I TRODUCTORY

FU CTIO 'AL A 'ALYSIS

WITH

APPLICATIONS

INTRODUCTORY

FUNCTIONAL ANALYSIS

WITH

APPLICATIONS

Erwin KreyszigUniversity of Windsor

Wiley Classics Library Edition Published 1989

IIWILEY

..JOHN WILEY & SONSNew York· Chichester· Brisbane· Toronto· Singapore

Copyright © 1978, by John Wiley & Sons. Inc.

All rights reserved. Published simultaneously in Canada.

Reproduction or translation of any part of this work beyondthat permitted by Sections 107 or 108 of the 1976 United StatesCopyright Act without the permission of the copyright owneris unlawful. Requests for permission or further informationshould be addressed to the Permissions Department, JohnWiley & Sons, Inc.

Library of Congress Cataloging in Publication Data:

Kreyszig, Erwin.Introductory functional analysis with applications.

Bibliography: p.l. Functional analysis. l. Title.

QA320.K74 515'.7 77-2560ISBN 0-471-50731-80-471-50459-9 (pbk)Printed and bound in the United States of America

30 29 28 27 26 25 24 23 22

PREFACE

Purpose of the book. Functional analysis plays an increasing role inthe applied sciences as well as in mathematics itself. Consequently, itbecomes more and more desirable to introduce the student to the fieldat an early stage of study. This book is intended to familiarize thereader with the basic concepts, principles and methods of functionalanalysis and its applications.

Since a textbook should be written for the student, I have soughtto bring basic parts of the field and related practical problems withinthe comfortable grasp of senior undergraduate students or beginninggraduate students of mathema,tics and physics. I hope that graduateengineering students may also profit from the presentation.

Prerequisites. The book is elementary. A background in under­graduate mathematics, in particular, linear algebra and ordinary cal­culus, is sufficient as a prerequisite. Measure theory is neither assumednor discussed. No knowledge in topology is required; the few consider­ations involving compactness are self-contained. Complex analysis isnot needed, except in one of the later sections (Sec. 7.5), which isoptional, so that it can easily be omitted. Further help is given inAppendix 1, which contains simple material for review and reference.

The book should therefore be accessible to a wide spectrum ofstudents and may also facilitate the transition between linear algebraand advanced functional analysis.

Courses. The book is suitable for a one-semester course meeting fivehours per week or for a two-semester course meeting three hours perweek.

The book can also be utilized for shorter courses. In fact, chapterscan be omitted without destroying the continuity or making the rest ofthe book a torso (for details see below). For instance:

Chapters 1 to 4 or 5 makes a very short course.Chapters 1 to 4 and 7 is a course that includes spectral theory and

other topics.

Content and arrangement. Figure 1 shows that the material has beenorganized into five major blocks.

v

vi

Spaces and Operators Chaps. 1 to 3

Metric spacesNormed and Banach spaces '1Li near operators IInner product and Hilbert spaces ~ I

I

!I II II I

Fundamental Theorems Chap. 4 I II IHahn-Banach theorem I IUniform boundedness theorem I IOpen mapping theorem I IClosed graph theorem I I

I I

I !I II II I

I Further Applications Chaps. 5 to 6 I II I

I Applications of contractionsj

I Approximation theory

t !Spectral Theory Chaps, 7 to 9

Basic conceptsOperators on normed spaces ,

ICompact operators I

Self~adjoint operators

tUnbounded Operators Chaps. 10 to 11

Unbounded operatorsQuantum mechanics

Fig. 1. Content and arrangement of material

Preface

Hilbert space theory (Chap. 3) precedes the basic theorems onnormed and Banach spaces (Chap. 4) because it is simpler, contributesadditional examples in Chap. 4 and, more important, give the studenta better feeling for the difficulties encountered in the transition fromHilbert spaces to general Banach spaces.

Chapters 5 and 6 can be omitted. Hence after Chap. 4 one canproceed directly to the remaining chapters (7 to 11).

Preface vii

Spectral theory is included in Chaps. 7 to 11. Here one has greatflexibility. One may only consider Chap. 7 or Chaps. 7 and 8. Or onemay focus on the basic concepts from Chap. 7 (Secs. 7.2. and 7.3) andthen immediately move to Chap. 9, which deals with the spectraltheory of bounded self-adjoint operators.

Applications are given at various places in the text. Chapters 5 and 6are separate chapters on applications. They can be considered insequence, or earlier if so desired (see Fig. 1):

Chapter 5 may be taken up immediately after Chap. 1.Chapter 6 may be taken up immediately after Chap. 3.

Chapters 5 and 6 are optional since they are not used as a prerequisitein other chapters.

Chapter 11 is another separate chapter on applications; it dealswith unbounded operators (in quantum physics), but is kept practicallyindependent of Chap. 10.

Presentation. The material in this book has formed the basis oflecture courses and seminars for undergraduate and graduate studentsof mathematics, physics and engineering in this country, in Canada andin Europe. The presentation is detailed, particularly in the earlierchapters, in order to ease the way for the beginner. Less demandingproofs are often preferred over slightly shorter but more advancedones.

In a book in which the concepts and methods are necessarilyabstract, great attention should be paid to motivations. I tried to do soin the general discussion, also in carefully selecting a large number ofsuitable examples, which include many simple ones. I hope that thiswill help the student to realize that abstract concepts, ideas andtechniques were often suggested by more concrete matter. The studentshould see that practical problems may serve as concrete models forillustrating the abstract theory, as objects for which the theory canyield concrete results and, moreover, as valuable sources of new ideasand methods in the further development of the theory.

Problems and solutions. The book contains more than 900 care­fully selected problems. These are intended to help the reader in betterunderstanding the text and developing skill and intuition in functionalanalysis and its applications. Some problems are very simple, toencourage the beginner. Answers to odd-numbered problems are givenin Appendix 2. Actually, for many problems, Appendix 2 containscomplete solutions.

viii Preface

The text of the book is self-contained, that is, proofs of theoremsand lemmas in the text are given in the text, not in the problem set.Hence the development of the material does not depend on theproblems and omission of some or all of them does not destroy thecontinuity of the presentation.

Reference material is included in Appendix 1, which contains someelementary facts about sets, .mappings, families, etc.

References to literature consisting of books and papers are collected inAppendix 3, to help the reader in further study of the text material andsome related topics. All the papers and most of the books are quotedin the text. A quotation consists of a name and a year. Here are twoexamples. "There are separable Banach spaces without Schauderbases; d. P. Enflo (1973)." The reader will then find a correspondingpaper listed in Appendix 3 under Enflo, P. (1973). "The theorem wasgeneralized to complex vector spaces by H. F. Bohnenblust and A.Sobczyk (1938)." This indicates that Appendix 3 lists a paper by theseauthors which appeared in 1938.

Notations are explained in a list included after the table of contents.

Acknowledgments. I want to thank Professors Howard Anton (Dre­xel University), Helmut Florian (Technical University of Graz, Au­stria), Gordon E. Latta (University at Virginia), Hwang-Wen Pu(Texas A and M University), Paul V. Reichelderfer (Ohio University),Hanno Rund (University of Arizona), Donald Sherbert (University ofIllinois) and Tim E. Traynor (University of Windsor) as well as manyof my former and present students for helpful comments and construc­tive criticism.

I thank also John Wiley and Sons for their effective cooperationand great care in preparing this edition of the book.

ERWIN KREYSZIG

CONTENTS

Chapter 1. Metric Spaces . . . . 1

1.1 Metric Space 21.2 Further Examples of Metric Spaces 91.3 Open Set, Closed Set, Neighborhood 171.4 Convergence, Cauchy Sequence, Completeness 251.5 Examples. Completeness Proofs 321.6 Completion of Metric Spaces 41

Chapter 2. Normed Spaces. Banach Spaces . 49

2.1 Vector Space 502.2 Normed Space. Banach Space 582.3 Further Properties of Normed Spaces 672.4 Finite Dimensional Normed Spaces and Subspaces 722.5 Compactness and Finite Dimension 772.6 Linear Operators 822.7 Bounded and Continuous Linear Operators 912.8 Linear Functionals 1032.9 Linear Operators and Functionals on Finite Dimen­

sional Spaces 1112.10 Normed Spaces of Operators. Dual Space 117

Chapter 3. Inner Product Spaces. Hilbert Spaces. . .127

3.1 Inner Product Space. Hilbert Space 1283.2 Further Properties of Inner Product Spaces 1363.3 Orthogonal Complements and Direct Sums 1423.4 Orthonormal Sets and Sequences 1513.5 Series Related to Orthonormal Sequences and Sets 1603.6 Total Orthonormal Sets and Sequences 1673.7 Legendre, Hermite and Laguerre Polynomials 1753.8 Representation of Functionals on Hilbert Spaces 1883.9 Hilbert-Adjoint Operator 195

3.10 Self-Adjoint, Unitary and Normal Operators 201

ix

x Contents

Chapter 4. Fundamental Theorems for Normedand Banach Spaces. . . . . . . . . . . 209

4.1 Zorn's Lemma 2104.2 Hahn-Banach Theorem 2134.3 Hahn-Banach Theorem for Complex Vector Spaces and

Normed Spaces 2184.4 Application to Bounded Linear Functionals on

C[a, b] 2254.5 Adjoint Operator 2314.6 Reflexive Spaces 2394.7 Category Theorem. Uniform Boundedness Theorem 2464.8 Strong and Weak Convergence 2564.9 Convergence of Sequences of Operators and

Functionals 2634.10 Application to Summability of Sequences 2694.11 Numerical Integration and Weak* Convergence 2764.12 Open Mapping Theorem 2854.13 Closed Linear Operators. Closed Graph Theorem 291

Chapter 5. Further Applications: Banach FixedPoint Theorem . . . . . . . . . . . . 299

5.1 Banach Fixed Point Theorem 2995.2 Application of Banach's Theorem to Linear Equations 3075.3 Applications of Banach's Theorem to Differential

Equations 3145.4 Application of Banach's Theorem to Integral

Equations 319

Chapter 6. Further Applications: ApproximationTheory ..... ..... . . . 327

6.1 Approximation in Normed Spaces 3276.2 Uniqueness, Strict Convexity 3306.3 Uniform Approximation 3366.4 Chebyshev Polynomials 3456.5 Approximation in Hilbert Space 3526.6 Splines 356

Chapter 7. Spectral Theory of Linear Operatorsin Normed Spaces . . . . . . . . . . . 363

7.1 Spectral Theory in Finite Dimensional Normed Spaces 3647.2 Basic Concepts 370

Contents

7.3 Spectral Properties of Bounded Linear Operators 3747.4 Further Properties of Resolvent and Spectrum 3797.5 Use of Complex Analysis in Spectral Theory 3867.6 Banach Algebras 3947.7 Further Properties of Banach Algebras 398

xi

Chapter 8. Compact Linear Operators on NormedSpaces and Their Spectrum . 405

8.1 Compact Linear Operators on Normed Spaces 4058.2 Further Properties of Compact Linear Operators 4128.3 Spectral Properties of Compact Linear Operators on

Normed Spaces 4198.4 Further Spectral Properties of Compact Linear

Operators 4288.5 Operator Equations Involving Compact Linear

Operators 4368.6 Further Theorems of Fredholm Type 4428.7 Fredholm Alternative 451

Chapter 9. Spectral Theory of BoundedSelf-Adjoint Linear Operators .. . 459

9.1 Spectral Properties of Bounded Self-Adjoint LinearOperators 460

9.2 Further Spectral Properties of Bounded Self-AdjointLinear Operators 465

9.3 Positive Operators 4699.4 Square Roots of a Positive Operator 4769.5 Projection Operators 4809.6 Further Properties of Projections 4869.7 Spectral Family 4929.8 Spectral Family of a Bounded Self-Adjoint Linear

Operator 4979.9 Spectral Representation of Bounded Self-Adjoint Linear

Operators 5059.10 Extension of the Spectral Theorem to Continuous

Functions 5129.11 Properties of the Spectral Family of a Bounded Self­

Adjoint Linear Operator 516

xii Contents

Chapter 10. Unbounded Linear Operators'inHilbert Space . . . . . . . . . . . . 523

10.1 Unbounded Linear Operators and theirHilbert-Adjoint Operators 524

10.2 Hilbert-Adjoint Operators, Symmetric and Self-AdjointLinear Operators 530

10.3 Closed Linear Operators and Closures 53510.4 Spectral Properties of Self-Aajo"int Linear Operators 54110.5 Spectral Representation of Unitary Operators 54610.6 Spectral Representation of Self-Adjoint Linear Operators

55610.7 Multiplication Operator and Differentiation Operator

562

Chapter 11. Unbounded Linear Operators inQuantum Mechanics . . . . . . 571

II. I Basic Ideas. States, Observables, Position Operator 57211.2 Momentum Operator. Heisenberg Uncertainty Principle

57611.3 Time-Independent Schrodinger Equation 58311.4 Hamilton Operator 59011.5 Time-Dependent Schrodinger Equation 598

Appendix 1. Some Material for Review andReference . . . . . . . . . . . . . . 609

A 1.1 Sets 609A1.2 Mappings 613A1.3 Families 617A1.4 Equivalence Relations 618A1.5 Compactness 618A1.6 Supremum and Infimum 619A 1.7 Cauchy Convergence Criterion 620A1.8 Groups 622

Appendix 2. Answers to Odd-Numbered Problems. 623

Appendix 3. References.

Index . . . . . . . . . . .

. . 675

. .681

NOTATIONS

In each line we give the number of the page on which the symbol isexplained.

AC

ATB[a, b]B(A)BV[a, b]B(X, Y)B(x; r)B(x;r)cCoeenC[a, b]C,[a, b]C(X, Y)9?J(T)d(x, y)dimX8jk

~ = (E),)

IltllCfj(T)IinfU[a, b]/P/~

L(X, Y)MJ..N(T)oo

Complement of a set A 18, 609Transpose of a matrix A 113Space of bounded functions 228Space of bounded functions 11Space of functions of bounded variation 226Space of bounded linear operators 118Open baIl 18Closed ball 18A sequence space 34A sequence space 70Complex plane or the field of complex numbers 6, 51Unitary n-space 6Space of continuous functions 7, 61Space of continuously differentiable functions 110Space of compact linear operators 411Domain of an operator T 83Distance from x to y 3Dimension of a space X 54Kronecker delta 114Spectral family 494Norm of a bounded linear functional t 104Graph of an operator T 292Identity operator 84Infimum (greatest lower bound) 619A function space 62A sequence -space 11A se"quence space 6A space of linear operators 118Annihilator of a set M 148Null space of an operator T 83Zero operator 84Empty set 609

xiii

xiv

RR"<Jt(T)R).(T)ru(T)p(T)su(T)uc(T)up(T)ur(T)spanMsupIITIIT*TX

T+,lT).+, T).­TI /2

Var(w)w~

X*X'Ilxll(x, y)x.lyy.i.

Notations

Real line or the field of real numbers 5, 51Euclidean n-space 6Range of an operator T 83Resolvent of an operator T 370Spectral radius of an operator T 378Resolvent set of an operator T 371A sequence space 9Spectrum of an operator T 371Continuous spectrum of T 371Point spectrum of T 371Residual spectrum of T 371Span of a set M 53Supremum (least upper bound) 619Norm of a bounded linear operator T 92Hilbert-adjoint operator of T 196Adjoint operator of T 232Positive and negative parts of T 498Positive and negative parts of T). = T - AI 500Positive square root of T 476Total variation of w 225Weak convergence 257Algebraic dual space of a vector space X 106Dual space of a normed space X 120Norm of x 59Inner product of x and y 128x is orthogonal to y 131Orthogonal complement of a closed subspace Y 146

I TRODUCTORY

FU CTIO ~L A ~LYSIS

WITH

APPLICATIONS

CHAPTER IMETRIC SPACES

Functional analysis is an abstract branch of mathematics that origi­nated from classical analysis. Its development started about eightyyears ago, and nowadays functional analytic methods and results areimportant in various fields of mathematics and its applications. Theimpetus came from linear algebra, linear ordinary and partial differen­tial equations, calculus of variations, approximation theory and, inparticular, linear integral equations, whose theory had the greatesteffect on the development and promotion of the modern ideas.Mathematicians observed that problems from different fields oftenenjoy related features and properties. This fact was used for aneffective unifying approach towards such problems, the unificationbeing obtained by the omission of unessential details. Hence theadvantage of uch an abstract approach is that it concentrates on theessential facts, so that these facts become clearly visible since theinvestigator's attention is not disturbed by unimportant details. In thisrespect the abstract method is the simplest and most economicalmethod for treating mathematical systems. Since any such abstractsystem will, in general, have various concrete realizations (concretemodels), we see that the abstract method is quite versatile in itsapplication to concrete situations. It helps to free the problem fromisolation and creates relations and transitions between fields whichhave at first no contact with one another.

In the abstract approach, one usually starts from a set of elementssatisfying certain axioms. The nature of the elements is left unspecified.This is done on purpose. The theory then consists of logical conse­quences which result from the axioms and are derived as theorems onceand for all. This means that in this axiomatic fashion one obtains amathematical structure whose theory is developed in an abstract way.Those general theorems can then later be applied to various specialsets satisfying those axioms.

For example, in algebra this approach is used in connection withfields, rings and groups. In functional analysis we use it in connectionwith abstract spaces; these are of basic importance, and we shallconsider some of them (Banach spaces, Hilbert spaces) in great detail.We shall see that in this connection the concept of a "space" is used in

1

2 Metric Spaces

a very wide and surprisingly general sense. An abstract space will be aset of (unspecified) elements satisfying certain axioms. And by choos­ing different sets of axioms we shall obtain different types of abstractspaces.

The idea of using abstract spaces in a systematic fashion goes backto M. Frechet (1906)1 and is justified by its great success.

In this chapter we consider metric spaces. These are fundamentalin functional analysis because they playa role similar to that of the realline R in calculus. In fact, they generalize R and have been created inorder to provide a basis for a unified treatment of important problemsfrom various branches of analysis.

We first define metric spaces and related concepts and illustratethem with typical examples. Special spaces of practical importance arediscussed in detail. Much attention is paid to the concept of complete­ness, a property which a metric space mayor may not have. Complete­ness will play a key role throughout the book.

Important concepts, brief orientation about main content

A metric space (d. 1.1-1) is a set X with a metric on it. The metricassociates with any pair of elements (points) of X a distance. Themetric is defined axiomatically, the axioms being suggested by certainsimple properties of the familiar distance between points on the realline R and the complex plane C. Basic examples (1.1-2 to 1.2-3) showthat the concept of a metric space is remarkably general. A veryimportant additional property which a metric space may have iscompleteness (d. 1.4-3), which is discussed in detail in Sees. 1.5 and1.6. Another concept of theoretical and practical interest i separabilityof a metric space (d. 1.3-5). Separable metric spaces are simpler thannonseparable ones.

1.1 Metric Space

In calculus we study functions defined on the real line R. A littlereflection shows that in limit processes and many other considerationswe use the fact that on R we have available a distance function, call itd, which associates a distance d(x, Y)=lx-YI with every pair of points

I References are given in Appendix 3, and we shall refer to books and papers listedin Appendix 3 as is shown here.

1.1 Metric Space

~--5--~ I-<---- 4.2~

I I I

3

3 8 -2.5 0 1.7

d(3, 8) = 13 - 8 I = 5 d(1.7, - 2.5) = /1.7 - (-2.5) 1=4.2

Fig. 2. Distance on R

x, YE R. Figure 2 illustrates the notation. In the plane and in "ordi­nary" three-dimensional space the situation is similar.

In functional analysis we shall study more general "spaces" and"functions" defined on them. We arrive at a sufficiently general andflexible concept of a "space" as follows. We replace the set of realnumbers underlying R by an abstract set X (set of elements whosenature is left unspecified) and introduce on X a "distance function"which has only a few of the most fundamental properties of thedi tance function on R. But what do we mean by "most fundamental"?This question is far from being trivial. In fact, the choice and formula­tion of axioms in a definition always needs experience, familiarity withpractical problems and a clear idea of the goal to be reached. In thepresent case, a development of over sixty years has led to the followingconcept which is basic and very useful in functional analysis and itsapplications.

1.1-1 Definition (Metric space, metric). A metric space is a pair(X, d), where X is a set and d is a metric on X (or distance function onX), that is, a function defined2 on X x X such that for all x, y, Z E X wehave:

(M1) d is real-valued, finite and nonnegative.

d(x, y) = dey, x)

(M2)

(M3)

d(x, y)=O if and only if x = y.

(Symmetry).

(M4) d(x, y):2id(x, z)+d(z, y) (Triangle inequality). I

2 The symbol x denotes the Cartesian product of sets: A x B is the set of all orderedpairs Ca, b), where a E A and bE B. Hence X x X is the set of aU ordered pairs ofelements of X.

4 Metric Spaces

A few related terms are as follows. X is usually called theunderlying set of (X, d). Its elements are called points. For fixed x, y wecall the nonnegative number d(x, y) the distance from x to y. Proper­ties (Ml) to (M4) are the axioms of a metric. The name "triangleinequality" is motivated by elementary geometry as shown in Fig. 3.

x

Fig. 3. Triangle inequality in the plane

From (M4) we obtain by induction the generalized triangle in­equality

Instead of (X, d) we may simply write X if there is no danger ofconfusion.

A subspace (Y, d) of (X, d) is obtained if we take a subset Y c Xand restrict d to Yx Y; thus the metric on Y is the restriction3

d = d!yxY.

d is called the metric induced on Y by d.We shall now list examples of metric spaces, some of which are

already familiar to the reader. To prove that these are metric spaces,we must verify in each case that the axioms (Ml) to (M4) are satisfied.Ordinarily, for (M4) this requires more work than for (Ml) to (M3).However, in our present examples this will not be difficult, so that wecan leave it to the reader (d. the problem set). More sophisticated

3 Appendix 1 contains a review on mappings which also includes the concept of arestriction.

1.1 . Metric Space 5

metric spaces for which (M4) is not so easily verified are included inthe next section.

Examples

1.1-2 Reallioe R. This is the set of all real numbers, taken with theusual metric defined by

(2) d(x, Y) = Ix - yl·

1.1-3 Euclidean plane R2• The metric space R2

, called the Euclideanplane, is obtained if we take the set of ordered pairs of real numbers,written4 x = (~h ~2), Y = (Tjl, Tj2), etc., and the Euclidean metricdefined by

(3) (~O).

See Fig. 4.Another metric space is obtained if we choose the same set as

before but another metric d 1 defined by

(4)

Y = (fJ,. fJ2)

III 1~2 - fJ2 III___________.J

I ~, - fJ, I

Fig. 4. Euclidean metric on the plane

4 We do not write x = (Xl> X2) since Xl> X2 • ••• are needed later in connection withsequences (starting in Sec. 1.4).