泛函分析导论

Preface

ListofSymbols

Chapter1MetricSpaces

1.1Preliminaries

1.2DefinitionsandExamples

1.3ConvergenceinaMetricSpace

1.4SetsinaMetricSpace

1.5CompleteMetricSpaces

1.6ContinuousMappingsonMetricSpaces

1.7CompactMetricSpaces

1.8TheContractionMappingPrinciple

Chapter2NormedLinearSpaces.BanachSpaces

2.1ReviewofLinearSpaces

2.2NorminaLinearSpace

2.3ExamplesofNormedLinearSpaces

2.4FiniteDimensionalNormedLinearSpaces

2.5LinearSubspacesofNormedLinearSpaces

2.6QuotientSpaces

2.7TheWeierstrassApproximationTheorem

Chapter3InnerProductSpaces.HilbertSpaces

3.1InnerProducts

3.2Orthogonality

3.3OrthonormalSystems

3.4FourierSeries

Chapter4LinearOperators.FundamentalTheorems

4.1ContinuousLinearOperatorsandFunctionals

4.2SpacesofBoundedLinearOperatorsandDualSpaces

4.3TheBanach-SteinhausTheorem

4.4InversesofOperators.TheBanachTheorem

4.5TheHahn-BanachTheorem

4.6StrongandWeakConvergence

Chapter5LinearOperatorsonHilbertSpaces

5.1AdjointOperators.TheLax-MilgramTheorem

5.2SpectralTheoremforSelf-adjointCompactOperators

Bibliography

Index

　　Functionalanalysisisprimarilyconcernedwithinfinite-dimensionallinear(vector)spaces,mainlyfunctionspaceswhose"points"arefunctions,andmappingsbetweenthem,usuallycalledoperatorsor,functionalsiftherangeisonthereallineorinthecomplexplane.Itwasinventedanddevelopedinthelastyearsofthenineteenthcenturyandthefirstfewdecadesofthetwentiethcentury.Duringtheearlyperiodofitsdevelopment,theoriginalpurposeoffunctionalanalysiswastouseaframeworkwhichallowsthestudyofdifferentialandintegralequationstobeconsideredinthesameformulation(cf.[6]).Later,functionalanalysisdevelopedrapidlyasin-depthstudyandin-terconnectiononspectraltheoryofordinaryandpartialdifferentialequations,potentialtheory,Fourierexpansions,andappliedmathematicaltechniques,especially,ontheinfluenceofmathematicalphysicsandquantummechan-ics.