Foreword
1.LinearSpaces
Axiomsforlinearspaces-Infinite-dimensionalexamples-Subspace,linearspan-Quotientspace-Isomorphism-Convexsets-Extremesubsets
2.LinearMaps
2.1Algebraoflinearmaps,
Axiomsforlinearmaps-Sumsandcomposites-Invertiblelinearmaps-Nullspaceandrange-Invariantsubspaces
2.2.Indexofalinearmap,
Degeneratemaps-Pseudoinverse-IndexmProductformulafortheindex-Stabilityoftheindex
3.TheHahn,BanachTheorem
3.1Theextensiontheorem,
Positivehomogeneous,subadditivefunctionals-Extensionoflinearfunctionals-Gaugefunctionsofconvexsets
3.2GeometricHahn-Banachtheorem,
Thehyperplaneseparationtheorem
3.3ExtensionsoftheHahn-Banachtheorem,
TheAgnew-Morsetheorem-The
Bohnenblust-Sobczyk-Soukhomlinovtheorem
4.ApplicationsoftheHahn-Banachtheorem
4.1Extensionofpositivelinearfunctionals,
4.2Banachlimits.
4.3Finitelyadditiveinvariantsetfunctions,
Historicalnote,
5.NormedLinearSpaces
5.1Norms,
Normsforquotientspaces-Completenormedlinearspaces-ThespacesC,B-LpspacesandH61dersinequality-Sobolevspaces,embeddingtheorems-Separablespaces
5.2Noncompactnessoftheunitbail,
Uniformconvexity-TheMazur-Ulamtheoremonisometrics
5.3Isometrics,
6.HilbertSpace
6.1Scalarproduct,
SchwarzinequalityParallelogramidentity——Completeness,closure-e2,L
6.2Closestpointinaclosedconvexsubset,54Orthogonalcomplementofasubspace-Orthogonaldecomposition
6.3Linearfunctionals,
TheRiesz-Frechetrepresentationtheorem-Lax-Milgramlemma
6.4Linearspan,
Orthogonalprojection-Orthonormalbases,Gram-Schmidtprocess-IsometriesofaHilbertspace
7.ApplicationsofHilbertSpaceResults
7.1Radon-Nikodymtheorem,
7.2Dirichletsproblem,
UseoftheRiesz-Frechettheorem-UseoftheLax-MilgramtheoremUseoforthogonaldecomposition
8.DualsofNormedLinearSpaces
8.1Boundedlinearfunctionals,
Dualspace
8.2Extensionofboundedlinearfunctionals,
Dualcharacterizationofnorm-Dualcharacterizationofdistancefromasubspace-Dualcharacterizationoftheclosedlinearspanofaset
8.3Reflexivespaces,
ReflexivityofLp,1
8.4Supportfunctionofaset,
Dualcharacterizationofconvexhull-Dualcharacterizationofdistancefromaclosed,convexset
9.ApplicationsofDuality
9.1Completenessofweightedpowers,
9.2TheMuntzapproximationtheorem,
9.3Rungestheorem,
9.4Dualvariationalproblemsinfunctiontheory,
9.5ExistenceofGreensfunction,
10.WeakConvergence
10.1Uniformboundednessofweaklyconvergentsequences,101Principleofuniformboundedness-Weaklysequentiallyclosedconvexsets
10.2Weaksequentialcompactness,104Compactnessofunitballinreflexivespace
10.3Weak*convergence,105Hellystheorem
11.ApplicationsofWeakConvergence
11.1Approximationofthefunctionbycontinuousfunctions,108Toeplitzstheoremonsummability
11.2DivergenceofFourierseries,
11.3Approximatequadrature,
11.4Weakandstronganalyticityofvector-valuedfunctions,
11.5Existenceofsolutionsofpartialdifferentialequations,112Galerkinsmethod
11.6Therepresentationofanalyticfunctionswithpositiverealpart,115Hergiotz-Riesztheorem
12.TheWeakandWeak*Topologies
Comparisonwithweaksequentialtopology-Closedconvexsetsintheweaktopology——Weakcompactness-Alaoglustheorem
13.LocallyConvexTopologiesandtheKrein-MilmanTheorem
13.1Separationofpointsbylinearfunctionals,
13.2TheKrein-Milmantheorem,
13.3TheStone-Weierstrasstheorem,
13.4Choquetstheorem,
14.ExamplesofConvexSetsandTheirExtremePoints
14.1Positivefunctionals,
14.2Convexfunctions,
14.3Completelymonotonefunctions,
14.4TheoremsofCaratheodoryandBochner,
14.5AtheoremofKrein,
14.6Positiveharmonicfunctions,
14.7TheHamburgermomentproblem,
14.8G.Birkhoffsconjecture,
14.9DeFinettistheorem,
14.10Measure-preservingmappings,
Historicalnote,
15.BoundedLinearMaps
15.1Boundednessandcontinuity,
Normofaboundedlinearmap-Transpose
15.2Strongandweaktopologies,
Strongandweaksequentialconvergence
15.3Principleofuniformboundedness,
15.4Compositionofboundedmaps,
15.5Theopenmappingprinciple,
ClosedgraphtheoremHistoricalnote,
16.ExamplesofBoundedLinearMaps
16.1Boundednessofintegraloperators,
IntegraloperatorsofHilbert-Schmidttype-IntegraloperatorsofHolmgrentype
16.2TheconvexitytheoremofMarcelRiesz,
16.3Examplesofboundedintegraloperators,
TheFouriertransform,ParsevalstheoremandHausdorff-Younginequality-TheHilberttransformTheLaplacetransform-TheHilbert-Hankeltransform
……
A.Riesz-Kakutanirepresentationtheorem
B.Theoryofdistributions
C.ZornsLemma
AuthorIndex
SubjectIndex