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简介:本文档为《Advanced_Calculus_An_Intro_to_Linear_Analysispdf》,可适用于高等教育领域,主题内容包含•leonardFRichardsonADVANCEDCALCULUSAnIntroductiontoLinearAnalysisLeonardFR符等。

•leonardFRichardsonADVANCEDCALCULUSAnIntroductiontoLinearAnalysisLeonardFRichardsonWILEYINTERSCIENCEAJOHNWILEYSONS,INC,PUBLICATIONThispageintentionallyleftblankADVANCEDCALCULUSThispageintentionallyleftblankADVANCEDCALCULUSAnIntroductiontoLinearAnalysisLeonardFRichardsonWILEYINTERSCIENCEAJOHNWILEYSONS,INC,PUBLICATIONCopyrightbyJohnWileySons,IncAllrightsreservedPublishedbyJohnWileySons,Inc,Hoboken,NewJerseyPublishedsimultaneouslyinCanadaNopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformorbyanymeans,electronic,mechanical,photocopying,recording,scanning,orotherwise,exceptaspermittedunderSectionIorIoftheUnitedStatesCopyrightAct,withouteitherthepriorwrittenpermissionofthePublisher,orauthorizationthroughpaymentoftheappropriatepercopyfcctotheCopyrightClearanceCenter,Inc,RosewoodDrive,Danvers,MA,(),fax(),oronthewebatwwwcopyrightcomRequeststothePublisherforpermissionshouldbeaddressedtothePermissionsDepartment,JohnWileySons,Inc,IllRiverStreet,Hoboken,NJ,(I),fax(I),oronlineathttp:wwwwilcycomgopcrmissionLimitofLiabilityDisclaimerofWarranty:Whilethepublisherandauthorhaveusedtheirbesteffortsinpreparingthisbook,theymakenorepresentationsorwarrantieswithrespecttotheaccuracyorcompletenessofthecontentsofthisbookandspecificallydisclaimanyimpliedwarrantiesofmerchantabilityorfitnessforaparticularpurposeNowarrantymaybecreatedorextendedbysalesrepresentativesorwrittensalesmaterialsTheadviceandstrategiescontainedhereinmaynotbesuitableforyoursituationYoushouldconsultwithaprofessionalwhereappropriateNeitherthepublishernorauthorshallbeliableforanylossofprofitoranyothercommercialdamages,includingbutnotlimitedtospecial,incidental,consequential,orotherdamagesForgeneralinformationonourotherproductsandservicesorfortechnicalsupport,pleasecontactourCustomerCareDepartmentwithintheUnitedStatesat(),outsidetheUnitedStatesat()orfax()WileyalsopublishesitsbooksinavarietyofelectronicformatsSomecontentthatappearsinprintmaynotbeavailableinelectronicformatsFormoreinformationaboutWileyproducts,visitourwebsiteatwwwwilcycomLibraryofCongressCataloginginPublicationData:Richardson,LeonardFAdvancedcalculus:anintroductiontolinearanalysisILeonardFRichardsonpcmIncludesbibliographicalreferencesandindexISBN(cloth)ICalculusITitleQARdcPrintedinMexicoToJoan,Daniel,andJosephThispageintentionallyleftblankCONTENTSPrefaceAcknowledgmentsIntroductionPARTIADVANCEDCALCULUSINONEVARIABLERealNumbersandLimitsofSequencesTheRealNumberSystemExercisesLimitsofSequencesCauchySequencesExercisesTheCompletenessAxiomandSomeConsequencesExercisesAlgebraicCombinationsofSequencesExercisesTheBolzanoWeierstrassTheoremExercisesTheNestedIntervalsTheoremXlllXIXxxiviiviiiCONTENTSExercisesTheHeineBorelCoveringTheoremExercisesCountabilityoftheRationalNumbersExercisesTestYourselfExercisesContinuousFunctionsLimitsofFunctionsExercisesContinuousFunctionsExercisesSomePropertiesofContinuousFunctionsExercisesExtremeValueTheoremandItsConsequencesExercisesTheBanachSpaceCa,bExercisesTestYourselfExercisesRiemannIntegralDefinitionandBasicPropertiesExercisesTheDarbouxIntegrabilityCriterionExercisesIntegralsofUniformLimitsExercisesTheCauchySchwarzInequalityExercisesTestYourselfExercisesTheDerivativeDerivativesandDifferentialsExercisesTheMeanValueTheoremCONTENTSixExercisesTheFundamentalTheoremofCalculusExercisesUniformConvergenceandtheDerivativeExercisesCauchy'sGeneralizedMeanValueTheoremExercisesTaylor'sTheoremExercisesTestYourselfExercisesInfiniteSeriesSeriesofConstantsExercisesConvergenceTestsforPositiveTermSeriesExercisesAbsoluteConvergenceandProductsofSeriesExercisesTheBanachSpacelandItsDualSpaceExercisesSeriesofFunctions:TheWeierstrassMTestExercisesPowerSeriesExercisesRealAnalyticFunctionsandc=FunctionsExercisesWeierstrassApproximationTheoremExercisesTestYourselfExercisesPARTIIADVANCEDTOPICSINONEVARIABLEFourierSeriesTheVibratingStringandTrigonometricSeriesExercisesEuler'sFormulaandtheFourierTransformExercisesXCONTENTSBessel'sInequalityandlzExercisesUniformConvergenceRiemannLocalizationExercisesLConvergencetheDualoflExercisesTestYourselfExercisesTheRlemannStieltjesIntegralFunctionsofBoundedVariationExercisesRiemannStieltjesSumsandIntegralsExercisesRiemannStieltjesIntegrabilityTheoremsExercisesTheRieszRepresentationTheoremExercisesTestYourselfExercisesPARTIllADVANCEDCALCULUSINSEVERALVARIABLESEuclideanSpaceEuclideanSpaceasaCompleteNorrnedVectorSpaceExercisesOpenSetsandClosedSetsExercisesCompactSetsExercisesConnectedSetsExercisesTestYourselfExercisesContinuousFunctionsonEuclideanSpaceLimitsofFunctionsExercisesCONTENTSxiContinuousFunctionsExercisesContinuousImageofaCompactSetExercisesContinuousImageofaConnectedSetExercisesTestYourselfExercisesTheDerivativeinEuclideanSpaceLinearTransformationsandNormsExercisesDifferentiableFunctionsExercisesTheChainRuleinEuclideanSpaceTheMeanValueTheoremTaylor'sTheoremExercisesInverseFunctionsExercisesImplicitFunctionsExercisesTangentSpacesandLagrangeMultipliersExercisesTestYourselfExercisesRiemannIntegrationinEuclideanSpaceDefinitionoftheIntegralExercisesLebesgueSetsandJordanSetsExercisesLebesgue'sCriterionforRiemannIntegrabilityExercisesFubini'sTheoremExercisesJacobianTheoremforChangeofVariablesExercisesXiiCONTENTSTestYourselfExercisesAppendixA:SetTheoryAITerminologyandSymbolsExercisesAParadoxesProblemSolutionsReferencesIndexPREFACEWhythisBookwasWrittenThecourseknownasAdvancedCalculus(orIntroductoryAnalysis)standsatthesummitoftherequirementsforseniormathematicsmajorsAnimportantobjectiveofthiscourseistopreparethestudentforacriticalchallengethatheorshewillfaceinthefirstyearofgraduatestudy:thecoursecalledAnalysisI,LebesgueMeasureandIntegration,orIntroductoryFunctionalAnalysisWeliveinaneraofrapidchangeonaglobalscaleAndtheauthorandhisdepartmenthavebeentestingwaystoimprovethepreparationofmathematicsmajorsforthechallengestheywillfaceDuringthepastquartercenturytheUnitedStateshasemergedasthedestinationofchoiceforgraduatestudyinmathematicsTheinfluxofwellprepared,talentedstudentsfromaroundtheworldbringsconsiderablebenefittoAmericangraduateprogramsTheinternationalstudentsusuallyarrivebetterpreparedforgraduatestudyinmathematicsinparticularbetterpreparedinanalysisthantheirtypicalUScounterpartsTherearemanyreasonsforthis,including(a)schoolsystemsabroadthatareorientedtowardteachingonlythebrighteststudents,and(b)theselfselectionthatispartofastudenttakingthestepoftravelabroadtostudyinaforeigncultureThepresenceofstronglypreparedinternationalstudentsintheclassroomraisesthelevelatwhichcoursesaretaughtThusitisappropriateatthepresenttime,intheearlyyearsofthenewmillennium,forcollegeanduniversitymathematicsdepartmentstoxiiiXiVPREFACEreconsidertheiradvancedcalculuscourseswithaneyetowardpreparinggraduatesfortheinternationalenvironmentinAmericangraduateschoolsThisisachallenge,butitisalsoanopportunityforAmericanstudentsandinternationalstudentstolearnsidebysidewith,andalsoabout,oneanotherItismoreimportantthanevertoteachundergraduateadvancedcalculusoranalysisinsuchawayastoprepareandreorientthestudentforgraduatestudyasitistodayinmathematicsAnotherrecentchangeisthatappliedmathematicshasemergedonalargescaleasanimportantcomponentofmanymathematicsdepartmentsInappliedandnumericalmathematics,functionalanalysisatthegraduatelevelplaysaveryimportantroleYetanotherchangethatisemergingisthatundergraduatesplanningcareersinthesecondaryteachingofmathematicsarebeingrequiredtomajorinmathematicsinsteadofeducationThesestudentsmustbepreparedtoteachthenextgenerationofyoungpeoplefortheworldinwhichtheywillliveWhetherornotthemathematicsmajorisplanninganacademiccareer,heorshewillbenefitfrombetterpreparationinadvancedcalculusforcareersintheemergingworldTheauthorhastaughtmathematicsmajorsandgraduatestudentsforthirtysevenyearsHehasservedasdirectorofhisdepartment'sgraduateprogramfornearlytwodecadesAllthechangesdescribedabovearepresenttodayintheauthor'sdepartmentThisbookhasbeenwritteninthehopeofaddressingthefollowingneedsStudentsofmathematicsshouldacquireasenseoftheunityofmathematicsHenceacoursedesignedforseniormathematicsmajorsshouldhaveanintegrativeeffectSuchacourseshoulddrawuponatleasttwobranchesofmathematicstoshowhowtheymaybecombinedwithilluminatingeffectStudentsshouldlearntheimportanceofrigorousproofanddevelopskillincoherentwrittenexpositiontocountertheuniversaltemptationtoengageinwishfulthinkingStudentsneedpracticecomposingandwritingproofsoftheirown,andthesemustbecheckedandcorrectedThefundamentaltheoremsoftheintroductorycalculuscoursesneedtobeestablishedrigorously,alongwiththetraditionaltheoremsofadvancedcalculus,whicharerequiredforthispurposeThetaskofestablishingtherigorousfoundationsofcalculusshouldbeenlivenedbytakingthisopportunitytointroducethestudenttomodernmathematicalstructuresthatwerenotpresentedinintroductorycalculuscoursesStudentsshouldlearntherigorousfoundationsofcalculusinamannerthatreorient<thinkinginthedirectionstakenbymodernanalysisTheclassictheoremsshouldbecouchedinamannerthatreflectstheperspectivesofmodemanalysisFeaturesofthisTextPREFACEXVTheauthorhasattemptedtoaddresstheseneedspresentedaboveinthefollowingmannerThetwopartsofmathematicsthathavebeenstudiedbynearlyeverymathematicsmajorpriortothesenioryearareintroductorycalculus,includingcalculusofseveralvariables,andlinearalgebraThustheauthorhaschosentohighlighttheinterplaybetweenthecalculusandlinearalgebra,emphasizingtheroleoftheconceptsofavectorspace,alineartransformation(includingalinearfunctional),anorm,andascalarproductForexample,thecustomarytheoremconcerninguniformlimitsofcontinuousfunctionsisinterpretedasacompletenesstheoremforCa,basavectorspaceequippedwiththesupnormTheelementarypropertiesoftheRiemannintegralgaincoherenceexpressedasatheoremestablishingtheintegralasaboundedlinearfunctionalonaconvenientfunctionspaceSimilarly,thefamilyofabsolutelyconvergentseriesispresentedfromtheperspectivethatitisacompletenormedvectorspaceequippedwiththehnormManyexercisesareofferedforeachsectionofthetextTheseareessentialtothecourseAnexerciseprecededbyadaggersymboltiscitedatsomepointinthetextSuchcitationsrefertotheexercisebysectionandnumberAnexerciseprecededbyadiamondsymbolisahardproblemIfahardproblemwillbecitedlaterinthetext,thentherewillbeafootnotetosaypreciselywhereitwillbecitedThisisintendedtohelptheprofessordecidewhetherornotanexerciseshouldbeassignedtoaparticularclassbaseduponhisorherplannedcoverageforthecourseTopicsthatcanbeomittedattheprofessor'sdiscretionwithoutdisturbingcontinuityofthecoursearesoindicatedbymeansoffootnotesAttheendofeachchapterthereisabriefsectioncalledTestYourself,consistingofshortquestionstotestthestudent'scomprehensionofthebasicconceptsandtheoremsTheanswerstotheseshortquestions,andalsotootherselectedshortquestions,appearinanappendixTherearenoproofsprovidedamongthoseanswerstoselectedquestionsThereasonisthattherearemanypossiblecorrectproofsforeachexerciseOnlytheprofessorortheprofessor'sdesignatedassistantwillbeabletoproperlyevaluateandcorrectthestudent'swritinginexercisesrequiringproofsTheIntroductiontothisbookisintendedtointroducethestudenttoboththeimportanceandthechallengesofwritingproofsTheguidanceprovidedintheintroductionisfollowedbycorrespondingillustrativeremarksthatappearafterthefirstproofineachofthefivechaptersofPartIofthistextWhetheraprofessorchoosestocollectwrittenassignmentsortohavestudentspresentproofsattheboardinfrontoftheclass,eachstudentmustregularlyconstructandwriteproofsThecoherenceandthepresentationoftheargumentsmustbecriticizedXViPREFACEMostofthetraditionaltheoremsofelementarydifferentialandintegralcalculusaredevelopedrigorouslySincetheorientationofthecourseistowardtheroleofnormedvectorspaces,CauchycompletenessisthemostnaturalformofthecompletenessconcepttouseThuswepresentthesystemofrealnumbersasaCauchycompleteArchimedeanorderedfieldThetraditionaltheoremsofadvancedcalculusarepresentedTheseincludetheelementsofthestudyofintegrableanddifferentiablefunctions,extremevaluetheorems,MeanValueTheorems,andconvergencetheorems,thepolynomialapproximationtheoremofWeierstrass,theinverseandimplicitfunctiontheorems,Lebesgue'stheoremforRiemannintegrability,andtheJacobiantheoremforchangeofvariablesStudentslearninthiscoursesuchconceptsasthoseofacompletenormedvectorspace(realBanachspace)andaboundedlinearfunctionalThisisnotacourseinfunctionalanalysisRatherthecentraltheoremsandexamplesofadvancedcalculusaretreatedasinstancesandmotivationsfortheconceptsoffunctionalanalysisForexample,thespaceofboundedsequencesisshowntobethedualspaceofthespaceofabsolutelysummablesequencesTheconceptofthisbookisthatthestudentisguidedgraduallyfromthestudyofthetopologyofthereallinetothebeginningtheoremsandconceptsofgraduateanalysis,expressedfromamodernviewpointManytraditionaltheoremsofadvancedcalculuslistpropertiesthatamounttostatingthatacertainsetoffunctionsformsavectorspaceandthatthisspaceiscompletewithrespecttoanormByphrasingthetraditionaltheoremsinthislight,wehelpthestudenttomentallyorganizetheknowledgeofadvancedcalculusinacoherentandmeaningfulmannerwhileacquiringahelpfulreorientationtowardmoderngraduatelevelanalysisCoursePlansthatAreSupportedbythisBookPartIofthisbookconsistsoffivechapterscoveringmostofthestandardonevariabletopicsfoundintwosemesteradvancedcalculuscoursesThesechaptersarearrangedinorderofdependence,withthelaterchaptersdependingontheearlieronesThoughthetopicsaremainlytheonestypicallyfound,theyhavebeenreorientedherefromtheviewpointoflinearspaces,norms,completeness,andlinearfunctionalsPartIIoffersachoiceoftwomutuallyindependentadvancedonevariabletopics:eitherFourierseriesorStieltjesintegrationItisespeciallythecaseinPartIIthateachprofessor'sindividualjudgmentaboutthereadinessofhisorherclassshouldguidewhatistaughtSomeofthesetopicswillnotbefortheaveragestudent,butwillmakeexcellentreadingmaterialforthestudentseekinghonorscreditorwritingaseniorthesisIndividualreadingcoursescanbeemployedveryeffectivelytoprovideadvancedexperiencefortheprospectivegraduatestudentInChaptertheintroductionofFourierseriesisaidedbyinclusionofcomplexvaluedfunctionsofarealvariableThisistheonlychapterinwhichcomplexvaluedfunctionsappear,andwiththesetheHermitianinnerproductisintroducedThePREFACEXViichapterincludeslanditsselfduality,convergenceinthenorm,theuniformconvergenceofFourierseriesofsmoothfunctions,andtheRiemannlocalizationtheoremThestudyofavibratingstringispresentedtomotivatethechapterChapter,whichisaboutStieltjesintegration,includesfunctionsofboundedvariationandtheRieszRepresentationTheorem,presentingthedualspaceofCa,bintermsofStieltjesintegrationThelattertheoremofFRieszisthehardestonepresentedinthisbookItisnotrequiredforthelaterchaptersHowever,itisanexcellenttheoremforapromisingstudentplanningsubsequentdoctoralstudy,anditrequiresonlywhathasbeenlearnedpreviouslyinthiscourseItisacenturysincethediscoveryoftheRieszRepresentationTheoremTheauthorthinksitistimeforittotakeitsplaceinanundergraduatetextforthetwentyfirstcenturyPartIIIisaboutseveralvariableadvancedcalculus,includingtheinverseandimplicitfunctiontheorems,andtheJacobiantheoremsformultipleintegralsWherethefirsttwopartsplaceemphasisoninfinitedimensionallinearspacesoffunctions,thethirdpartemphasizesfinitedimensionalspacesandthederivativeasalineartransformationAtLouisianaStateUniversity,AdvancedCalculusisofferedasathreesemestertriadofcoursesThefirstsemesteristakenbyallandisthestar

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