Function Analysis 泛函分析 Walter Rudin 英文清晰版

ABOUTTHEAUTHOR InadditiontoFunctionalanalysis,secondedition,Walterrudinisthe authoroftwootherbooks:Principlesofmathematicalanalysisandreal andComplexanalysis,whosewidespreaduseisillustratedbythefactthat theyhavebeentranslatedintoatotalof13languages.HewrotePrinciples ofmathematicalAnalysiswhilehewasaC.L.E.MooreInstructoratthe MassachusettsInstituteofTechnology-justtwoyearsafterreceivinghis Ph.D.atDukeUniversity.Later,hetaughtattheUniversityofrochester, andisnowavilasResearchProfessorattheUniversityofWisconsin- Madison.Inthepast,hehasspentleavesatYaleUniversitytheUniversity ofCaliforniainLaJolla,andtheUniversityofHawaii Dr.Rudin'sresearchhasdealtmainlywithharmonicanalysisand withcomplexvariables.Hehaswrittenthreeresearchmonographsonthese topics:FourierAnalysisonGroups,FunctionTheoryinPolydiscs,and FunctionTheoryintheunitBallofcn CONTENTS Preface XIll PartIGeneralTheory 1TopologicalVectorSpaces Introduction Separationproperties 10 Linearmappings 14 Finite-dimensionalspaces Metrization Boundednessandcontinuity 23 Seminormsandlocalconvexity 25 Quotientspaces 30 E Xam 33 Exercises 38 2Completeness 42 Bairecategory TheBanach-Steinhaustheorem 43 Theopenmappingtheorem 47 Theclosedgraphtheorem 50 Bilinearmappings 52 Exercises 53 3Convexity 56 TheHahn-Banachtheorems Weaktopologies 62 Compactconvexsets 68 Vector-valuedintegration 77 Holomorphicfunctions 82 Exercises 85 IX XCONTENTS 4DualityinBanachSpaces 92 Thenormeddualofanormedspace 92 Adjoints 97 Compactoperators 103 E mercies 111 5SomeApplications 116 Acontinuitytheorem 116 ClosedsubspacesofL'-spaces 117 Therangeofavector-valuedmeasure 120 AgeneralizedStone-Weierstrasstheorem 121 Twointerpolationtheorems 124 Kakutani'sfixedpointtheorem 126 Haarmeasureoncompactgroups 128 Uncomplementedsubspaces 132 SumsofPoissonkernels 138 Twomorefixedpointtheorems 139 Exercises Partiidistributionsandfouriertransforms 6TestFunctionsandDistributions 49 Introduction Testfunctionspaces 151 Calculuswithdistributions 157 calization 162 Supportsofdistribution 164 Distributionsasderivatives 167 Convolutions 170 Exercises 177 7FourierTransforms 182 Basicproperties 182 Tempereddistributions 189 Paley-Wienertheorems 196 Sobolev'slemma 202 Exercises 204 8ApplicationstoDifferentialEquations Fundamentalsolutions 210 Ellipticequations 215 Exercises 222 CoNTENTS 9TauberianTheory 226 Wiener'stheorem 226 Theprimenumbertheorem 230 Therenewalequation 236 Exercises 239 PartIiiBanachAlgebrasandspectralTheory 10BanachAlgebras 45 Introduction 245 Complexhomomorphisms 249 Basicpropertiesofspectra 252 Symboliccalculus 258 Thegroupofinvertibleelements 267 Lomonosov'sinvariantsubspacetheorem Exercises 271 11CommutativeBanachalgebras 275 Idealsandhomomorphisms 275 Gelfandtransforms Involutions 287 applicationstononcommutativealgebras 292 Positivefunctionals 296 Exercis 301 12BoundedOperatorsonahilbertspace 306 Basicfacts 306 Boundedoperators 309 Acommutativitytheorem 315 Resolutionsoftheidentity 316 Thespectraltheorem 321 Eigenvaluesofnormaloperators 327 Positiveoperatorsandsquareroots 330 Thegroupofinvertibleoperators 333 AcharacterizationofB*-algebras 336 Anergodictheorem 339 Exercises 341 13UnboundedOperators 347 Introduction 347 Graphsandsymmetricoperators 351 TheCayleytransform 356 Resolutionsoftheidentity 360 Thespectraltheorem 368 Semigroupsofoperators 375 Exercises 385 XICONTENTS AppendixACompactnessandContinuity 391 AppendixbnotesandComments 397 Bibliography 412 ListofSpecialSymbols 414 Index 417 PREFACE Functionalanalysisisthestudyofcertaintopological-algebraicstructures andofthemethodsbywhichknowledgeofthesestructurescanbeapplied toanalyticproblems agoodintroductorytextonthissubjectshouldincludeapresentation ofitsaxiomatics(i.e.,ofthegeneraltheoryoftopologicalvectorspaces),it shouldtreatatleastafewtopicsinsomedepth,anditshouldcontainsome interestingapplicationstootherbranchesofmathematics.ihopethatthe presentbookmeetsthesecriteria Thesubjectishugeandisgrowingrapidly.Thebibliographyin volumeIof[4]contains96pagesandgoesonlyto1957.Inordertowrite abookofmoderatesizeitwasthereforenecessarytoselectcertainareas andtoignoreothers.Ifullyrealizethatalmostanyexpertwholooksatthe tableofcontentswillfindthatsomeofhisorher(andmy)favoritetopics aremissing,butthisseemsunavoidable.Itwasnotmyintentiontowritean encyclopedictreatise.Iwantedtowriteabookthatwouldopenthewayto furtherexploration Thisisthereasonforomittingmanyofthemoreesoterictopicsthat mighthavebeenincludedinthepresentationofthegeneraltheoryoftopo logicalvectorspaces.Forinstance,thereisnodiscussionofuniformspaces ofMoore-Smithconvergence,ofnets,oroffilters.Thenotionofcomplete nessoccursonlyinthecontextofmetricspaces.Bornologicalspacesare notmentioned,norarebarreledones.Dualityisofcoursepresented,but notinitsutmostgenerality.Integrationofvector-valuedfunctionsistreated strictlyasatool;attentionisconfinedtocontinuousintegrandswithvalues InaFrechetspace Nevertheless,thematerialofPartiisfullyadequateforalmostall applicationstoconcreteproblems.Andthisiswhatoughttobestressedin suchacourse:Thecloseinterplaybetweentheabstractandtheconcreteis XIVPREFACE notonlythemostusefulaspectofthewholesubjectbutalsothemost fascinatingone Herearesomefurtherfeaturesoftheselectedmaterial.afairlylarge partofthegeneraltheoryispresentedwithouttheassumptionoflocalcon- vexity.Thebasicpropertiesofcompactoperatorsarederivedfromthe dualitytheoryinBanachspaces.TheKrein-Milmantheoremontheexis- tenceofextremepointsisusedinseveralwaysinChapter5.Thetheoryof distributionsandFouriertransformsisworkedoutinfairdetailandis applied(intwoverybriefchapters)totwoproblemsinpartialdifferential equationsaswellastoWienerstauberiantheoremandtwoofitsapplica tions.ThespectraltheoremisderivedfromthetheoryofBanachalgebras (specifically,fromtheGelfand-Naimarkcharacterizationofcommutative BW-algebras);thisisperhapsnottheshortestway,butitisaneasyone.The symboliccalculusinBanachalgebrasisdiscussedinconsiderabledetail:so areinvolutionsandpositivefunctionals IassumefamiliaritywiththetheoryofmeasureandLebesgueintegra tion(includingsuchfactsasthecompletenessoftheIP-spaces),withsome basicpropertiesofholomorphicfunctions(suchasthegeneralformof Cauchystheorem,andRungestheorem),andwiththeelementarytopo- logicalbackgroundthatgoeswiththesetwoanalytictopics.Someother topologicalfactsarebrieflypresentedinAppendixA.almostnoalgebraic backgroundisneeded,beyondtheknowledgeofwhatahomomorphismis HistoricalreferencesaregatheredinAppendixB.Someoftheserefer totheoriginalsources,andsometomorerecentbooks,papers,orexposi toryarticlesinwhichfurtherreferencescanbefoundThereare,ofcourse manyitemsthatarenotdocumentedatall.innocasedoestheabsenceofa specificreferenceimplyanyclaimtooriginalityonmypart MostoftheapplicationsareinChapters5,8,and9.Somearein Chapter11andinthemorethan250exercises;manyofthesearesupplied withhints.Theinterdependenceofthechaptersisindicatedinthediagram onthefollowingpage MostoftheapplicationscontainedinChapter5canbetakenupwell beforethefirstfourchaptersarecompletedIthasthereforebeensuggested thatitmightbegoodpedagogytoinsertthemintothetextearlier,assoon astherequiredtheoreticalbackgroundisestablished.However,inorder nottointerruptthepresentationofthetheoryinthisway,Ihaveinstead startedChapter5withashortindicationofthebackgroundthatisneeded foreachitem.Thisshouldmakeiteasytostudytheapplicationsasearlyas possible,ifsodesired Inthefirstedition,afairlylargepartofChapter10dealtwithdiffer entiationinBanachalgebras.Twentyyearsagothis(thenrecent)material lookedinterestingandpromising,butitdoesnotseemtohaveledany- where,andIhavedeletedit.Ontheotherhand.Ihaveaddedafewitems whichwereeasytofitintotheexistingtext:themeanergodictheoremof