An Introduction to Functional Analysis.pdf

An Introduction to Functional Analysis.pdf

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AnIntroductiontoFunctionalAnalysisThisaccessibletextcoverskeyresultsinfunctionalanalysisthatareessentialforfur-therstudyinthecalculusofvariations,analysis,dynamicalsystems,andthetheoryofpartialdifferentialequations.ThetreatmentofHilbertspacescoversthetopicsrequiredtoprovetheHilbert-SchmidtTheorem,includingorthonormalbases,theRieszRepre-sentationTheorem,andthebasicsofspectraltheory.ThematerialonBanachspacesandtheirdualsincludestheHahn-BanachTheorem,theKrein-MilmanTheorem,andresultsbasedontheBaireCategoryTheorem,beforeculminatinginaproofofsequen-tialweakcompactnessinreflexivespaces.Argumentsarepresentedindetail,andmorethan200fully-workedexercisesareincludedtoprovidepracticeapplyingtechniquesandideasbeyondthemajortheorems.Familiaritywiththebasictheoryofvectorspacesandpoint-settopologyisassumed,butknowledgeofmeasuretheoryisnotrequired,makingthisbookidealforupperundergraduate-levelandbeginninggraduate-levelcourses.JAMEsRoBINsoNisaprofessorintheMathematicsInstituteattheUniversityofWarwick.HehasbeentherecipientofaRoyalSocietyUniversityResearchFellowshipandanEPSRCLeadershipFellowship.Hehaswrittensixbooksinadditiontohismanypublicationsininfinite-dimensionaldynamicalsystems,dimensiontheory,andpartialdifferentialequations.AnIntroductiontoFunctionalAnalysisJAMESC.ROBINSONUniversityofWarwickCAMBRIDGEUNIVERSITYPRESSCAMBRIDGEUNIVERSITYPRESSUniversityPrintingHouse,CambridgeCB28BS,UnitedKingdomOneLibertyPlaza,20thFloor,NewYork,NY10006,USA477WilliamstownRoad,PortMelbourne,VIC3207,Australia314-321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,NewDelhi-110025,India79AnsonRoad,#06-04/06,Singapore079906CambridgeUniversityPressispartoftheUniversityofCambridge.ItfurtherstheUniversitysmissionbydisseminatingknowledgeinthepursuitofeducation,learning,andresearchatthehighestinternationallevelsofexcellence.www.cambridge.orgInformationonthistitle:

www.cambridge.org/9780521899642DOI:

10.1017/9781139030267JamesC.Robinson2020Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisionsofrelevantcollectivelicensingagreements,noreproductionofanypartmaytakeplacewithoutthewrittenpermissionofCambridgeUniversityPress.Firstpublished2020PrintedintheUnitedKingdombyTJInternationalLtd.PadstowCornwallAcataloguerecordforthispublicationisavailablefromtheBritishLibrary.ISBN978-0-521-89964-2HardbackISBN978-0-521-72839-3PaperbackCambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublicationanddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate.ToMum&Dad123ContentsPrefacePARTIPRELIMINARIESVectorSpacesandBases1.1DefinitionofaVectorSpace1.2ExamplesofVectorSpaces1.3LinearSubspacespagexm133461.4SpanningSets,LinearIndependence,andBases71.5LinearMapsbetweenVectorSpacesandTheirInverses101.6ExistenceofBasesandZornsLemma13ExercisesMetricSpaces2.1MetricSpaces2.2OpenandClosedSets2.3ContinuityandSequentialContinuity2.4Interior,Closure,Density,andSeparability2.5CompactnessExercisesPARTIINORMEDLINEARSPACESNormsandNormedSpaces3.1Norms3.2ExamplesofNormedSpaces3.3ConvergenceinNormedSpaces3.4EquivalentNorms3.5IsomorphismsbetweenNormedSpaces3.6SeparabilityofNormedSpacesExercisesVll1517171922232530333535384243464850VlllContents4CompleteNormedSpaces534.1BanachSpaces534.2ExamplesofBanachSpaces564.2.1SequenceSpaces574.2.2SpacesofFunctions584.3SequencesinBanachSpaces614.4TheContractionMappingTheorem63Exercises645Finite-DimensionalNormedSpaces665.1EquivalenceofNormsonFinite-DimensionalSpaces665.2CompactnessoftheClosedUnitBall68Exercises706SpacesofContinuousFunctions716.1TheWeierstrassApproximationTheorem716.2TheStone-WeierstrassTheorem776.3TheArzela-AscoliTheorem83Exercises867CompletionsandtheLebesgueSpacesLP（Q）897.1Non-completenessofC（O,l）withtheL1Norm897.2TheCompletionofaNormedSpace917.3DefinitionoftheLPSpacesasCompletions94Exercises97PARTIIIHILBERTSPACES998HilbertSpaces1018.1InnerProducts1018.2TheCauchy-SchwarzInequality1038.3PropertiesoftheInducedNorms1058.4HilbertSpaces107Exercises1089OrthonormalSetsandOrthonormalBasesforHilbertSpaces1109.1SchauderBasesinNormedSpaces1109.2OrthonormalSets1129.3ConvergenceofOrthogonalSeries1159.4OrthonormalBasesforHilbertSpaces1179.5SeparableHilbertSpaces122Exercises123ContentsIX10ClosestPointsandApproximation12610.1ClosestPointsinConvexSubsetsofHilbertSpaces12610.2LinearSubspacesandOrthogonalComplements12910.3BestApproximations131Exercises13411LinearMapsbetweenNormedSpaces13711.1BoundedLinearMaps13711.2SomeExamplesofBoundedLinearMaps14111.3CompletenessofB（X,Y）WhenYIsComplete14511.4KernelandRange14611.5InversesandInvertibility147Exercises15012DualSpacesandtheRieszRepresentationTheorem15312.1TheDualSpace15312.2TheRieszRepresentationTheorem155Exercises15713TheHilbertAdjointofaLinearOperator15913.1ExistenceoftheHilbertAdjoint15913.2SomeExamplesoftheHilbertAdjoint162Exercises16414TheSpectrumofaBoundedLinearOperator16514.1TheResolventandSpectrum16514.2TheSpectralMappingTheoremforPolynomials169Exercises17115CompactLinearOperators17315.1CompactOperators17315.2ExamplesofCompactOperators17515.3TwoResultsforCompactOperators177Exercises17816TheHilbert-SchmidtTheorem18016.1EigenvaluesofSelf-AdjointOperators18016.2EigenvaluesofCompactSelf-AdjointOperators18216.3TheHilbert-SchmidtTheorem184Exercises18817Application:

Sturm-LiouvilleProblems19017.1SymmetryofLandtheWronskian191xContents17.2TheGreensFunction19317.3EigenvaluesoftheSturm-LiouvilleProblem195PARTIVBANACHSPACES19918DualSpacesofBanachSpaces20118.1TheYoungandHolderInequalities20218.2TheDualSpacesof,f,P20418.3DualSpacesofLP（Q）207Exercises20819TheHahn-BanachTheorem21019.1TheHahn-BanachTheorem:

RealCase21019.2TheHahn-BanachTheorem:

ComplexCase214Exercises21720SomeApplicationsoftheHahn-BanachTheorem21920.1ExistenceofaSupportFunctional21920.2TheDistanceFunctional22020.3SeparabilityofX*ImpliesSeparabilityofX22120.4AdjointsofLinearMapsbetweenBanachSpaces22220.5GeneralisedBanachLimits224Exercises22621ConvexSubsetsofBanachSpaces22821.1TheMinkowskiFunctional22821.2SeparatingConvexSets23021.3LinearFunctionalsandHyperplanes23321.4CharacterisationofClosedConvexSets23421.5TheConvexHull23521.6TheKrein-MilmanTheorem236Exercises23922ThePrincipleofUniformBoundedness24022.1TheBaireCategoryTheorem24022.2ThePrincipleofUniformBoundedness24222.3FourierSeriesofContinuousFunctions244Exercises24723TheOpenMapping,InverseMapping,andClosedGraphTheorems24923.1TheOpenMappingandInverseMappingTheorems24923.2SchauderBasesinSeparableBanachSpaces252Contents23.3TheClosedGraphTheoremExercises24SpectralTheoryforCompactOperators24.1PropertiesofT-IWhenTIsCompact24.2PropertiesofEigenvaluesXl25525625825826225UnboundedOperatorsonHilbertSpaces26425.1AdjointsofUnboundedOperators26525.2ClosedOperatorsandtheClosureofSymmetricOperators26725.3TheSpectrumofClosedUnboundedSelf-AdjointOperators26926ReflexiveSpaces27326.1TheSecondDual27326.2SomeExamplesofReflexiveSpaces26.3XIsReflexiveIfandOnlyIfX*IsReflexiveExercises27WeakandWeak-*Convergence27.1WeakConvergence27527728028228227.2ExamplesofWeakConvergenceinVariousSpaces28527.2.1WeakConvergenceinf,P,1p.xEVfor.EIK,xEV,（1.1）（i）additiveandmultiplicativeidentitiesexist:

thereexistsazeroelement0EVsuchthatx+0=xforallxEV;and1EIKistheidentityforscalarmultiplication,Ix=xforallxEV;（ii）thereareadditiveinverses:

foreveryxEVthereexistsanelement-xEVsuchthatx+（-x）=0;34VectorSpacesandBases（iii）additioniscommutativeandassociative,andforallx,y,zEV;andx+（y+z）=（x+y）+z,（iv）multiplicationisassociative,a（f3x）=（af3）xanddistributive,a（x+y）=ax+ayforalla,f3EIK,x,yEV.forallCi,f3EIK,XEV,and（a+f3）x=ax+f3xIncheckingthataparticularcollectionVisavectorspaceoverIK,properties（i）-（iv）areoftenimmediate;oneusuallyhastocheckonlythatVisclosedunderadditionandscalarmultiplication（i.e.that（1.1）holds）.1.2ExamplesofVectorSpacesOfcourse,IR.nisarealvectorspaceoverIR;butisnotavectorspaceoverC,sinceixIRnforany1xEIRn.Incontrast,ccncanbeavectorspaceoverbothIRandC;thespaceccnoverIRis（accordingtotheterminologyintro-ducedabove）arealvectorspace.Thisexampleisausefulillustrationthatthereal/complexlabelreferstothefieldIK,i.e.theallowablescalarmultiples,ratherthantotheelementsofthespaceitself.GivenanytwovectorspacesViandV2overIK,theproductspaceVixV2consistingofallpairs（v1,v2）withv1EViandv2EV2isanothervectorspaceifwedefineandforv1,u1EVi,v2,u2EV2,aEIK.Wenowintroducesomelesstrivialexamples.Example1.2ThespaceJ（U,V）ofallfunctionsf:

U-+V,whereUandVarebothvectorspacesoverthesamefieldIK,isitselfavectorspace,ifweusetheobviousdefinitionsofwhatadditionandscalarmultiplicationshouldmeanforfuncti