应用泛函分析（第1卷）
出版时间:
2009-10
版次:
1
ISBN:
9787510005442
定价:
59.00
装帧:
平装
开本:
24开
纸张:
胶版纸
页数:
481页
正文语种:
英语
-
Moreprecisely，by（i），Imeanasystematicpresentationofthematerialgovernedbythedesireformathematicalperfectionandcompletenessoftheresults.Incontrastto（i），approach（ii）startsoutfromthequestion"Whatarethemostimportantapplications？"andthentriestoanswerthisquestionasquicklyaspossible.Here，onewalksdirectlyonthemainroadanddoesnotwanderintoalltheniceandinterestingsideroads.
Thepresentbookisbasedonthesecondapproach.Itisaddressedtoundergraduateandbeginninggraduatestudentsofmathematics，physics，andengineeringwhowanttolearnhowfunctionalanalysiselegantlysolvesmahematicalproblemsthatarerelatedtoourrealworldazldthathaveplayedanimportantroleinthehistoryofmathematics.Thereadershouldsensethatthetheoryisbeingdeveloped，notsimplyforitsownsake，butfortheeffectivesolutionofconcreteproblems. Preface
Prologue
ContentsofAMSVolume109
1BanachSpacesandFixed-PointTheorems
1.1LinearSpacesandDimension
1.2NormedSpacesandConvergence
1.3BanachSpacesandtheCauchyConvergenceCriterion
1.4OpenandClosedSets
1.5Operators
1.6TheBanachFixed-PointTheoremandtheIterationMethod
1.7ApplicationstoIntegralEquations
1.8ApplicationstoOrdinaryDifferentialEquations
1.9Continuity
1.10Convexity
1.11Compactness
1.12Finite-DimensionalBanachSpacesandEquivalentNorms
1.13TheMinkowskiFunctionalandHomeomorphisms
1.14TheBrouwerFixed-PointTheorem
1.15TheSchauderFixed-PointTheorem
1.16ApplicationstoIntegralEquations
1.17ApplicationstoOrdinaryDifferentialEquations
1.18TheLeray-SchauderPrincipleandaprioriEstimates
1.19Sub-andSupersolutions，andtheIterationMethodinOrderedBanachSpaces
1.20LinearOperators
1.21TheDualSpace
1.22InfiniteSeriesinNormedSpaces
1.23BanachAlgebrasandOperatorFunctions
1.24ApplicationstoLinearDifferentialEquationsinBanachSpaces
1.25ApplicationstotheSpectrum
1.26DensityandApproximation
1.27SummaryofImportantNotions
2HilbertSpaces，Orthogonality，andtheDirichlet
Principle
2.1HilbertSpaces
2.2StandardExamples
2.3BilinearForms
2.4TheMainTheoremonQuadraticVariationalProblems
2.5TheFunctionalAnalyticJustificationoftheDirichletPrinciple
2.6TheConvergenceoftheRitzMethodforQuadraticVariationalProblems
2.7ApplicationstoBoundary-ValueProblems，theMethodofFiniteElements，andElasticity
2.8GeneralizedFunctionsandLinearFunctionals
2.9OrthogonalProjection
2.10LinearFunctionalsandtheRieszTheorem
2.11TheDualityMap
2.12DualityforQuadraticVariationalProblems
2.13TheLinearOrthogonalityPrinciple
2.14NonlinearMonotoneOperators
2.15ApplicationstotheNonlinearLax-MilgramTheoremandtheNonlinearOrthogonalityPrinciple
3HilbertSpacesandGeneralizedFourierSeries
3.1OrthonormalSeries
3.2ApplicationstoClassicalFourierSeries
3.3TheSchmidtOrthogonalizationMethod
3.4ApplicationstoPolynomials
3.5UnitaryOperators
3.6TheExtensionPrinciple
3.7ApplicationstotheFourierTransformation
3.8TheFourierTransformofTemperedGeneralizedFunctions
4EigenvalueProblemsforLinearCompactSymmetricOperators
4.1SymmetricOperators
4.2TheHilbert-SchmidtTheory
4.3TheFredholmAlternative
4.4ApplicationstoIntegralEquations
4.5ApplicationstoBoundary-EigenvalueValueProblems
5Self-AdjointOperators，thePriedrichsExtensionandthePartialDifferentialEquationsofMathematical
Physics
5.1ExtensionsandEmbeddings
5.2Self-AdjointOperators
5.3TheEnergeticSpace
5.4TheEnergeticExtension
5.5TheFriedrichsExtensionofSymmetricOperators
5.6ApplicationstoBoundary-EigenvalueProblemsfortheLaplaceEquation
5.7ThePoincar6InequalityandRellichsCompactnessTheorem
5.8FunctionsofSelf-AdjointOperators
5.9Semigroups，One-ParameterGroups，andTheirPhysicalRelevance
5.10ApplicationstotheHeatEquation
5.11ApplicationstotheWaveEquation
5.12ApplicationstotheVibratingStringandtheFourierMethod
5.13ApplicationstotheSchrSdingerEquation
5.14ApplicationstoQuantumMechanics
5.15GeneralizedEigenfunctions
5.16TraceClassOperators
5.17ApplicationstoQuantumStatistics
5.18C*-AlgebrasandtheAlgebraicApproachtoQuantumStatistics
5.19TheFockSpaceinQuantumFieldTheoryandthePauliPrinciple
5.20ALookatScatteringTheory
5.21TheLanguageofPhysicistsinQuantumPhysicsandtheJustificationoftheDiracCalculus
5.22TheEuclideanStrategyinQuantumPhysics
5.23ApplicationstoFeynmansPathIntegral
5.24TheImportanceofthePropagatorinQuantumPhysics
5.25ALookatSolitonsandInverseScatteringTheory
Epilogue
Appendix
References
HintsforFurtherReading
ListofSymbols
ListofTheorems
ListoftheMostImportantDefinitions
SubjectIndex
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内容简介:
Moreprecisely，by（i），Imeanasystematicpresentationofthematerialgovernedbythedesireformathematicalperfectionandcompletenessoftheresults.Incontrastto（i），approach（ii）startsoutfromthequestion"Whatarethemostimportantapplications？"andthentriestoanswerthisquestionasquicklyaspossible.Here，onewalksdirectlyonthemainroadanddoesnotwanderintoalltheniceandinterestingsideroads.
Thepresentbookisbasedonthesecondapproach.Itisaddressedtoundergraduateandbeginninggraduatestudentsofmathematics，physics，andengineeringwhowanttolearnhowfunctionalanalysiselegantlysolvesmahematicalproblemsthatarerelatedtoourrealworldazldthathaveplayedanimportantroleinthehistoryofmathematics.Thereadershouldsensethatthetheoryisbeingdeveloped，notsimplyforitsownsake，butfortheeffectivesolutionofconcreteproblems.
-
目录:
Preface
Prologue
ContentsofAMSVolume109
1BanachSpacesandFixed-PointTheorems
1.1LinearSpacesandDimension
1.2NormedSpacesandConvergence
1.3BanachSpacesandtheCauchyConvergenceCriterion
1.4OpenandClosedSets
1.5Operators
1.6TheBanachFixed-PointTheoremandtheIterationMethod
1.7ApplicationstoIntegralEquations
1.8ApplicationstoOrdinaryDifferentialEquations
1.9Continuity
1.10Convexity
1.11Compactness
1.12Finite-DimensionalBanachSpacesandEquivalentNorms
1.13TheMinkowskiFunctionalandHomeomorphisms
1.14TheBrouwerFixed-PointTheorem
1.15TheSchauderFixed-PointTheorem
1.16ApplicationstoIntegralEquations
1.17ApplicationstoOrdinaryDifferentialEquations
1.18TheLeray-SchauderPrincipleandaprioriEstimates
1.19Sub-andSupersolutions，andtheIterationMethodinOrderedBanachSpaces
1.20LinearOperators
1.21TheDualSpace
1.22InfiniteSeriesinNormedSpaces
1.23BanachAlgebrasandOperatorFunctions
1.24ApplicationstoLinearDifferentialEquationsinBanachSpaces
1.25ApplicationstotheSpectrum
1.26DensityandApproximation
1.27SummaryofImportantNotions
2HilbertSpaces，Orthogonality，andtheDirichlet
Principle
2.1HilbertSpaces
2.2StandardExamples
2.3BilinearForms
2.4TheMainTheoremonQuadraticVariationalProblems
2.5TheFunctionalAnalyticJustificationoftheDirichletPrinciple
2.6TheConvergenceoftheRitzMethodforQuadraticVariationalProblems
2.7ApplicationstoBoundary-ValueProblems，theMethodofFiniteElements，andElasticity
2.8GeneralizedFunctionsandLinearFunctionals
2.9OrthogonalProjection
2.10LinearFunctionalsandtheRieszTheorem
2.11TheDualityMap
2.12DualityforQuadraticVariationalProblems
2.13TheLinearOrthogonalityPrinciple
2.14NonlinearMonotoneOperators
2.15ApplicationstotheNonlinearLax-MilgramTheoremandtheNonlinearOrthogonalityPrinciple
3HilbertSpacesandGeneralizedFourierSeries
3.1OrthonormalSeries
3.2ApplicationstoClassicalFourierSeries
3.3TheSchmidtOrthogonalizationMethod
3.4ApplicationstoPolynomials
3.5UnitaryOperators
3.6TheExtensionPrinciple
3.7ApplicationstotheFourierTransformation
3.8TheFourierTransformofTemperedGeneralizedFunctions
4EigenvalueProblemsforLinearCompactSymmetricOperators
4.1SymmetricOperators
4.2TheHilbert-SchmidtTheory
4.3TheFredholmAlternative
4.4ApplicationstoIntegralEquations
4.5ApplicationstoBoundary-EigenvalueValueProblems
5Self-AdjointOperators，thePriedrichsExtensionandthePartialDifferentialEquationsofMathematical
Physics
5.1ExtensionsandEmbeddings
5.2Self-AdjointOperators
5.3TheEnergeticSpace
5.4TheEnergeticExtension
5.5TheFriedrichsExtensionofSymmetricOperators
5.6ApplicationstoBoundary-EigenvalueProblemsfortheLaplaceEquation
5.7ThePoincar6InequalityandRellichsCompactnessTheorem
5.8FunctionsofSelf-AdjointOperators
5.9Semigroups，One-ParameterGroups，andTheirPhysicalRelevance
5.10ApplicationstotheHeatEquation
5.11ApplicationstotheWaveEquation
5.12ApplicationstotheVibratingStringandtheFourierMethod
5.13ApplicationstotheSchrSdingerEquation
5.14ApplicationstoQuantumMechanics
5.15GeneralizedEigenfunctions
5.16TraceClassOperators
5.17ApplicationstoQuantumStatistics
5.18C*-AlgebrasandtheAlgebraicApproachtoQuantumStatistics
5.19TheFockSpaceinQuantumFieldTheoryandthePauliPrinciple
5.20ALookatScatteringTheory
5.21TheLanguageofPhysicistsinQuantumPhysicsandtheJustificationoftheDiracCalculus
5.22TheEuclideanStrategyinQuantumPhysics
5.23ApplicationstoFeynmansPathIntegral
5.24TheImportanceofthePropagatorinQuantumPhysics
5.25ALookatSolitonsandInverseScatteringTheory
Epilogue
Appendix
References
HintsforFurtherReading
ListofSymbols
ListofTheorems
ListoftheMostImportantDefinitions
SubjectIndex
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