Partition, 85
of unity, 147
Pearcy, Carl M., 381, 382
Pettis, Billy J., 375
Pick-Nevanlinna problem, 118
Pitt, Harry R., 211, 379
Pitt's theorem, 212, 220
Plancherel, M., 378
Plancherel theorem, 172
Point spectrum, 247, 312, 325
Polar of a set, 66
Polar decomposition, 315, 364
Pole, 261
Polydisc, 267
Polydisc algebra, 288
Polynomial convexity, 272
Positive-definite function, 290
Positive functional, 283, 319 .
Positive operator, 313, 349
Preimage, 13
Prime number theorem, 212
Principal component, 257
Principal part of operator, 198
Principal value integral, 165

INDEX 395
Probability measure, 74
Product topology, 49
Projection, 126, 298, 299
Ptak, Vlastimil, 381
Putnam, Calvin R., 300, 381
Quotient algebra, 264
Quotient map, 29
Quotient norm, 30
Quotient space, 29
Quotient topology, 29
Radical, 268
Range, 94
Rapidly decreasing function, 168
Real vector space, 5
Reflexive space, 90, 104, 382
Regularity theorem, 197, 201
Renewal equation, 218
Residual spectrum, 325
Resolution of identity, 301, 341, 348
Resolvent set, 234, 346
Riemann, Bernhard, 373
Riemann-Lebesgue lemma, 378
Riemann zeta function, 214
Riesz, Frederic, 117, 372, 376, 382
Riesz, Marcel, 117, 130
Riesz representation theorem, 53
Right continuity, 228
Right multiplication, 229
Right shift, 259
Right translate, 122
Riviere, Nestor M., 380
Robinson, Abraham, 381
Rogosinski, Werner W., 376
Romberg, Bernard W., 375
Root, 246
Rosenblum, Marvin, 300, 381
Rosenthal, Haskell, P., 377
Royden, Halsey L., 380
Runge's theorem, 244
Scalar, 5
Scalar field, 5
Scalar multiplication, 5
Schaffer, Juan J., 381
Schauder, J., 376
Schneeberger, Charles M., 382
Schwartz, Laurent, 373, 378
Schwarz inequality, 293
Second category, 41
Second dual, 90, 105
Seid, Howard A., 379
Selberg, Atle, 379
Self-adjoint algebra, 115
Self-adjoint element, 275
Self-adjoint operator, 298
Semigroup, 355
of normal operators, 360
unitary, 360
Seminorm, 24
Semisimple algebra, 268
Separable space, 68
Separate continuity, 51
Separating family, 24
Separation theorems, 9, 58, 70
Sequential continuity, 370
Sesquilinear functional, 292
Set:
absorbing, 24
antisymmetric, 115
balanced, 6
Borel, 74
bounded, 8, 22
closed, 6
compact, 7
convex, 6
dense, 14
extreme, 70
of first category, 41
normal, 281
nowhere dense, 41
open, 6
partially ordered, 367
of second category, 41
totally ordered, 367
weakly bounded, 64
Shapiro, riarold S., 376
Shapiro, Joel H., 375
Shields, Allen L., 375, 376
Shift operator, 106
Shilov boundary, 290
Siddiqui, Jamil A., 379

396 INDEX
Slowly oscillating function, 211
Smith, Kennan Т., 381
Smulian, V., 376
Sobczyk, Andrew, 375, 377
Sobolev, S. L., 373, 378
Sobolev spaces, 199, 378
Sobolev's lemma, 185
Soukhomlinoff, G. A., 375
Space:
Banach, 4
barreled, 375
complete metric, 20
extremally disconnected, 274
Frechet, 8
with Heine-Borel property, 9
Hilbert, 293
Lipschitz, 40
locally bounded, 8
locally compact, 8
locally convex, 8
metric, 4
Montel, 375
normable, 8
normed, 4
quotient, 29
reflexive, 90, 104, 382
separable, 68
topological, 6
totally disconnected, 370
uniformly convex, 327, 382
unitary, 292
vector, 5
Spectral decomposition, 309, 348, 351
Spectral mapping theorem, 244, 247
Spectral radius, 234
formula, 235
Spectral theorem, 305, 348, 351
Spectrum, 98, 234, 346
of compact operator, 103
continuous, 325
of differential operator, 365
point, 247, 312, 325
residual, 325
of self-adjoint operator, 310, 348
of unitary operator, 310
Square root, 278, 314, 349
Stone, Marshall H., 373, 374, 382
Stone-Weierstrass theorem, 115, 377
Stout, Edgar Lee, 380
Strong topology, 64л.
Strongly holomorphic function, 78
Subadditivity, 24
Subbase, 27
Subbase theorem, 368
Subspace, 6
complemented, 100
translation-invariant, 211
uncomplemented, 125, 130
Support, 33
of a distribution, 149
Support function, 374
Surrounding contour, 241
Symbolic calculus, 240, 278, 309
Symmetric involution, 285
Symmetric neighborhood, 9
Symmetric operator, 110, 331
Tauber, A., 208, 378
Tauberian condition, 208
Tauberian theorem, 208
Ingham's, 215
Littlewood's, 222
Pitt's, 212
Wiener's, 210, 211
Taylor, Angus E., 374, 375
Taylor, Joseph L, 380
Tempered distribution, 174
Test function space, 136
topology of, 137
Topological divisor of zero, 259
Topological group, 122
Topological space, 6
Topology, 6
compact, 61, 368
compatible, 7
Hausdorff, 7, 61
inherited, 7
invariant, 8
metrizable, 17
norm, 4
original, 63
strong, 64/i.
stronger, 60

INDEX 397
Topology:
weak, 63
weak*, 66
weaker, 60
Torus, 193
Totally bounded set, 71, 72, 369
Totally disconnected space, 370
Totally ordered sqU 367
Transformation (see Mapping; Operator)
Translate, 8, 122, 154
of a distribution, 156
Translation-invariant subspace, 211
Translation-invariant topology, 8
Translation operator, 8
Treves, Francois, 374
Tychonoff, A., 383
Tychonoffs theorem, 61, 368
Uhl, J. Jerry, 377
Uniform boundedness principle, ^43
Uniform continuity, 14, 122
Uniformly convex space, 327, 382
Unit ball, 4
Unit element, 227, 228
Unitary operator, 298
Unitary space, 292
Vector space, 5
complex, 5
real, 5
Vector space:
topological, 7
locally bounded, 8
locally compact, 8
locally convex, 8
metrizable, 8
normable, 8
Vector topology, 7
Veech, William A., 377
Volterra, Vito, 372
Weak closure, 64
Weak neighborhood, 64
Weak sequential closure, 83
Weak topology, 63
Weak*-topology, 66
Weakly bounded set, 64
Weakly convergent sequence, 64
Weakly holomorphic function, 78
Wielandt, Helmut W., 332, 382
Wiener, Norbert, 372-374, 379
Wiener's lemma, 266
Wiener's theorem, 210, 211
Williamson, John H., 376
Wintner, Aurel, 382
Yood, Bertram, 381, 382
Zelazko, W., 233, 379