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[美] 斯坦恩 (Stein E.M.) 著 / 世界图书出版公司 / 2013-01 / 平装
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上书时间2024-04-09
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实分析(英文)
《实分析(英文)》在Princeton大学使用,同时在其它学校,比如UCLA等名校也在本科生教学中得到使用。其教学目的是,用统一的、联系的观点来把现代分析的“核心”内容教给本科生,力图使本科生的分析学课程能接上现代数学研究的脉络。
ForewordIntroduction1Fourierseries:completion2Limitsofcontinuousfunctions3Lengthofcurves4Differentiationandintegration5TheproblemofmeasureChapter1.MeasureTheory1Prelhninaries2Theexteriormeasure3MeasurablesetsandtheLebesguemeasure4Measurablefunctions4.1Definitionandbasicproperties4.2Approximationbysimplefunctionsorstepfunctions4.3Littlewood'sthreeprinciples5TheBrunn-Minkowskiinequality6Exercises7ProblemsChapter2.IntegrationTheory1TheLebesgueintegral:basicpropertiesandconvergencetheorems2ThespaceL1ofintegrablefunctions3Fubini'stheorem3.1Statementandproofofthetheorem3.2ApplicationsofFubini'stheorem4*AFourierinversionformula5Exercises6ProblemsChapter3.DifferentiationandIntegration1Differentiationoftheintegral1.1TheHardy-Littlewoodmaximalfunction1.2TheLebesguedifferentiationtheorem2Goodkernelsandapproximationstotheidentity3Differentiabilityoffunctions3.1Functionsofboundedvariation3.2Absolutelycontinuousfunctions3.3Differentiabilityofjumpfunctions4Rectifiablecurvesandtheisoperimetricinequality4.1Minkowskicontentofacurve4.2Isoperimetricinequality5Exercises6ProblemsChapter4.HilbertSpaces:AnIntroduction1TheHilbertspaceL22Hilbertspaces2.1Orthogonality2.2Unitarymappings2.3Pre-Hilbertspaces3FourierseriesandFatou'stheorem3.1Fatou'stheorem4Closedsubspacesandorthogonalprojections5Lineartransformations5.1LinearfunctionalsandtheRieszrepresentationtheorem5.2Adjoints5.3Examples6Compactoperators7Exercises8ProblemsChapter5.HilbertSpaces:SeveralExamples1TheFouriertransformonL22TheHardyspaceoftheupperhalf-plane3Constantcoefficientpartialdifferentialequations3.1Weaaksolutions3.2Themaintheoremandkeyestimate4TheDirichletprinciple4.1Harmonicfunctions4.2TheboundaryvalueproblemandDirichlet'sprinciple5Exercises6ProblemsChapter6.AbstractMeasureandIntegrationTheoryChapter7.HausdorffMeasureandFractalsNotesandReferencesBibliographySymbolGlossaryIndex
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