Principles of functional analysis

450 Pages·2002·1.571 MB·English

by  Martin Schechter

Principles of
Functional Analysis
SECOND EDITION
Martin Schechter
Graduate Studies
in Mathematics
Volume 36
American Mathematical Society
Selected Titles in This Series
36 Martin Schechter, Principlesoffunctionalanalysis,secondedition,2002
35 James F. Davis and Paul Kirk, Lecturenotesinalgebraictopology,2001
34 Sigurdur Helgason, Differentialgeometry,Liegroups,andsymmetricspaces,2001
33 Dmitri Burago, Yuri Burago, and Sergei Ivanov, Acourseinmetricgeometry,2001
32 Robert G. Bartle, Amoderntheoryofintegration,2001
31 Ralf Korn and Elke Korn, Optionpricingandportfoliooptimization:Modernmethods
offinancialmathematics,2001
30 J. C. McConnell and J. C. Robson, NoncommutativeNoetherianrings,2001
29 Javier Duoandikoetxea, Fourieranalysis,2001
28 Liviu I. Nicolaescu, NotesonSeiberg-Wittentheory,2000
27 Thierry Aubin, Acourseindifferentialgeometry,2001
26 Rolf Berndt, Anintroductiontosymplecticgeometry,2001
25 Thomas Friedrich, DiracoperatorsinRiemanniangeometry,2000
24 Helmut Koch, Numbertheory:Algebraicnumbersandfunctions,2000
23 Alberto Candel and Lawrence Conlon, FoliationsI,2000
22 Gu¨nter R. Krause and Thomas H. Lenagan, GrowthofalgebrasandGelfand-Kirillov
dimension,2000
21 John B. Conway, Acourseinoperatortheory,2000
20 Robert E. Gompf and Andr´as I. Stipsicz, 4-manifoldsandKirbycalculus,1999
19 Lawrence C. Evans, Partialdifferentialequations,1998
18 Winfried Just and Martin Weese, Discoveringmodernsettheory.II:Set-theoretic
toolsforeverymathematician,1997
17 Henryk Iwaniec, Topicsinclassicalautomorphicforms,1997
16 Richard V. Kadison and John R. Ringrose, Fundamentalsofthetheoryofoperator
algebras.VolumeII:Advancedtheory,1997
15 Richard V. Kadison and John R. Ringrose, Fundamentalsofthetheoryofoperator
algebras.VolumeI:Elementarytheory,1997
14 Elliott H. Lieb and Michael Loss, Analysis,1997
13 Paul C. Shields, Theergodictheoryofdiscretesamplepaths,1996
12 N. V. Krylov, LecturesonellipticandparabolicequationsinH¨olderspaces,1996
11 Jacques Dixmier, Envelopingalgebras,1996Printing
10 Barry Simon, Representationsoffiniteandcompactgroups,1996
9 Dino Lorenzini, Aninvitationtoarithmeticgeometry,1996
8 Winfried Just and Martin Weese, Discoveringmodernsettheory.I:Thebasics,1996
7 Gerald J. Janusz, Algebraicnumberfields,secondedition,1996
6 Jens Carsten Jantzen, Lecturesonquantumgroups,1996
5 Rick Miranda, AlgebraiccurvesandRiemannsurfaces,1995
4 Russell A. Gordon, TheintegralsofLebesgue,Denjoy,Perron,andHenstock,1994
3 William W. Adams and Philippe Loustaunau, AnintroductiontoGro¨bnerbases,
1994
2 Jack Graver, Brigitte Servatius, and Herman Servatius, Combinatorialrigidity,
1993
1 Ethan Akin, Thegeneraltopologyofdynamicalsystems,1993

Principles of
Functional Analysis

Principles of
Functional Analysis
SECOND EDITION
Martin Schechter
Graduate Studies
in Mathematics
Volume 36
American Mathematical Society
Providence,Rhode Island
Editorial Board
Steven G. Krantz
David Saltman (Chair)
David Sattinger
Ronald Stern
2000 Mathematics Subject Classification. Primary 46–01, 47–01, 46B20, 46B25, 46C05,
47A05, 47A07, 47A12, 47A53, 47A55.
Abstract. The book is intended for a one-year course for beginning graduate or senior under-
graduatestudents. However,itcanbeusedatanylevelwherethestudentshavetheprerequisites
mentionedbelow. Becauseofthecrucialroleplayedbyfunctionalanalysisintheappliedsciences
aswellasinmathematics,theauthorattemptedtomakethisbookaccessibletoaswideaspec-
trumofbeginningstudentsaspossible. Muchofthebookcanbeunderstoodbyastudenthaving
taken a course in advanced calculus. However, in several chapters an elementary knowledge of
functionsofacomplexvariableisrequired. TheseincludeChapters6,9,and11. Onlyrudimen-
tary topological or algebraic concepts are used. They are introduced and proved as needed. No
measuretheoryisemployedormentioned.
Library of Congress Cataloging-in-Publication Data
Schechter,Martin.
Principlesoffunctionalanalysis/MartinSchechter.—2nded.
p.cm. —(Graduatestudiesinmathematics,ISSN1065-7339;v.36)
Includesbibliographicalreferencesandindex.
ISBN0-8218-2895-9(alk.paper)
1.Functionalanalysis. I.Title. II.Series.
QA320.S32 2001
515(cid:2).7—dc21 2001031601
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actingforthem,arepermittedtomakefairuseofthematerial,suchastocopyachapterforuse
in teaching or research. Permission is granted to quote brief passages from this publication in
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Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication
is permitted only under license from the American Mathematical Society. Requests for such
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P.O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to
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BS D
To my wife, children, and grandchildren.
May they enjoy many happy years.

Contents
PREFACE TO THE REVISED EDITION xv
FROM THE PREFACE TO THE FIRST EDITION xix
Chapter 1. BASIC NOTIONS 1
§1.1. A problem from differential equations 1
§1.2. An examination of the results 6
§1.3. Examples of Banach spaces 9
§1.4. Fourier series 17
§1.5. Problems 24
Chapter 2. DUALITY 29
§2.1. The Riesz representation theorem 29
§2.2. The Hahn-Banach theorem 33
§2.3. Consequences of the Hahn-Banach theorem 36
§2.4. Examples of dual spaces 39
§2.5. Problems 51
Chapter 3. LINEAR OPERATORS 55
§3.1. Basic properties 55
§3.2. The adjoint operator 57
§3.3. Annihilators 59
§3.4. The inverse operator 60
§3.5. Operators with closed ranges 66
§3.6. The uniform boundedness principle 71
ix