实分析（影印版）

图书标准信息

• 作者 [美]德贝内代托（DiBenedetto E.） 著
• 出版社 高等教育出版社
• 出版时间 2007-10
• 版次 1
• ISBN 9787040226652
• 定价 39.50元
• 装帧 平装
• 开本 16开
• 纸张 胶版纸
• 页数 485页
• 字数 560千字
• 正文语种 英语
• 丛书 天元基金影印数学丛书

【内容简介】

　　《实分析（影印版）》是一本内容十分翔实的实分析教材。它包含集论，点集拓扑。测度与积分，Lebesgue函数空间，Banach空间与Hilbert空间，连续函数空间，广义函数与弱导数，Sobolev空间与Sobolev嵌入定理等；同时还包含Lebesgue微分定理，Stone-Weierstrass逼近定理，Ascoli—Arzela定理，Calderon—Zygmund分解定理，Fefferman—Stein定理。Marcinkiewlcz插定理等实分析中有用的内容。
　　《实分析（影印版）》内容由浅入深。读者具有扎实的数学分析知识基础便可学习《实分析（影印版）》，学完《实分析（影印版）》的读者将具备学习分析所需要的实变与泛函（不包括算子理论）的准备知识和训练。

【目录】

Preface
Acknowledgments
Preliminaries
1Countablesets
2TheCantorset
3Cardinality
3.1Someexamples
4CardinalityofsomeinfiniteCartesianproducts
5Orderings,themaximalprinciple,andtheaxiomofchoice
6Well-ordering
6.1Thefirstuncountable
ProblemsandComplements

ⅠTopologiesandMetricSpaces
1Topologicalspaces
1.1Hausdorffandnormalspaces
2Urysohnslemma
3TheTietzeextensiontheorem
4Bases,axiomsofcountability,andproducttopologies
4.1Producttopologies

5Compacttopologicalspaces
5.1Sequentiallycompacttopologicalspaces
6CompactsubsetsofRN
7Continuousfunctionsoncountablycompactspaces
8Productsofcompactspaces

9Vectorspaces
9.1Convexsets
9.2Linearmapsandisomorphisms

10Topologicalvectorspaces
10.1Boundednessandcontinuity
11Linearfunctionals

12Finite-dimensionaltopologicalvectorspaces
12.1Locallycompactspaces

13Metricspaces
13.1Separationandaxiomsofcountability
13.2Equivalentmetrics
13.3Pseudometrics

14Metricvectorspaces
14.1Mapsbetweenmetricspaces

15Spacesofcontinuousfunctions
15.1Spacesofcontinuouslydifferentiablefunctions
16Onthestructureofacompletemetricspace

17Compactandtotallyboundedmetricspaces
17.1PrecompactsubsetsofX
ProblemsandComplements

ⅡMeasuringSets
1PartitioningopensubsetsofRN
2Limitsofsets,characteristicfunctions,andor-algebras
3Measures
3.1Finite,a-finite,andcompletemeasures
3.2Someexamples

4Outermeasuresandsequentialcoverings
4.1TheLebesgueoutermeasureinRN
4.2TheLebesgue-Stieltjesoutermeasure
5TheHausdorffoutermeasureinRN
6Constructingmeasuresfromoutermeasures

7TheLebesgue——StieltjesmeasureonR
7.1Borelmeasures
8TheHausdorffmeasureonRN
9Extendingmeasuresfromsemialgebrastoa-algebras
9.1OntheLebesgue-StieltjesandHausdorffmeasures
10Necessaryandsufficientconditionsformeasurability
11Moreonextensionsfromsemialgebrastoa-algebras
12TheLebesguemeasureofsetsinRN
12.1Anecessaryandsufficientconditionofnaeasurability
13Anonmeasurableset

14Borelsets,measurablesets,andincompletemeasures
14.1Acontinuousincreasingfunctionf:[0,1]→[0,1]
14.2Onthepreimageofameasurableset
14.3ProofofPropositions14.1and14.2

15MoreonBorelmeasures
15.1SomeextensionstogeneralBorelmeasures
15.2RegularBorelmeasuresandRadonmeasures

16RegularoutermeasuresandRadonmeasures
16.1MoreonRadonmeasures
17Vitalicoverings
18TheBesicovitchcoveringtheorem
19ProofofProposition18.2
20TheBesicovitchmeasure-theoreticalcoveringtheorem
ProblemsandComplements

ⅢTheLebesgueIntegral
1Measurablefunctions
2TheEgorovtheorem
2.1TheEgorovtheoreminRN
2.2MoreonEgorovstheorem
3Approximatingmeasurablefunctionsbysimplefunctions
4Convergenceinmeasure
5Quasi-continuousfunctionsandLusinstheorem
6Integralofsimplefunctions
7TheLebesgueintegralofnonnegativefunctions
8Fatouslemmaandthemonotoneconvergencetheorem
9BasicpropertiesoftheLebesgueintegral
10Convergencetheorems
11Absolutecontinuityoftheintegral
12Productofmeasures
13Onthestructureof（A*p）
14TheFubini-Tonellitheorem
14.1TheTonelliversionoftheFubinitheorem

15SomeapplicationsoftheFubini-Tonellitheorem
15.1Integralsintermsofdistributionfunctions
15.2Convolutionintegrals
15.3TheMarcinkiewiczintegral
16SignedmeasuresandtheHahndecomposition
17TheRadon-Nikodymtheorem

18Decomposingmeasures
18.1TheJordandecomposition
18.2TheLebesguedecomposition
18.3AgeneralversionoftheRadon-Nikodymtheorem
ProblemsandComplements

IVTopicsonMeasurableFunctionsofRealVariables
1Functionsofboundedvariations
2Diniderivatives
3Differentiatingfunctionsofboundedvariation
4Differentiatingseriesofmonotonefunctions
5Absolutelycontinuousfunctions
6Densityofameasurableset
7Derivativesofintegrals
8DifferentiatingRadonmeasures
9ExistenceandmeasurabilityofDvv
9.1ProofofProposition9.2
10RepresentingDvv
10.1RepresentingDuvforv<<#
10.2RepresentingDuvforvu

11TheLebesguedifferentiationtheorem
11.1Pointsofdensity
11.2Lebesguepointsofanintegrablefunction
12Regularfamilies
13Convexfunctions
14Jensensinequality
15Extendingcontinuousfunctions
16TheWeierstrassapproximationtheorem
17TheStone-Weierstrasstheorem

18ProofoftheStone-Weierstrasstheorem
18.1ProofofStonestheorem
19TheAscoli-Arzelatheorem
19.1PrecompactsubsetsofC（E）
ProblemsandComplements

VTheLP（E）Spaces
1FunctionsinLp（E）andtheirnorms
1.1ThespacesLPfor01.2ThespacesLqforq<0
2TheHOlderandMinkowskiinequalities
3ThereverseHolderandMinkowskiinequalities
4MoreonthespacesLpandtheirnorms
4.1Characterizingthenormfpfor14.2ThenormIII1forEoffinitemeasure
4.3ThecontinuousversionOftheMinkowskiinequality

5LP（E）for15.1Lp（E）for1
6AmetrictopologyforLP（E）when06.1OpenconvexsubsetsofLP（E）when07ConvergenceinLP（E）andcompleteness
8SeparatingLP（E）bysimplefunctions

ⅥBanachSpaces
ⅦSpacesofContinuousFunctions,Distributions,andWeak
ⅧTopicsonIntegrableFunctionsofRealVariables
ⅨEmbeddingsofW1,p（E)intoLq（E)
References
Index