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The.Way.of.Analysis.Revised.Edi…

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简介:本文档为《The.Way.of.Analysis.Revised.Edition_Robert.S.Strichartzpdf》,可适用于高等教育领域,主题内容包含TheWayRobertSStrichartzOfAnalysisTheWayofAnalysisRevisedEditionRobertSStri符等。

TheWayRobertSStrichartzOfAnalysisTheWayofAnalysisRevisedEditionRobertSStrichartzCornellUniversitynJONESANDBARTLETTPUBLISHERSSudbury,MassachusettsBOSTONTORONTOLONDONSINGAPOREWorldHeadquartersJonesandBartlettPublishersTallPineDriveSudbury,MAinfojbpubcomwwwjbpubcomJonesandBartlettPublishersCANADAWESTBloorStSuiteToronto,ONMACANADAJonesandBartlettPublishersInternationalBarbHouse,BarbMewsLondonWPAUKCopyrightbyJonesandBartlettPublishers,IncAllrightsreservedNopartofthematerialprotectedbythiscopyrightnoticemaybereproducedorutilizedinanyform,electronicormechanical,includingphotocopying,recording,orbyanyinformationstorageandretrievalsystem,withoutwrittenpermissionfromthecopyrightownerISBN:Thequotationsinprobleminproblemset(page)areofproverbialorbiblicalorigin,exceptforafrom"HomeontheRange,"byBrewsterMHigley,andgfromMacbeth,byWilliamShakespearePrintedintheUnitedStatesofAmericaContentsPrefacePreliminariesTheLogicofQuantifiersRulesofQuantifiersExamplesExercisesInfiniteSetsCountableSetsUncountableSetsExercisesProofsHowtoDiscoverProofsHowtoUnderstandProofsTheRationalNumberSystemTheAxiomofChoice*ConstructionoftheRealNumberSystemCauchySequencesMotivationTheDefinitionExercisesTheRealsasanOrderedFieldDefiningArithmeticTheFieldAxiomsOrderExercisesvxiiiviLimitsandCompletenessProofofCompletenessSquareRootsExercisesOtherVersionsandVisionsInfiniteDecimalExpansionsDedekindCuts*NonStandardAnalysis*ConstructiveAnalysis*ExercisesSummaryContentsTopologyoftheRealLineTheTheoryofLimitsLimits,Sups,andInfsLimitPointsExercisesOpenSetsandClosedSetsOpenSetsClosedSetsExercisesCompactSetsExercisesSummaryContinuousFunctionsConceptsofContinuityDefinitionsLimitsofFunctionsandLimitsofSequencesInverseImagesofOpenSetsRelatedDefinitionsExercisesPropertiesofContinuousFunctionsBasicPropertiesContinuousFunctionsonCompactDomainsMonotoneFunctionsExercisesSummaryContentsDifferentialCalculusConceptsoftheDerivativeEquivalentDefinitionsContinuityandContinuousDifferentiabilityExercisesPropertiesoftheDerivativeLocalPropertiesIntermediateValueandMeanValueTheoremsGlobalPropertiesExercisesTheCalculusofDerivativesProductandQuotientRulesTheChainRuleInverseFunctionTheoremExercisesHigherDerivativesandTaylor'sTheoremInterpretationsoftheSecondDerivativeTaylor'sTheoremL'Hopital'sRule*LagrangeRemainderFormula*OrdersofZeros*ExercisesSummaryIntegralCalculusIntegralsofContinuousFunctionsExistenceoftheIntegralFundamentalTheoremsofCalculusviiUsefulIntegrationFormulasNumericalIntegrationExercisesTheRiemannIntegralDefinitionoftheIntegralElementaryPropertiesoftheIntegralFunctionswithaCountableNumberofDiscontinuities*ExercisesImproperIntegrals*viiiDefinitionsandExamplesExercisesSummaryContentsSequencesandSeriesofFunctionsComplexNumbersBasicPropertiesofCComplexValuedFunctionsExercisesNumericalSeriesandSequencesConvergenceandAbsoluteConvergenceRearrangementsSummationbyParts*ExercisesUniformConvergenceUniformLimitsandContinuityIntegrationandDifferentiationofLimitsUnrestrictedConvergence*ExercisesPowerSeriesTheRadiusofConvergenceAnalyticContinuationAnalyticFunctionsonComplexDomains*ClosurePropertiesofAnalyticFunctions*ExercisesApproximationbyPolynomialsLagrangeInterpolationConvolutionsandApproximateIdentitiesTheWeierstrassApproximationTheoremApproximatingDerivativesExercisesEquicontinuityTheDefinitionofEquicontinuityTheArzelaAscoliTheoremExercisesSummaryTranscendentalFunctionsTheExponentialandLogarithmFiveEquivalentDefinitionsExponentialGlueandBlipFunctionsFunctionswithPrescribedTaylorExpansions*ExercisesTrigonometricFunctionsDefinitionofSineandCosineRelationshipBetweenSines,Cosines,andComplexExponentialsExercisesSummaryEuclideanSpaceandMetricSpacesStructuresonEuclideanSpaceVectorSpaceandMetricSpaceNormandInnerProductTheComplexCaseExercisesTopologyofMetricSpacesOpenSetsLimitsandClosedSetsCompletenessCompactnessExercisesContinuousFunctionsonMetricSpacesThreeEquivalentDefinitionsContinuousFunctionsonCompactDomainsConnectednessTheContractiveMappingPrincipleTheStoneWeierstrassTheorem*NowhereDifferentiableFunctions,andWorse*ExercisesSummaryDifferentialCalculusinEuclideanSpaceTheDifferential:DefinitionofDifferentiabilityXPartialDerivativesTheChainRuleDifferentiationofIntegralsExercisesHigherDerivativesEqualityofMixedPartialsLocalExtremaTaylorExpansionsExercisesSummaryOrdinaryDifferentialEquationsExistenceandUniquenessMotivationPicardIterationLinearEquationsLocalExistenceandUniqueness*HigherOrderEquations*ExercisesOtherMethodsofSolution*DifferenceEquationApproximationPeanoExistenceTheoremPowerSeriesSolutionsExercisesVectorFieldsandFlows*IntegralCurvesHamiltonianMechanicsFirstOrderLinearPDE'sExercisesSummaryFourierSeriesOriginsofFourierSeriesFourierSeriesSolutionsofPDE'sSpectralTheoryHarmonicAnalysisExercisesConvergenceofFourierSeriesContentsIIIIIContentsUniformConvergenceforCFunctionsSummabilityofFourierSeriesConvergenceintheMeanDivergenceandGibb'sPhenomenon*SolutionoftheHeatEquation*ExercisesSummaryImplicitFunctions,Curves,andSurfacesTheImplicitFunctionTheoremStatementoftheTheoremTheProof*ExercisesCurvesandSurfacesMotivationandExamplesImmersionsandEmbeddingsParametricDescriptionofSurfacesImplicitDescriptionofSurfacesExercisesMaximaandMinimaonSurfacesLagrangeMultipliersASecondDerivativeTest*ExercisesArcLengthRectifiableCurvesTheIntegralFormulaforArcLengthArcLengthParameterization*ExercisesSummaryTheLebesgueIntegralTheConceptofMeasureMotivationPropertiesofLengthMeasurableSetsBasicPropertiesofMeasuresAFormulaforLebesgueMeasureOtherExamplesofMeasuresxixiiExercisesProofofExistenceofMeasures*OuterMeasuresMetricOuterMeasureHausdorffMeasures*ExercisesTheIntegralNonnegativeMeasurableFunctionsTheMonotoneConvergenceTheoremIntegrableFunctionsAlmostEverywhereExercisesTheLebesgueSpacesLandLLasaBanachSpaceLasaHilbertSpaceFourierSeriesforLFunctionsExercisesSummaryContentsMultipleIntegralsInterchangeofIntegralsIntegralsofContinuousFunctionsFubini'sTheoremTheMonotoneClassLemma*ExercisesoChangeofVariableinMultipleIntegralsDeterminantsandVolumeoTheJacobianFactor*ooPolarCoordinatesooooChangeofVariableforLebesgueIntegrals*ooExercisesoSummaryooooooooooooooooooIndexPrefaceDonotaskpermissiontounderstandDonotwaitforthewordofauthoritySeizereasoninyourownhandWithyourownteethsavorthefruitMathematicsismorethanacollectionoftheorems,definitions,problemsandtechniquesitisawayofthoughtThesamecanbesaidaboutanindividualbranchofmathematics,suchasanalysisAnalysishasitsrootsintheworkofArchimedesandotherancientGreekgeometers,whodevelopedtechniquestofindareas,volumes,centersofgravity,arclengths,andtangentstocurvesIntheseventeenthcenturythesetechniqueswerefurtherdeveloped,culminatingintheinventionofthecalculusofNewtonandLeibnizDuringtheeighteenthcenturythecalculuswasfashionedintoatoolofboldcomputationalpowerandappliedtodiverseproblemsofpracticalandtheoreticalinterestAtthesametimethefoundationofanalysisthelogicaljustificationforthesuccessofthemethodswasleftinlimboThishadpracticalconsequences:forexample,EulertheleadingmathematicianoftheeighteenthcenturydevelopedallthetechniquesneededforthestudyofFourierseries,buthenevercarriedouttheprojectOnthecontrary,hearguedinprintagainstthepossibilityofrepresentingfunctionsasFourierseries,whenthisproposalwasputforthbyDanielBernoulli,andhisargumentwasbasedonfundamentalmisconceptionsconcerningthenatureoffunctionsandinfiniteseriesInthenineteenthcentury,theproblemofthefoundationofanalysiswasfacedsquarelyandresolvedThetheorythatwasdevelopedformsmostofthecontentofthisbookWewilldescribeitinitslogicalxiiixivPrefaceorder,startingfromthemostbasicconceptssuchassetsandnumbersandbuildinguptothemoreinvolvedconceptsoflimits,continuity,derivative,andintegralTheactualhistoricalorderofdiscoverywasalmostthereversemuchlikepeelingacabbage,mathematiciansbeganwiththeoutermostlayersandworkedtheirwayinwardCauchyandBolzanobegantheprocessinthesbydevelopingthetheoryoffunctionswithoutdefiningtherealnumbersThefirstrigorousdefinitionoftherealnumbersystemcameintheworkofDedekind,Weierstrass,andHeineinthesSettheorycamelaterintheworkofCantor,Peano,andFregeTheconsequencesofthenineteenthcenturyfoundationalworkwereenormousandarestillbeingfelttodayPerhapstheleastimportantconsequencewastheestablishmentofalogicallyvalidexplanationofthecalculusNioreimportant,withtheclearingawayoftheconceptualmurk,newproblemsemergedwithclarityandweredevelopedintoimportanttheoriesWewillgivesomeillustrationsofthesenewnineteenthcenturydiscoveriesinourdiscussionsofdifferentialequations,Fourierseries,higherdimensionalcalculus,andmanifoldsMostimportantofall,however,thenineteenthcenturyfoundationalworkpavedthewayfortheworkofthetwentiethcenturyAnalysistodayisasubjectofvastscopeandbeauty,rangingfromtheabstracttotheconcrete,characterizedbothbytheboldcomputationalpoweroftheeighteenthcenturyandthelogicalsubtletyofthenineteenthcenturyMostofthesedevelopmentsarebeyondthescopeofthisbookoratbestmerelyhintedatStill,itismyhopethatthereader,afterhavingenteredsodeeplyalongthewayofanalysis,willbeencouragedtocontinuethestudyMygoalinwritingthisbookistocommunicatethemathematicalideasofthesubjecttothereaderIhavetriedtobegenerouswithexplanationsPerhapstherewillbeplaceswhereIbelabortheobvious,nevertheless,IthinkthereisenoughtrulychallengingmaterialheretoinspireeventhestrongeststudentsOntheotherhand,therewillinevitablybeplaceswhereeachreaderwillfinddifficultiesinfollowingtheargumentsWhenthishappens,IsuggestthatyouwriteyourquestionsinthemarginsLater,whenyougooverthematerial,youmayfindthatyoucananswerthequestionIfnot,besuretoaskyourinstructororanotherstudentoften,itisaminormisunderstandingthatcausesconfusionandcaneasilybeclearedupSometimes,theinPrefaceXVherentdifficultyofthematerialwilldemandconsiderableeffortonyourparttoattainunderstandingIhopeyouwillnotbecomefrustratedintheprocessitissomethingwhichallstudentsofmathematicsmustconfrontIbelievethatwhatyoulearnthroughaprocessofstruggleismorelikelytostickwithyouthanwhatyoulearnwithouteffortUnderstandingmathematicsisacomplexprocessItinvolvesnotonlyfollowingthedetailsofanargumentandverifyingitscorrectness,butseeingtheoverallstrategyoftheargument,theroleplayedbyeveryhypothesis,andunderstandinghowdifferenttheoremsanddefinitionsfittogethertocreatethewholeItisalongtermprocessinasense,youcannotappreciatethesignificanceofthefirsttheoremuntilyouhavelearnedthelasttheoremSopleasebesuretoreviewoldmaterialyoumayfindthechaptersummariesusefulforthispurposeThemathematicalideaspresentedinthisbookareoffundamentalimportance,andyouaresuretoencounterthemagaininfurtherstudiesinbothpureandappliedmathematicsLearnthemwellandtheywillserveyouwellinthefutureItmaynotbeaneasytask,butitisaworthyoneTotheInstructorThisbookisdesignedsothatitmaybeusedinseveralways,includingaonesemesterintroductoryrealanalysiscourse,atwosemesterrealanalysiscoursenotincludingLebesgueintegration,atwosemesterrealanalysiscourseincludinganintroductiontoLebesgueintegrationTherearemanyoptionalsections,markedwithanasterisk(*),thatcanbecoveredoromittedatyourdiscretionThereissomeflexibilityintheorderingofthelaterchaptersThusyoucandesignacourseinaccordancewithyourinterestsandrequirementsTherearethreechaptersonapplications(Chapter,OrdinaryDifferentialEquationsChapter,FourierSeriesandChapter,ImplicitFunctions,CurvesandSurfaces)Thesetopicsareoftenomitted,ortreatedverybriefly,inxviPrefacearealanalysiscoursebecausetheyarecoveredinothercoursesHowever,theyserveanimportantpurposeinillustratinghowtheabstracttheorymaybeappliedtomoreconcretesituationsIwouldurgeyoutotrytofitasmuchofthismaterialastimeallowsintoyourcourseThechaptersmaybedividedintofourgroupings:functionsofonevariable:,,,,,,,functionsofseveralvariables:,,applications:,,Lebesgueintegration:,NotethatChapter,MultipleIntegrals,maybeusedeitherwithorwithouttheLebesgueintegralThefirstchaptersaredesignedtobeusedinthegivenorder(sectionsmarkedwithanasteriskmaybeomittedorpostponed)IfyouarenotcoveringtheLebesgueintegral,thenselectionsfromChapter(,,and)canbecoveredanytimeafterChapter

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