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 Copying and reprinting. Individual readers of this publication, and nonprofit libraries actingforthem,arepermittedtomakefairuseofthematerial,suchastocopyachapterforuse in teaching or research. Permission is granted to quote brief passages from this publication in reviews,providedthecustomaryacknowledgmentofthesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication is permitted only under license from the American Mathematical Society. Requests for such permissionshouldbeaddressedtotheAssistanttothePublisher,AmericanMathematicalSociety, P.O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to [email protected]. (cid:2)c 2002bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageatURL:http://www.ams.org/ 1098765432 17161514(cid:160)1312 (cid:2)(cid:2) 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