# 实分析（英文版·第4版）

8.9

2010-08

ISBN: 9787111313052

• 《实分析（英文版·第4版）》是实分析课程的优秀教材，被国外众多著名大学（如斯坦福大学、哈佛大学等）采用。全书分为三部分：第一部分为实变函数论.介绍一元实变函数的勒贝格测度和勒贝格积分:第二部分为抽象空间。介绍拓扑空间、度量空间、巴拿赫空间和希尔伯特空间；第三部分为一般测度与积分理论。介绍一般度量空间上的积分.以及拓扑、代数和动态结构的一般理论。书中不仅包含数学定理和定义，而且还提出了富有启发性的问题，以便读者更深入地理解书中内容。 LebesgueIntegrationforFunctionsofaSingleRealVariable
PreliminariesonSets,Mappings,andRelations
UnionsandIntersectionsofSets
EquivalenceRelations,theAxiomofChoice,andZorn'sLemma
1TheRealNumbers:Sets.Sequences,andFunctions
TheField,Positivity,andCompletenessAxioms
TheNaturalandRationalNumbers
CountableandUncountableSets
OpenSets,ClosedSets,andBorelSetsofRealNumbers
SequencesofRealNumbers
ContinuousReal-ValuedFunctionsofaRealVariable
2LebesgneMeasure
Introduction
LebesgueOuterMeasure
Theo'-AlgebraofLebesgueMeasurableSets
OuterandInnerApproximationofLebesgueMeasurableSets
NoumeasurableSets
TheCantorSetandtheCantorLebesgueFunction
3LebesgReMeasurableFunctions
Sums,Products,andCompositions
SequentialPointwiseLimitsandSimpleApproximation
Littlewood'sThreePrinciples,Egoroff'sTheorem,andLusin'sTheorem
4LebesgueIntegration
TheRiemannIntegral
TheLebesgueIntegralofaBoundedMeasurableFunctionoveraSetof
FiniteMeasure
TheLebesgueIntegralofaMeasurableNonnegativeFunction
TheGeneralLebesgueIntegral
UniformIntegrability:TheVifaliConvergenceTheorem
viiiContents
5LebusgueIntegration:Fm'therTopics
UniformIntegrabilityandTightness:AGeneralVitaliConvergenceTheorem
ConvergenceinMeasure
CharacterizationsofRiemaunandLebesgueIntegrability
6DifferentiationandIntegration
ContinuityofMonotoneFunctions
DifferentiabilityofMonotoneFunctions:Lebesgue'sTheorem
FunctionsofBoundedVariation:Jordan'sTheorem
AbsolutelyContinuousFunctions
IntegratingDerivatives:DifferentiatingIndefiniteIntegrals
ConvexFunction
7TheLpSpaces:CompletenessandAppro~umation
Nor/nedLinearSpaces
TheInequalitiesofYoung,HOlder,andMinkowski
LvIsComplete:TheRiesz-FiseherTheorem
ApproximationandSeparability
8TheLPSpacescDeailtyandWeakConvergence
TheRieszRepresentationfortheDualof
WeakSequentialConvergenceinLv
WeakSequentialCompactness
TheMinimizationofConvexFunctionals
IIAbstractSpaces:Metric,Topological,Banach,andHiibertSpaces
9.MetricSpaces:GeneralProperties
ExamplesofMetricSpaces
OpenSets,ClosedSets,andConvergentSequences
ContinuousMappingsBetweenMetricSpaces
CompleteMetricSpaces
CompactMetricSpaces
SeparableMetricSpaces
10MetricSpaces:ThreeFundamentalThanreess
TheArzelb.-AscoliTheorem
TheBaireCategoryTheorem
TheBanaehContractionPrinciple
HTopologicalSpaces:GeneralProperties
OpenSets,ClosedSets,Bases,andSubbases
TheSeparationProperties
CountabilityandSeparability
ContinuousMappingsBetweenTopologicalSpaces

CompactTopologicalSpaces
ConnectedTopologicalSpaces
12TopologicalSpaces:ThreeFundamentalTheorems
Urysohn'sLemmaandtheTietzeExtensionTheorem
TheTychonoffProductTheorem
TheStone-WeierstrassTheorem
13ContinuousLinearOperatorsBetweenBauschSpaces
NormedLinearSpaces
LinearOperators
CompactnessLost:InfiniteDimensionalNormodLinearSpaces
TheOpenMappingandClosedGraphTheorems
TheUniformBoundednessPrinciple
14DualityforNormedIinearSpaces
LinearFtmctionals,BoundedLinearFunctionals,andWeakTopologies
TheHahn-BanachTheorem
ReflexiveBanachSpacesandWeakSequentialConvergence
LocallyConvexTopologicalVectorSpaces
TheSeparationofConvexSetsandMazur'sTheorem
TheKrein-MiimanTheorem
15CompactnessRegained:TheWeakTopology
Alaoglu'sExtensionofHelley'sTheorem
ReflexivityandWeakCompactness:Kakutani'sTheorem
CompactnessandWeakSequentialCompactness:TheEberlein-mulian
Theorem
MemzabilityofWeakTopologies
16ContinuousLinearOperatorsonHilbertSpaces
TheInnerProductandOrthogonality
TheDualSpaceandWeakSequentialConvergence
BessersInequalityandOrthonormalBases
CompactOperators
TheHilbert-SchmidtTheorem
TheRiesz-SchauderTheorem:CharacterizationofFredholmOperators
MeasureandIntegration:GeneralTheory
17GeneralMeasureSpaces:TheirPropertlesandConstruction
MeasuresandMeasurableSets
SignedMeasures:TheHahnandJordanDecompositions
TheCaratheodoryMeasureInducedbyanOuterMeasure
18IntegrationOeneralMeasureSpaces
19GengralLSpaces:Completeness,DualityandWeakConvergence
20TheConstrucitonofParticularMeasures
21MeasureandTopbogy
22InvariantMeasures
Bibiiography
index
• ##### 内容简介:
《实分析（英文版·第4版）》是实分析课程的优秀教材，被国外众多著名大学（如斯坦福大学、哈佛大学等）采用。全书分为三部分：第一部分为实变函数论.介绍一元实变函数的勒贝格测度和勒贝格积分:第二部分为抽象空间。介绍拓扑空间、度量空间、巴拿赫空间和希尔伯特空间；第三部分为一般测度与积分理论。介绍一般度量空间上的积分.以及拓扑、代数和动态结构的一般理论。书中不仅包含数学定理和定义，而且还提出了富有启发性的问题，以便读者更深入地理解书中内容。
• ##### 目录:
LebesgueIntegrationforFunctionsofaSingleRealVariable
PreliminariesonSets,Mappings,andRelations
UnionsandIntersectionsofSets
EquivalenceRelations,theAxiomofChoice,andZorn'sLemma
1TheRealNumbers:Sets.Sequences,andFunctions
TheField,Positivity,andCompletenessAxioms
TheNaturalandRationalNumbers
CountableandUncountableSets
OpenSets,ClosedSets,andBorelSetsofRealNumbers
SequencesofRealNumbers
ContinuousReal-ValuedFunctionsofaRealVariable
2LebesgneMeasure
Introduction
LebesgueOuterMeasure
Theo'-AlgebraofLebesgueMeasurableSets
OuterandInnerApproximationofLebesgueMeasurableSets
NoumeasurableSets
TheCantorSetandtheCantorLebesgueFunction
3LebesgReMeasurableFunctions
Sums,Products,andCompositions
SequentialPointwiseLimitsandSimpleApproximation
Littlewood'sThreePrinciples,Egoroff'sTheorem,andLusin'sTheorem
4LebesgueIntegration
TheRiemannIntegral
TheLebesgueIntegralofaBoundedMeasurableFunctionoveraSetof
FiniteMeasure
TheLebesgueIntegralofaMeasurableNonnegativeFunction
TheGeneralLebesgueIntegral
UniformIntegrability:TheVifaliConvergenceTheorem
viiiContents
5LebusgueIntegration:Fm'therTopics
UniformIntegrabilityandTightness:AGeneralVitaliConvergenceTheorem
ConvergenceinMeasure
CharacterizationsofRiemaunandLebesgueIntegrability
6DifferentiationandIntegration
ContinuityofMonotoneFunctions
DifferentiabilityofMonotoneFunctions:Lebesgue'sTheorem
FunctionsofBoundedVariation:Jordan'sTheorem
AbsolutelyContinuousFunctions
IntegratingDerivatives:DifferentiatingIndefiniteIntegrals
ConvexFunction
7TheLpSpaces:CompletenessandAppro~umation
Nor/nedLinearSpaces
TheInequalitiesofYoung,HOlder,andMinkowski
LvIsComplete:TheRiesz-FiseherTheorem
ApproximationandSeparability
8TheLPSpacescDeailtyandWeakConvergence
TheRieszRepresentationfortheDualof
WeakSequentialConvergenceinLv
WeakSequentialCompactness
TheMinimizationofConvexFunctionals
IIAbstractSpaces:Metric,Topological,Banach,andHiibertSpaces
9.MetricSpaces:GeneralProperties
ExamplesofMetricSpaces
OpenSets,ClosedSets,andConvergentSequences
ContinuousMappingsBetweenMetricSpaces
CompleteMetricSpaces
CompactMetricSpaces
SeparableMetricSpaces
10MetricSpaces:ThreeFundamentalThanreess
TheArzelb.-AscoliTheorem
TheBaireCategoryTheorem
TheBanaehContractionPrinciple
HTopologicalSpaces:GeneralProperties
OpenSets,ClosedSets,Bases,andSubbases
TheSeparationProperties
CountabilityandSeparability
ContinuousMappingsBetweenTopologicalSpaces

CompactTopologicalSpaces
ConnectedTopologicalSpaces
12TopologicalSpaces:ThreeFundamentalTheorems
Urysohn'sLemmaandtheTietzeExtensionTheorem
TheTychonoffProductTheorem
TheStone-WeierstrassTheorem
13ContinuousLinearOperatorsBetweenBauschSpaces
NormedLinearSpaces
LinearOperators
CompactnessLost:InfiniteDimensionalNormodLinearSpaces
TheOpenMappingandClosedGraphTheorems
TheUniformBoundednessPrinciple
14DualityforNormedIinearSpaces
LinearFtmctionals,BoundedLinearFunctionals,andWeakTopologies
TheHahn-BanachTheorem
ReflexiveBanachSpacesandWeakSequentialConvergence
LocallyConvexTopologicalVectorSpaces
TheSeparationofConvexSetsandMazur'sTheorem
TheKrein-MiimanTheorem
15CompactnessRegained:TheWeakTopology
Alaoglu'sExtensionofHelley'sTheorem
ReflexivityandWeakCompactness:Kakutani'sTheorem
CompactnessandWeakSequentialCompactness:TheEberlein-mulian
Theorem
MemzabilityofWeakTopologies
16ContinuousLinearOperatorsonHilbertSpaces
TheInnerProductandOrthogonality
TheDualSpaceandWeakSequentialConvergence
BessersInequalityandOrthonormalBases
CompactOperators
TheHilbert-SchmidtTheorem
TheRiesz-SchauderTheorem:CharacterizationofFredholmOperators
MeasureandIntegration:GeneralTheory
17GeneralMeasureSpaces:TheirPropertlesandConstruction
MeasuresandMeasurableSets
SignedMeasures:TheHahnandJordanDecompositions
TheCaratheodoryMeasureInducedbyanOuterMeasure
18IntegrationOeneralMeasureSpaces
19GengralLSpaces:Completeness,DualityandWeakConvergence
20TheConstrucitonofParticularMeasures
21MeasureandTopbogy
22InvariantMeasures
Bibiiography
index

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