泛函分析

　　he present book is based on lectures given by the author at the University of Tokyo during the past ten years. It is intended as a textbook to be studied by students on their own or to be used in a course on Functional Analysis， i.e.， the general theory of linear operators infunction spaces together with salient features of its application to diverse fields of modem and classical analysis. Necessary prerequisites for the reading of this book are summarized，with or without proof， in Chapter 0 under titles： Set Theory， Topological Spaces， Measure Spaces and Linear Spaces. Then， starting with the chapter on Semi-norms， a general theory of Banach and Hilbert spaces is presented in connection with the theory of generalized functions of S. L. SOBOLEV and L. SCHWARTZ. While the book is primarily addressed to graduate students， it is hoped it might prove useful to research mathematicians， both pure and applied. The reader may pass， e.g.， fromChapter IX （Analytical Theory. of Semi-groups） directly to Chapter XIII （Ergodic Theory and Diffusion Theory） and to Chapter XIV （Integration of the Equation of Evolution）. Such materials as "Weak Topologies and Duality in Locally Convex Spaces" and "Nuclear Spaces" are presented in the form of the appendices to Chapter V and Chapter X，respectively. These might be skipped for the first reading by those who are interested rather in the application of linear operators.

暂缺《泛函分析》作者简介

O.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.TheTermwiseDifferentiabilityofaSequenceofGeneralized
Functions
4.ThePrincipleoftheCondensationofSingularities
5.TheOpenMappingTheorem
6.TheClosedGraphTheorem
7.AnApplicationoftheClosedGraphTheorem(H6rmander'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.TheWeakConvergenceandTheWeak*Convergence
2.TheLocalSequentialWeakCompactnessofReflexiveB-
spaces.TheUniformConvexity
3.Dunford'sTheoremandTheGelfand-MazurTheorem
4.TheWeakandStrongMeasurability.Pettis'Theorem
5.Bochner'sIntegral
AppendixtoChapterV.WeakTopologiesandDualityinLocally
ConvexLinearTopologicalSpaces
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.TheHypoellipticity(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.ErgodicTheoremsoftheHilleTypeConcerningPseudoresolvents
5.TheMeanValueofanAlmostPeriodicFunction
6.TheResolventofaDualOperator
7.Dunford'sIntegral
8.TheIsolatedSingularitiesofaResolvent
IX.AnalyticalTheoryofSemi-groups
1.TheSemi-groupofClass(Co)
2.TheEqui-continuousSemi-groupofClass(Co)inLocally
CofivexSpaces.ExamplesofSemi-groups
3.TheInfinitesimalGeneratorofanEqui-continuousSemigroupofClass(Co)
4.TheResolventoftheInfinitesimalGeneratorA
5.ExamplesofInfinitesimalGenerators
6.TheExponentialofaContinuousLinearOperatorwhose
PowersareEqui-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-
ciple,andtheKrylov-WeinsteinTheorem.TheMultiplicity
oftheSpectrum
9.TheGeneralExpansionTheorem.AConditionforthe
AbsenceoftheContinuousSpectrum
10.ThePeter-Weyl-NeumannTheorem
11.Tannaka'sDualityTheoremforNon-commutativeCompactGroups
12.FunctionsofaSelf-adjointOperator
13.Stone'sTheoremandBochner'sTheorem
14.ACanonicalFormofaSelf-adjointOperatorwithSimpleSpectrum
15.TheDefectIndicesofaSymmetricOperator.TheGeneralized
ResolutionoftheIdentity
16.TheGroup-ringL1andWiener'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.TheErgodicDecompositionofaMarkovProcesswitha
LocallyCompactPhaseSpace
5.TheBrownianMotiononaHomogeneousRiemannianSpace
6.TheGeneralizedLaplacianofW.FELLER
7.AnExtensionoftheDiffusionOperator
8.MarkovProcessesandPotentials
9.AbstractPotentialOperatorsandSemi-groups
XIV.TheIntegrationoftheEquationofEvolution
1.IntegrationofDiffusionEquationsinL2(Rm)
2.IntegrationofDiffusionEquationsinaCompactRiemannianSpace
3.IntegrationofWaveEquationsinaEuclideanSpaceRm
4.IntegrationofTemporallyInhomogeneousEquationsof
EvolutioninaB-space
5.TheMethodofTANABEandSOBOLEVSKI
6.Non-linearEvolutionEquations1(TheKomura-KatoApproach)
7.Non-linearEvolutionEquations2(TheApproachthrough
theCrandall-LiggettConvergenceTheorem)
SupplementaryNotes
Bibliography
Index
NotationofSpaces