内容简介

泛函分析 第6版》是一部数学经典教材,初版于1965年,以作者在东京大学任教十余年所用的讲义为基础写成的。经过几次修订和增补,1980年出了第5版,本版(第6版)实际上是第5版的重印版。《泛函分析 第6版》论述了泛函空间的线性算子理论及其在现代分析和经典分析各领域中的许多应用。目次:预备知识;半范数;Baire-Hausdorff定理的应用;正交射影和riesz表示定理;Hahn-Banach定理;强收敛和弱收敛;傅里叶变换和微分方程;对偶算子;预解和谱;半群的解析理论;紧致算子;赋范环和谱表示;线性空间中的其他表示定理;遍历性理论和扩散理论;发展方程的积分。

读者对象:数学专业的研究生和科研人员。

作者简介

《泛函分析》(第6版)作者KôsakuYosida(吉田耕作,日)是东京大学教授,《泛函分析 第6版》依据作者多年的教学讲义集结而成。即可作为学生的自学读本,也可作为泛函分析教材。

目录

0.Preliminaries
1.SetTheory
2.TopologicalSpaces
3.MeasureSpaces
4.LinearSpaces
Ⅰ.Semi-norms
1.Semi-normsandLocallyConvexLinearTopologicalSpaces
2.NormsandQuasi-norms
3.ExamplesofNormedLinearSpaces
4.ExamplesofQuasi-normedLinearSpaces
5.Pre-HilbertSpaces
6.ContinuityofLinearOperators
7.BoundedSetsandBornologicSpaces
8.GeneralizedFunctionsandGeneralizedDerivatives
9.B-spacesandF-spaces
10.TheCompletion
11.FactorSpacesofaB-space
12.ThePartitionofUnity
13.GeneralizedFunctionswithCompactSupport
14.TheDirectProductofGeneralizedFunctions

Ⅱ.ApplicationsoftheBaire-HausdorffTheorem
1.TheUniformBoundednessTheoremandtheResonanceTheorem
2.TheVitali-Hahn-SaksTheorem
3.TheTermwiseDifferentiabilityofaSequenceofGeneralizedFunctions
4.ThePrincipleoftheCondensationofSingularities
5.TheOpenMappingTheorem
6.TheClosedGraphTheorem
7.AnApplicationoftheClosedGraphTheorem(Hormander'sTheorem)

Ⅲ.TheOrthogonalProjectionandF.Riesz'RepresentationTheorem
1.TheOrthogonalProjection
2."NearlyOrthogonal"Elements
3.TheAscoli-ArzelaTheorem
4.TheOrthogonalBase.Bessel'sInequalityandParseval'sRelation
5.E.Schmidt'sOrthogonalization
6.F.Riesz'RepresentationTheorem
7.TheLax-MilgramTheorem
8.AProofoftheLebesgue-NikodymTheorem
9.TheAronszajn-BergmanReproducingKernel
10.TheNegativeNormofP.LAX
11.LocalStructuresofGeneralizedFunctions

Ⅳ.TheHahn-BanachTheorems
1.TheHahn-BanachExtensionTheoreminRealLinearSpaces
2.TheGeneralizedLimit
3.LocallyConvex,CompleteLinearTopologicalSpaces
4.TheHahn-BanachExtensionTheoreminComplexLinearSpaces
5.TheHahn-BanachExtensionTheoreminNormedLinearSpaces
6.TheExistenceofNon-trivialContinuousLinearFunctionals
7.TopologiesofLinearMaps
8.TheEmbeddingofXinitsBidualSpaceX"
9.ExamplesofDualSpaces

Ⅴ.StrongConvergenceandWeakConvergence
1.TheWeakCosvergenceandTheWeak*Convergence
2.TheLocalSequentialWeakCompactnessofReflexiveB-spaces.TheUniformConvexity
3.Dunford'sTheoremandTheGelfand-MazurTheorem
4.TheWeakandStrong.Measurability.Pettis'Theorem
5.Bochner'sIntegral
AppendixtoChapterV.WeakTopologiesandDualityinLocallyConvexLinearTopologicalSpaces
1.PolarSets
2.BarrelSpaces
3.Semi-reflexivityandReflexivity
4.TheEberlein-ShmulyanTheorem

Ⅵ.FourierTransformandDifferentialEquations
1.TheFourierTransformofRapidlyDecreasingFunctions
2.TheFourierTransformofTemperedDistributions
3.Convolutions
4.ThePaley-WienerTheorems.TheOne-sidedLaplaceTransform
5.Titchmarsh'sTheorem
6.Mikusinski'sOperationalCalculus
7.Sobolev'sLemma
8.Garding'sInequality
9.Friedrichs'ThEorem
10.TheMalgrange-EhrenpreisTheorem
11.DifferentialOperatorswithUniformStrength
12.TheI-Iypoellipticity(Hormander'sTheorem)

Ⅶ.DualOperators
1.DualOperators
2.AdjointOperators
3.SymmetricOperatorsandSelf-adjointOperators
4.UnitaryOperators.TheCayleyTransform
5.TheClosedRangeTheorem

Ⅷ.ResolventandSpectrum
1.TheResolventandSpectrum
2.TheResolventEquationandSpectralRadius
3.TheMeanErgodicTheorem
4.ErgodicTheoremsoftheHilleTypeConcerningPseudo-resolvents
5.TheMeanValueofanAlmostPeriodicFunction
6.TheResolventofaDualOperator
7.Dunford'sIntegral
8.TheIsolatedSingularitiesofaResolvent

Ⅸ.AnalyticalTheoryofSemi-groups
1.TheSemi-groupofClass(Co)
2.TheEqui-continuousSemi-groupofClass(Co)inLocallyConvexSpaces,ExamplesofSemi-groups
3.TheInfinitesimalGeneratorofanEqui-continuousSemi-groupofClass(Co)
4.TheResolventoftheInfinitesimalGeneratorA
5.ExamplesofInfinitesimalGenerators
6.TheExponentialofaContinuousLinearOperatorwhosePowersareEqui-continuous
7.TheRepresentationandtheCharacterizationofEqui-con-tinuousSemi-groupsofClass(Co)inTermsoftheCorre-spondingInfinitesimalGenerators
8.ContractionSemi-groupsandDissipativeOperators
9.Equi-continuousGroupsofClass(Co).Stone'sTheorem
10.HolomorphicSemi-groups
11.FractionalPowersofClosedOperators
12.TheConvergenceofSemi-groups.TheTrotter-KatoTheorem
13.DualSemi-groups.Phillips'Theorem

Ⅹ.CompactOperators
1.CompactSetsinB-spaces
2.CompactOperatorsandNuclearOperators
3.TheRellich-GardingTheorem
4.Schauder'sTheorem
5.TheRiesz-SchauderTheory
6.Dirichlet'sProblem
AppendixtoChapterX.TheNuclearSpaceofA.GROTHENDIECK

Ⅺ.NormedRingsandSpectralRepresentation
1.MaximalIdealsofaNormedRing
2.TheRadical.TheSemi-simplicity
3.TheSpectralResolutionofBoundedNormalOperators
4.TheSpectralResolutionofaUnitaryOperator
5.TheResolutionoftheIdentity
6.TheSpectralResolutionofaSelf-adjointOperator
7.RealOperatorsandSemi-boundedOperators.Friedrichs'Theorem
8.TheSpectrumofaSelf-adjointOperator.Rayleigh'sPrin-cipleandtheKrylov-WeinsteinTheorem.TheMultiplicityoftheSpectrum
9.TheGeneralExpansionTheorem.AConditionfortheAbsenceoftheContinuousSpectrum
10.ThePeter-Weyl-NeumannTheorem
11.Tannaka'sDualityTheoremforNon-commutativeCompactGroups
12.FunctionsofaSelf-adjointOperator
13.Stone'sTheoremandBochner'sTheorem
14.ACanonicalFormofaSelf-adjointOperatorwithSimpleSpectrum
15.TheDefectIndicesofaSymmetricOperator.TheGeneralizedResolutionoftheldentity
16.TheGroup-ringL'andWiener'sTauberianTheorem

Ⅻ.OtherRepresentationTheoremsinLinearSpaces
1.ExtremalPoints.TheKrein-MilmanTheorem
2.VectorLattices
3.B-latticesandF-lattices
4.AConvergenceTheoremofBANACH
5.TheRepresentationofaVectorLatticeasPointFunctions
6.TheRepresentationofaVectorLatticeasSetFunctions

ⅩⅢ.ErgodicTheoryandDiffusionTheory
1.TheMarkovProcesswithanInvariantMeasure
2.AnIndividualErgodicTheoremandItsApplications
3.TheErgodicHypothesisandtheH-theorem
4.TheErgodicDecompositionofaMarkovProcesswithaLocallyCompactPhaseSpace
5.TheBrownianMotiononaHomogeneousRiemannianSpace
6.TheGeneralizedLaplacianofW.FELLER
7.AnExtensionoftheDiffusionOperator
8.MarkovProcessesandPotentials
9.AbstractPotentialOperatorsandSemi-groups

ⅩⅣ.TheIntegrationoftheEquationofEvolution
1.IntegrationofDiffusionEquationsinLS(Rm)
2.IntegrationofDiffusionEquationsinaCompactRiemannianSpace
3.IntegrationofWaveEquationsinaEuclideanSpaceRm
4.IntegrationofTemporallyInhomogeneousEquationsofEvolutioninaB-space
5.TheMethodofTANABEandSOBOI.EVSKI
6.Non-linearEvolutionEquations1(TheKomura-KatoApproach)
7.Non-linearEvolutionEquations2(TheApproachthroughtheCrandall-LiggettConvergenceTheorem)
SupplementaryNotes
Bibliography
Index
NotationofSpaces

其他推荐