Graduate Texts in Mathematics 192 Editorial Board

S. Axler F.W. Gehring K.A. Ribet

Springer Science+Business Media, LLC

Graduate Texts in Mathematics

TAKEunlZARING. Introduction to 33 HIRSCH. Differential Topology. Axiomatic Set Theory. 2nd ed. 34 SPITZER. Principles of Random Walk.

2 OXTOBY. Measure and Category. 2nd ed. 2nd ed. 3 SCHAEFER. Topological Vector Spaces. 35 ALEXANDERIWERMER. Several Complex 4 HILTON/STAMMBACH. A Course in Variables and Banach Algebras. 3rd ed.

Homological Algebra. 2nd ed. 36 KELLEy/NAMIOKA et al. Linear 5 MAC LANE. Categories for the Working Topological Spaces.

Mathematician. 2nd ed. 37 MONK. Mathematical Logic. 6 HUGHES/PiPER. Projective Planes. 38 GRAUERT/FRrrz.~CHE . Several Complex 7 SERRE. A Course in Arithmetic. Variables. 8 TAKEunlZARING. Axiomatic Set Theory. 39 ARVESON. An Invitation to C*-Algebra~ .

9 HUMPHREYS. Introduction to Lie Algebras 40 KEMENy/SNELu'KNAPP. Denumerable and Representation Theory. Markov Chains. 2nd ed.

10 COHEN. A Course in Simple Homotopy 41 APOSTOL. Modular Functions and Theory. Dirichlet Series in Number Theory.

II CONWAY. Functions of One Complex 2nd ed. Variable I. 2nd ed. 42 SERRE. Linear Representations of Finite

12 BEALS. Advanced Mathematical Analysis. Groups. 13 ANDERSON/FuLLER. Rings and Categories 43 GILLMAN/JERISON. Rings of Continuous

of Modules. 2nd ed. Functions. 14 GOLUBITSKy/GUILLEMIN. Stable Mappings 44 KENDIG. Elementary Algebraic Geometry.

and Their Singularities. 45 LOEVE. Probability Theory I. 4th ed. 15 BERBERIAN. Lectures in Functional 46 LOEVE. Probability Theory II. 4th ed.

Analysis and Operator Theory. 47 MOISE. Geometric Topology in 16 WINTER. The Structure of Fields. Dimensions 2 and 3. 17 ROSENBLATT. Random Processes. 2nd ed. 48 SACHslWu. General Relativity for 18 HALMOS. Measure Theory. Mathematicians. 19 HALMOS. A Hilbert Space Problem Book. 49 GRUENBERGlWEIR. Linear Geometry.

2nd ed. 2nd ed. 20 HUSEMOLLER. Fibre Bundles. 3rd ed. 50 EDWARDS. Fermat's Last Theorem. 21 HUMPHREYS. Linear Algebraic Groups. 51 KLINGENBERG. A Course in Differential 22 BARNES/MACK. An Algebraic Introduction Geometry.

to Mathematical Logic. 52 HARTSHORNE. Algebraic Geometry. 23 GREUB. Linear Algebra. 4th ed. 53 MANIN. A Course in Mathematical Logic. 24 HOLMES. Geometric Functional Analysis 54 GRA VERIW ATKINS. Combinatorics with

and Its Applications. Emphasis on the Theory of Graphs. 25 HEwm/STRoMBERG. Real and Abstract 55 BROWN/PEARCY. Introduction to Operator

Analysis. Theory I: Elements of Functional 26 MANES. Algebraic Theories. Analysis. 27 KELLEY. General Topology. 56 MASSEY. Algebraic Topology: An 28 ZARISKIISAMUEL. Commutative Algebra. Introduction.

VoU. 57 CROWELLlFox. Introduction to Knot 29 ZARISKIlSAMUEl. Commutative Algebra. Theory.

Vol.II. 58 KOBun. p-adic Numbers. p-adic 30 JACOBSON. Lectures in Abstract Algebra I. Analysis. and Zeta-Functions. 2nd ed.

Basic Concepts. 59 LANG. Cyclotomic Fields. 31 JACOBSON. Lectures in Abstract Algebra 60 ARNOLD. Mathematical Methods in

II. Linear Algebra. Classical Mechanics. 2nd ed. 32 JACOBSON. Lectures in Abstract Algebra 61 WHITEHEAD. Elements of Homotopy

III. Theory of Fields and Galois Theory. Theory.

(continued after index)

Francis Hirsch Gilles Lacombe

Elements of Functional Analysis

Translated by Silvio Levy

, Springer

Francis Hirsch Gilles Lacombe

Translator Silvio Levy

Departement de Mathematiques Universite d'Evry-Val d'Essonne Boulevard des coquibus

Mathematical Sciences Research Institute l ()()() Centennial Drive

Evry Cedex F-91025 Berkeley, CA 94720-5070

France

Editorial Board S. Axler Mathematics Department San Francisco State

University San Francisco, CA 94132 USA

USA

F.W. Gehring Mathematics Department East HaU University of Michigan Ann Arbor, MI 48109 USA

K.A. Ribet Mathematics Department University of California

at Berkeley Berkeley, CA 94720-3840 USA

Mathematics Subject Classification (1991): 46-01, 46Fxx, 47E05, 46E35

Library of Congress Cataloging-in-Publication Data Hirsch, F. (Francis)

Elements of functional analysis / Francis Hirsch, Gilles Lacombe. p. cm. - (Graduate texts in mathematics ; 192)

Includes bibliographical references and index. ISBN 978-1-4612-7146-8 ISBN 978-1-4612-1444-1 (eBook) DOI 10.1007/978-1-4612-1444-1 1. Functional analysis. 1. Lacombe, Gilles. D. Title.

ID. Series. QA320.H54 1999 5 1 5.7-ilc2 l 98-53153

Printed on acid-free paper.

French Edition: ELements d'analysefonctionnelle © Masson, Paris, 1997.

© 1999 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 1999 Softcover reprint ofthe hardcover 1st edition 1999 AII rights reserved. This work may not be translated or copied in whole or in part without the written pennission of the publisher Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any fonn of infonnation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the fonner are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Production managed by A. Orrantia; manufacturing supervised by Jacqui Ashri. Photocomposed copy prepared from the translator' s PostScript files.

9 8 7 6 5 432 1

ISBN 978-1-4612-7146-8 SPIN 10675899

Preface

This book arose from a course taught for several years at the Univer­sity of Evry-Val d'Essonne. It is meant primarily for graduate students in mathematics. To make it into a useful tool, appropriate to their knowl­edge level, prerequisites have been reduced to a minimum: essentially, basic concepts of topology of metric spaces and in particular of normed spaces (convergence of sequences, continuity, compactness, completeness), of "ab­stract" integration theory with respect to a measure (especially Lebesgue measure), and of differential calculus in several variables.

The book may also help more advanced students and researchers perfect their knowledge of certain topics. The index and the relative independence of the chapters should make this type of usage easy.

The important role played by exercises is one of the distinguishing fea­tures of this work. The exercises are very numerous and written in detail, with hints that should allow the reader to overcome any difficulty. Answers that do not appear in the statements are collected at the end of the volume.

There are also many simple application exercises to test the reader's understanding of the text, and exercises containing examples and coun­terexamples, applications of the main results from the text, or digressions to introduce new concepts and present important applications. Thus the text and the exercises are intimately connected and complement each other.

Functional analysis is a vast domain, which we could not hope to cover exhaustively, the more so since there are already excellent treatises on the subject. Therefore we have tried to limit ourselves to results that do not require advanced topological tools: all the material covered requires no more than metric spaces and sequences. No recourse is made to topological

vi Preface

vector spaces in general, or even to locally convex spaces or Frechet spaces. The Baire and Banach- Steinhaus theorems are covered and used only in some exercises. In particular, we have not included the "great" theorems of functional analysis, such as the Open Mapping Theorem, the Closed Graph Theorem, or the Hahn-Banach theorem. Similarly, Fourier transforms are dealt with only superficially, in exercises. Our guiding idea has been to limit the text proper to those results for which we could state significant applications within reasonable limits.

This work is divided into a prologue and three parts. The prologue gathers together fundamentals results about the use of

sequences and, more generally, of countability in analysis. It dwells on the notion of separability and on the diagonal procedure for the extraction of subsequences.

Part I is devoted to the description and main properties of fundamental function spaces and their duals. It covers successively spaces of continuous functions, functional integration theory (Daniell integration) and Radon measures, Hilbert spaces and L1' spaces.

Part II covers the theory of operators. We dwell particularly on spectral properties and on the theory of compact operators. Operators not every­where defined are not discussed.

Finally, Part III is an introduction to the theory of distributions (not in­cluding Fourier transformation of distributions, which is nonetheless an im­portant topic). Differentiation and convolution of distributions are studied in a fair amount of detail. We introduce explicitly the notion of a fundamen­tal solution of a differential operator, and give the classical examples and their consequences. In particular, several regularity results, notably those concerning the Sobolev spaces Wl,1'(JR d ), are stated and proved. Finally, in the last chapter, we study the Laplace operator on a bounded subset of JRd: the Dirichlet problem, spectra, etc. Numerous results from the preceding chapters are used in Part III, showing their usefulness. Prerequisites. We summarize here the main post-calculus concepts and re­sults whose knowledge is assumed in this work.

- Topology of metric spaces: elementary notions: convergence of sequences, lim sup and lim inf, continuity, compactness (in particular the Borel­Lebesgue defining property and the Bolzano-Weierstrass property), and completeness.

- Banach spaces: finite-dimensional normed spaces, absolute convergence of series, the extension theorem for continuous linear maps with values in a Banach space.

- Measure theory: measure spaces, construction of the integral, the Mono­tone Convergence and Dominated Convergence Theorems, the definition and elementary properties of L1' spaces (particularly the Holder and Minkowski inequalities, completeness of L1', the fact that convergence

Preface VII

of a sequence in LP implies the convergence of a subsequence almost everywhere), Fubini's Theorem, the Lebesgue integral.

- Differential calculus: the derivative of a function with values in a Banach space, the Mean Value Theorem.

These results can be found in the following references, among others: For the topology and normed spaces, Chapters 3 and 5 of J. Dieudonne's Foun­dations of Modern Analysis (Academic Press, 1960); for the integration theory, Chapters 1, 2, 3, and 7 of W. Rudin's Real and Complex Analysis, McGraw-Hill; for the differential calculus, Chapters 2 and 3 of H. Cartan's Cours de calcul differentiel (translated as Differential Calculus, Hermann).

We are thankful to Silvio Levy for his translation and for the opportunity to correct here certain errors present in the French original.

We thankfully welcome remarks and suggestions from readers. Please send them by email tohirsch@lamLuniv-evry.frorlacombe@lamLuniv-evry.fr.

Francis Hirsch Gilles Lacombe

Contents

Preface Notation

Prologue: Sequences 1 Count ability . . . . 2 Separability. . .. 3 The Diagonal Procedure 4 Bounded Sequences of Continuous Linear Maps

v xiii

1 1 7

12 18

I FUNCTION SPACES AND THEIR DUALS 25

1 The Space of Continuous Functions on a Compact Set 1 Generalities . . . .... . .... 2 The Stone-Weierstrass Theorems 3 Ascoli's Theorem . .... . . . .

2 Locally Compact Spaces and Radon Measures 1 Locally Compact Spaces. 2 Daniell's Theorem . .... . .. . . . 3 Positive Radon Measures . . .... .

3A Positive Radon Measures on IR and the Stieltjes Integral. . . .

3B Surface Measure on Spheres in IRd 4 Real and Complex Radon Measures ...

27 28 31 42

49 49 57 68

71 74 86

x Contents

3 Hilbert Spaces 97 1 Definitions, Elementary Properties, Examples 97 2 The Projection Theorem . . . . . . . . . . . . 105 3 The Riesz Representation Theorem . . . . . . 111

3A Continuous Linear Operators on a Hilbert Space 112 3B Weak Convergence in a Hilbert Space 114

4 Hilbert Bases . . . . . . . . . . . . . . . . . . . . . . . . 123

4 LP Spaces 143 1 Definitions and General Properties 143 2 Duality... 159 3 Convolution... 169

II OPERATORS 185

5 Spectra 187 1 Operators on Banach Spaces .. ... . . . .... 187 2 Operators in Hilbert Spaces . . . . . . . . . . . . . 201

2A Spectral Properties of Hermitian Operators 203 2B Operational Calculus on Hermitian Operators . 205

6 Compact Operators 213 1 General Properties . . . . . . . . . . . . . . . . . . 213

1A Spectral Properties of Compact Operators . 217 2 Compact Selfadjoint Operators . . . . . . . . . . . 234

2A Operational Calculus and the Fredholm Equation . 238 2B Kernel Operators . . . . . . . . . . . . . . . . . .. 240

III DISTRIBUTIONS

7 Definitions and Examples 1 Test Functions .. .. .

1A Notation .... . 1B Convergence in Function Spaces 1 C Smoothing.. . .... 1D Coo Partitions of Unity

2 Distributions. . ... 2A Definitions....... 2B First Examples . . . . . 2C Restriction and Extension of a Distribution to an

Open Set . ......... . ....... . 2D Convergence of Sequences of Distributions . 2E Principal Values 2F Finite Parts . . . . . . . . . . . . . . . . . .

255

257 257 257 259 261 262 267 267 268

271 272 272 273

3 Complements . . . . . . . . . . . . . . 3A Distributions of Finite Order . 3B The Support of a Distribution . 3C Distributions with Compact Support .

Contents Xl

280 280 281 281

8 Multiplication and Differentiation 287 1 Multiplication . . . . . . . . . . . . . . . . . . . . 287 2 Differentiation... . .. . . .. . . . ... . . . 292 3 Fundamental Solutions of a Differential Operator 306

3A The Laplacian ..... . .. .. 307 3B The Heat Operator . . . . . . . . 310 3C The Cauchy-Riemann Operator 311

9 Convolution of Distributions 317 1 Tensor Product of Distributions . 317 2 Convolution of Distributions . 324

2A Convolution in C' . . . . 324 2B Convolution in~' . . .. 325 2C Convolution of a Distribution with a Function 332

3 Applications . . .... . . . . .. . .. . . 337 3A Primitives and Sobolev's Theorem 337 3B Regularity........ ... 340 3C Fundamental Solutions and

3D Partial Differential Equations The Algebra ~~ . . . .

10 The Laplacian on an Open Set 1 The spaces Hl(O) and HJ(O) 2 The Dirichlet Problem . . . . .

2A The Dirichlet Problem . 2B The Heat Problem . 2C The Wave Problem .

Answers to the Exercises

Index

343 343

349 349 363 366 367 368

379

387

Notation

If A is a subset of X, we denote by A C the complement of A in X. If A c X and B eX, we set A \ B = An B C • The characteristic function of a subset A of X is denoted by 1 A. It is defined by

1 ( ) = {I if x E A, A X 0 if x¢: A .

N, Z, Q, and lR represent the nonnegative integers, the integers, the rationals, and the reals. If IE is one of these sets, we write IE* = IE \ {O}. We also write lR+ = {x E lR : x 2: O}. If a E lR we write a+ = max(O,a) and a- = - min(a, 0).

<C denotes the complex numbers. As usual, if z E <C, we denote by z the complex conjugate of z, and by Rez and Imz the real and imaginary parts of z.

If f is a function from a set X into lR and if a E lR, we write {f > a} = {x EX: f(x) > a} . We define similarly the sets {f < a}, {f 2: a}, {f ~ a}, etc.

As usual, a number x E lR is positive if x > 0, and negative if x < O. However, for the sake of brevity in certain statements, we adopt the con­vention that a real-valued function f is positive if it takes only nonnegative values (including zero), and we denote this fact by f 2: o.

Let (X, d) be a metric space. If A is a subset of X, we denote by A and A the closure and interior of A. If x E X, we write Y(x) for the set of neighborhoods of x (that is, subsets of X whose interior contains x). We set

B(x,r) = {y EX: d(x,y) < r}, B(x,r) = {y EX: d(x,y) ~ r}.

xiv Notation

(We do not necessarily have B(x, r) = B(x, r), but this equality does hold if, for example, X is a normed space with the associated metric.) If X is a normed vector space with norm 11·11, the closed unit ball of X is

B(X) = {x EX: IIxll ~ I} .

When no ambiguity is possible, we write B instead of B(X). If A is a subset of X, the diameter of A is

d(A) = sup d(x,y). x,yEA

If A C X and B eX, the distance between A and B is

d(A,B) = inf d(x,y), (x,y)EAxB

and d( x, A) = d( {x}, A) for x EX. We set OC = JR or C. All vector spaces are over one or the other OC. If

E is a vector space and A is a subset of E, we denote by [AJ the vector subspace generated by A. If E is a vector space, A, B are subsets of E, and >. E OC, we write A + B = {x + y : x E A, y E B} and >'A = {>.x : x E A} .

Lebesgue measure over JR d, considered as a measure on the Borel sets of JRd, is denoted by >'d. We also use the notations d>'d{X) = dx = dXl . . . dXd . We omit the dimension subscript d if there is no danger of confusion.

If x E JRd, the euclidean norm of x is denoted by Ixl.