Elements of Functional Analysis

399 Pages·1999·29.714 MB·English

by  Lacombe, Gilles;Hirsch, Francis

192
Graduate Texts in Mathematics
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. AC ourse 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. AC ourse 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 Translator
Gilles Lacombe Silvio Levy
Departement de Mathematiques Mathematical Sciences Research Institute
Universite d'Evry-Val d'Essonne l ()()() Centennial Drive
Boulevard des coquibus Berkeley, CA 94720-5070
Evry Cedex F-91025 USA
France
Editorial Board
S. Axler F.W. Gehring K.A. Ribet
Mathematics Department Mathematics Department Mathematics Department
San Francisco State East HaU University of California
University University of Michigan at Berkeley
San Francisco, CA 94132 Ann Arbor, MI 48109 Berkeley, CA 94720-3840
USA USA 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 15 .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
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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 Univer
sity of Evry-Val d'Essonne. It is meant primarily for graduate students
in mathematics. To make it into a useful tool, appropriate to their knowl
edge 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 "ab
stract" 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 fea
tures 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 coun
terexamples, 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 every
where defined are not discussed.
Finally, Part III is an introduction to the theory of distributions (not in
cluding Fourier transformation of distributions, which is nonetheless an im
portant topic). Differentiation and convolution of distributions are studied
in a fair amount of detail. We introduce explicitly the notion of a fundamen
tal 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'(JRd), 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 re
sults 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 Borel
Lebesgue 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 Mono
tone 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 Foun
dations 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 [email protected]@lamLuniv-evry.fr.
Francis Hirsch
Gilles Lacombe
Contents
Preface v
Notation xiii
Prologue: Sequences 1
1 Count ability . . . . 1
2 Separability. . .. 7
3 The Diagonal Procedure 12
4 Bounded Sequences of Continuous Linear Maps 18
I FUNCTION SPACES AND THEIR DUALS 25
1 The Space of Continuous Functions on a Compact Set 27
1 Generalities . . . .... . .... 28
2 The Stone-Weierstrass Theorems 31
3 Ascoli's Theorem . .... . . . . 42
2 Locally Compact Spaces and Radon Measures 49
1 Locally Compact Spaces. 49
2 Daniell's Theorem . .... . .. . . . 57
3 Positive Radon Measures . . .... . 68
3A Positive Radon Measures on IR
and the Stieltjes Integral. . . . 71
3B Surface Measure on Spheres in IRd 74
4 Real and Complex Radon Measures ... 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 . . . . . . . . . . . . . ... . . 2. 40
III DISTRIBUTIONS 255
7 Definitions and Examples 257
1 Test Functions .. .. . 257
1A Notation .... . 257
1B Convergence in Function Spaces 259
1C Smoothing.. . .... 261
1D Coo Partitions of Unity 262
2 Distributions. . ... 267
2A Definitions....... 267
2B First Examples . . . . . 268
2C Restriction and Extension of a Distribution to an
Open Set . ......... . ....... . 271
2D Convergence of Sequences of Distributions . 272
2E Principal Values 272
2F Finite Parts . . . . . . . . . . . . . . . . . . 273