实分析和泛函分析

　　This book is meant as a text for a first year graduate course in analysis. Any standard course in undergraduate analysis will constitute sufficient preparation for its understanding， for instance， my Undergraduate Analysis. I assume that the reader is acquainted with notions of uniform convergence and the like.　In this third edition， I have reorganized the book by covering integration before functional analysis. Such a rearrangement fits the way courses are taught in all the places I know of. I have added a number of examples and exercises， as well as some material about integration on the real line （e.g. on Dirac sequence approximation and on Fourier analysis）， and some material on functional analysis （e.g. the theory of the Gelfand transform in Chapter XVI）. These upgrade previous exercises to sections in the text.

暂缺《实分析和泛函分析》作者简介

PARTONE
GeneralTopology
CHAPTERI
Sets
1.SomeBasicTerminology
2.DenumerahleSets
3.Zorn'sLemma
CHAPTERII
TopologicalSpaces
1.OpenandClosedSets
2.ConnectedSets
3.CompactSpaces
4.SeparationbyContinuousFunctions
5.Exercises
CHAPTERIII
ContinuousFunctionsonCompactSets
1.TheStone-WeierstrassTheorem
2.IdealsofContinuousFunctions
3.Ascoli'sTheorem
4.Exercises
PARTTWO
BanachandHilbertSpaces
CHAPTERIV
BanachSpaces
1.Definitions,theDualSpace,andtheHahn-BanachTheorem
2.BanachAlgebras
3.TheLinearExtensionTheorem
4.CompletionofaNormedVectorSpace
5.SpaceswithOperators
Appendix:ConvexSets
1.TheKrein-MilmanTheorem
2.Mazur'sTheorem
6.Exercises
CHAPTERV
HIIbertSpace
1.HermitianForms
2.FunctionalsandOperators
3.Exercises
PARTTHREE
Integration
CHAPTERVI
TheGeneralIntegral
1.MeasuredSpaces,MeasurableMaps,andPositiveMeasures
2.TheIntegralofStepMaps
3.TheL1-Compledon
4.PropertiesoftheIntegral:FirstPart
5.PropertiesoftheIntegral:SecondPart
6.Approximations
7.ExtensionofPositiveMeasuresfromAlgebrastoq-Algebras
8.ProductMeasuresandIntegrationonaProductSpace
9.TheLebesgueIntegralinRp
10.Exercises
CHAPTERVII
DualityandRepresentationTheorems
1.TheHilbertSpaceL2(u)
2.DualityBetweenL1(u)andL(#)
3.ComplexandVectorialMeasures
4.ComplexorVectorialMeasuresandDuality
5.TheLBSpaces,16.TheLawofLargeNumbers
7.Exercises
CHAPTERVIII
SomeApplicationsofIntegration
1.Convolution
2.ContinuityandDifferentiationUndertheIntegralSign
3.DiracSequences
4.TheSchwartzSpaceandFourierTransform
5.TheFourierInversionFormula
6.ThePoissonSummationFormula
7.AnExampleofFourierTransformNotintheSchwartzSpace
8.Exercises
CHAPTERIX
IntegrationandMeasuresonLocallyCompactSpaces
1.PositiveandBoundedFunctionalsonCc(X)
2.PositiveFunctionalsasIntegrals
3.RegularPositiveMeasures
4.BoundedFunctionalsasIntegrals
5.LocalizationofaMeasureandoftheIntegral
6.ProductMeasuresonLocallyCompactSpaces
7Exercises
CHAPTERX
Riemann-StleltlesIntegralandMeasure
I.FunctionsofBoundedVariationandtheStieltjesIntegral
2.ApplicationstoFourierAnalysis
3.Exercises
CHAPTERXl
Distributions
I.DefinitionandExamples
2.SupportandLocalization
3.DerivationofDistributions
4.DistributionswithDiscreteSupport
CHAPTERXll
IntegrationonLocallyCompactGroups
1.TopologicalGroups
2.TheHaarIntegral,Uniqueness
3.ExistenceoftheHaarIntegral
4.MeasuresonFactorGroupsandHomogeneousSpaces
5.Exercises
PARTFOUR
Calculus
CHAPTERXIII
DifferentialCalculus
1.IntegrationinOneVariable
2.TheDerivativeasaLinearMap
3.PropertiesoftheDerivative
4.MeanValueTheorem
5.TheSecondDerivative
6.HigherDerivativesandTaylor'sFormula
7.PartialDerivatives
8.DifferentiatingUndertheIntegralSign
9.DifferentiationofSequences
10.Exercises
CHAPTERXlV
InverseMappingsandDifferentialEquations
1.TheInverseMappingTheorem
2.TheImplicitMappingTheorem
3.ExistenceTheoremforDifferentialEquations
4.LocalDependenceonInitialConditions
5.GlobalSmoothnessoftheFlow
6.Exercises
PARTFIVE
FunctionalAnalysis
CHAPTERXV
TheOpenMappingTheorem,FactorSpaces,andDuality
1.TheOpenMappingTheorem
2.Orthogonality
3.ApplicationsoftheOpenMappingTheorem
CHAPTERXVI
TheSpectrum
1.TheGelfand-MazurTheorem
2.TheGelfandTransform
3.C*-Algebras
4.Exercises
CHAPTERXVll
CompactandFredholmOperators
1.CompactOperators
2.FredholmOperatorsandtheIndex
3.SpectralTheoremforCompactOperators
4.ApplicationtoIntegralEquations
5.Exercises
CHAPTERXVlll
SpectralTheoremforBoundedHermifianOperators
1.HermitianandUnitaryOperators
2.PositiveHermitianOperators
3.TheSpectralTheoremforCompactHermitianOperators
4.TheSpectralTheoremforHermitianOperators
5.OrthogonalProjections
6.Schur'sLemma
7.PolarDecompositionofEndomorphisms
8.TheMorse-PalaisLemma
9.Exercises
CHAPTERXIX
FurtherSpectralTheorems
1.ProjectionFunctionsofOperators
2.Self-AdjointOperators
3.Example:TheLaplaceOperatorinthePlane
CHAPTERXX
SpectralMeasures
1.DefinitionoftheSpectralMeasure
2.UniquenessoftheSpectralMeasure:
theTitchmarshKodairaFormula
3.UnboundedFunctionsofOperators
4.SpectralFamiliesofProjections
5.TheSpectralIntegralasStiehjesIntegral
6.Exercises
PARTSIX
GlobalAnalysis
CHAPTERXXI
LocalIntegrationofDifferentialForms
1.SetsofMeasure0
2.ChangeofVariablesFormula
3.DifferentialForms
4.InverseImageofaForm
5.Appendix
CHAPTERXXII
Manifolds
1.Atlases,Charts,Morphisms
2.Submanifolds
3.TangentSpaces
4.PartitionsofUnity
5.ManifoldswithBoundary
6.VectorFieldsandGlobalDifferentialEquations
CHAPTERXXIII
IntegrationandMeasuresonManifolds
1.DifferentialFormsonManifolds
2.Orientation
3.TheMeasureAssociatedwithaDifferentialForm
4.Stokes'TheoremforaRectangularSimplex
5.Stokes'TheoremonaManifold
6.Stokes'TheoremwithSingularities
Bibliography
TableofNotation
Index