分析（第3卷）

图书标准信息

• 作者 [德]阿莫恩 著
• 出版社 世界图书出版公司
• 出版时间 2012-09
• 版次 1
• ISBN 9787510047985
• 定价 99.00元
• 装帧 平装
• 开本 16开
• 纸张 其他
• 页数 468页

【内容简介】

　　Thisthirdvolumeconcludesourintroductiontoanalysis,whereinwefinishlayingthegroundworkneededforfurtherstudyofthesubject.Aswiththefirsttwo,thisvolumecontainsmorematerialthancantreatedinasinglecourse.Itisthereforeimportantinpreparinglecturestochooseasuitablesubsetofitscontent;theremaindercanbetreatedinseminarsorlefttoindependentstudy.Foraquickoverviewofthiscontent,consultthetableofcontentsandthechapterintroductions.

【目录】

Foreword
ChapterⅨElementsofmeasuretheory
1Measurablespaces
σ-algebras
TheBorelσ-algebra
Thesecondcountabilityaxiom
GeneratingtheBorela-algebrawithintervals
Basesoftopologicalspaces
Theproducttopology
ProductBorela-algebras
Measurabilityofsections
2Measures
Setfunctions
Measurespaces
Propertiesofmeasures
Nullsets
Outermeasures
Theconstructionofoutermeasures
TheLebesgueoutermeasure
TheLebesgue-Stieltjesoutermeasure
Hausdorffoutermeasures
4Measurablesets
Motivation
Thea-algebraof/μ*-measurablesets
LebesguemeasureandHausdorffmeasure
Metricmeasures
5TheLebasguemeasure
TheLebesguemeasurespace
TheLebesguemeasureisregular
AcharacterizationofLebesguemeasurability
ImagesofLebesguemeasurablesets
TheLebesguemeasureistranslationinvariant
AcharacterizationofLebesguemeasure
TheLebesguemeasureisinvariantunderrigidmotions
Thesubstitutionruleforlinearmaps
SetswithoutLebesguemeasure
ChapterⅩIntegrationtheory
1Measurablefunctions
Simplefunctionsandmeasurablefunctions
Ameasurabilitycriterion
MeasurableR-valuedfunctions
ThelatticeofmeasurableT-valuedfunctions
Pointwiselimitsofmensurablefunctions
Radonmeasures
2Integrablefuuctions
Theintegralofasimplefunction
TheL1-seminorm
TheBochner-Lebesgueintegral
ThecompletenessofL1
Elementarypropertiesofintegrals
ConvergenceinL1
3Convergencetheorems
IntegrationofnonnegativeT-valuedfunctions
Themonotoneconvergencetheorem
Fatou'slemma
IntegrationofR-valuedfunctions
Lebesgue'sdominatedconvergencetheorem
Parametrizedintegrals
4Lebesguespaces
Essentiallyboundedfunctions
TheHolderandMinkowskiinequalities
Lebesguespacesarecomplete
Lp-spaces
Continuousfunctionswithcompactsupport
Embeddings
ContinuouslinearfunctionalsonLp
5Then-dimensionalBochner-Lebesgueintegral
Lebesguemeasurespaces
TheLebesgueintegralofabsolutelyintegrablefunctions
AcharacterizationofRiemannintegrablefunctions
6Fubiul'stheorem
Mapsdefinedalmosteverywhere
Cavalieri'sprinciple
ApplicationsofCavalieri'sprinciple
Tonelli'stheorem
Fubini'stheoremforscalarfunctions
Fubini'stheoremforvector-vainedfunctions
Minkowski'sinequalityforintegrals
AcharacterizationofLp（Rm+n,E）
Atracetheorem
7Theconvolution
Definingtheconvolution
Thetranslationgroup
Elementarypropertiesoftheconvolution
Approximationstotheidentity
Testfunctions
Smoothpartitionsofunity
ConvolutionsofE-valuedfunctions
Distributions
Lineardifferentialoperators
Weakderivatives
8Thesubstitutionrule
PullingbacktheLebesguemeasure
Thesubstitutionrule:generalcase
Planepolarcoordinates
Polarcoordinatesinhigherdimensions
Integrationofrotationallysymmetricfunctions
Thesubstitutionruleforvector-valuedfunctions
9TheFouriertransform
Definitionandelementaryproperties
Thespaceofrapidlydecreasingfunctions
TheconvolutionalgebraS
CalculationswiththeFouriertransform
TheFourierintegraltheorem
ConvolutionsandtheFouriertransform
Fouriermultiplicationoperators
Plancherel'stheorem
Symmetricoperators
TheHeisenberguncertaintyrelation
ChapterⅪManifoldsanddifferentialforms
1Submanifolds
Definitionsandelementaryproperties
Submersions
Submanifo]dswithboundary
Localcharts
Tangentsandnormals
Theregularvaluetheorem
One-dimensionalmanifolds
Partitionsofunity
2MultUinearalgebra
Exteriorproducts
Pullbacks
Thevolumeelement
TheRieszisomorphism
TheHodgestaroperator
Indefiniteinnerproducts
Tensors
3Thelocaltheoryofdifferentialforms
Definitionsandbasisrepresentations
Pullbacks
Theexteriorderivative
ThePoincarelemma
Tensors
4Vectorfieldsanddifferentialforms
Vectorfields
Localbasisrepresentation
Differentialforms
Localrepresentations
Coordinatetransformations
Theexteriorderivative
Closedandexactforms
Contractions
Orientability
Tensorfields
5Riemannianmetrics
Thevolumeelement
Riemannianmanifolds
TheHodgestar
Thecodifferential
6Vectoranalysis
TheRieszisomorphism
Thegradient
Thedivergence
TheLaplace-Beltramioperator
Thecurl
TheLiederivative
TheHodge-Laplaceoperator
Thevectorproductandthecurl
ChapterⅫIntegrationonmanifolds
1Volumemeasure
TheLebesguea-algebraofM
Thedefiaitionofthevolumemeasure
Properties
Integrability
Calculationofseveralvolumes
2Integrationofdifferentialforms
Integralsofm-forms
Restrictionstosubmanifolds
Thetransformationtheorem
Fubini'stheorem
Calculationsofseveralintegrals
Flowsofvectorfields
Thetransporttheorem
3Stokes'stheorem
Stokes'stheoremforsmoothmanifolds
Manifoldswithsingularities
Stokes'stheoremwithsingularities
Planardomains
Higher-dimensionalproblems
Homotopyinvarianceandapplications
Gauss'slaw
Green'sformula
TheclassicalStokes'stheorem
Thestaroperatorandthecoderivative
References