an introduction to functional analysis - vitali milman

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1、VitaliMilmanAnIntroductionToFunctionalAnalysisWORLD199923dedications4Contents1Linearspaces;normedspaces;firstexamples91.1Linearspaces.......................91.2Normedspaces;firstexamples.............111.2.1H¨olderinequality.................121.2.2Minkowskiinequality...........

2、...131.3Completeness;completion...............161.3.1Constructionofcompletion..........171.4Exercises.........................182Hilbertspaces212.1Basicnotions;firstexamples..............212.1.1Cauchy-Schwartzinequality..........222.1.2Bessel’sinequality...............232.1.3

3、Gram-Schmidtorthogonalizationprocedure.242.1.4Parseval’sequality...............252.2Projections;decompositions..............272.2.1Separablecase..................272.2.2Uniquenessofthedistancefromapointtoaconvexset:thegeometricmeaning......272.2.3Orthogonaldecomposition.....

4、......282.3Linearfunctionals....................292.3.1Linearfunctionalsinagenerallinearspace.292.3.2Boundedlinearfunctionalsinnormedspaces.Thenormofafunctional............312.3.3BoundedlinearfunctionalsinaHilbertspace322.3.4AnExampleofanon-separableHilbertspace:322.4Exercis

5、es.........................3356CONTENTS3Thedualspace
393.1Hahn-Banachtheoremanditsfirstconsequences.393.2DualSpaces.......................413.3Exercises:.........................424Boundedlinearoperators434.1Completenessofthespaceofboundedlinearopera-tors....................

6、........434.2Examplesoflinearoperators..............444.3Compactoperators...................454.3.1Compactsets..................464.3.2Thespaceofcompactoperators........484.4DualOperators......................484.5Differentconvergencesinthespaceofboundedoperators........

7、.......504.6InvertibleOperators..................524.7Exercises.........................525Spectraltheory575.1Classificationofspectrum...............575.2FredholmTheoryofcompactoperators........585.3Exercises.........................636Selfadjointcompactoperators656.1GeneralP

8、roperties...................656.2Exercises.........................727Self-