内容简介

《经典巴拿赫空间1和2》延续了该系列书的一贯风格,深入但不深沉。材料新颖,许多内容是同类书籍不具备的。对于学习Banach空间结构理论的学者来说,这是一本参考价值极高的书籍;对于学习该科目的读者,《经典巴拿赫空间1和2》也是同等重要。目次:schauder基;C0空间和lp空间;对称基;Orlicz序列空间。
读者对象:数学专业高年级的学生、老师和相关的科研人员。

目录

1.SchauderBases
a.ExistenceofBasesandExamples
b.SchauderBasesandDuality
c.UnconditionalBases
d.ExamplesofSpacesWithoutanUnconditionalBasis
e.TheApproximationProperty
f.BiorthogonalSystems
g.SchauderDecompositions

2.TheSpacescoandlp
a.ProjectionsincoandlpandCharacterizationsoftheseSpaces
b.AbsolutelySummingOperatorsandUniquenessofUnconditionalBases
c.FredholmOperators,StrictlySingularOperatorsandComplementedSubspacesoflplr
d.SubspacesofCoandlpandtheApproximationProperty,ComplementablyUniversalSpaces
e.BanachSpacesContainingIvorco
f.ExtensionandLiftingProperties,Automorphismsofloo,coandlx

3.SymmetricBases
a.PropertiesofSymmetricBases,ExamplesandSpecialBlockBases
b.SubspacesofSpaceswithaSymmetricBasis

4.OrliczSequenceSpaces
a.SubspacesofOrliczSequenceSpaceswhichhaveaSymmetricBasis
b.DualityandComplementedSubspaces
c.ExamplesofOrliczSequenceSpaces.
d.ModularSequenceSpacesandSubspacesofIplr
e.LorentzSequenceSpaces
References
SubjectIndex

前言/序言

  TheappearanceofBanachsbook[8]in1932signifiedthebeginningofasystematicstudyofnormedlinearspaces,whichhavebeenthesubjectofcontinuousresearcheversince.
  Inthesixties,andespeciallyinthelastdecade,theresearchactivityinthisarea
  grewconsiderably.Asaresult,Banachspacetheorygainedverymuchindepthaswellasinscope.Mostofitswellknownclassicalproblemsweresolved,manyinterestingnewdirectionsweredeveloped,anddeepconnectionsbetweenBanachspacetheoryandotherareasofmathematicswereestablished.
  ThepurposeofthisbookistopresentthemainresultsandcurrentresearchdirectionsinthegeometryofBanachspaces,withanemphasisonthestudyofthestructureoftheclassicalBanachspaces,thatisC(K)andLp()andrelatedspaces.WedidnotattempttowriteacomprehensivesurveyofBanachspacetheory,orevenonlyofthetheoryofclassicalBanachspaces,sincetheamountofinterestingresultsonthesubjectmakessuchasurveypracticallyimpossible.
  Apartofthesubjectmatterofthisbookappearedinoutlineinourlecturenotes[96].Incontrasttothosenotes,mostoftheresultspresentedherearegivenwithcompleteproofs.WethereforehopethatitwillbepossibletousethepresentbookbothasatextbookonBanachspacetheoryandasareferencebookforresearchworkersinthearea.Itcontainsmuchmaterialwhichwasnotdiscussedin[96],alargepartofwhichbeingtheresultofveryrecentresearchwork.AnindicationtotherapidrecentprogressinBanachspacetheoryisthefactthatmostofthemanyproblemsstatedin[96]havebeensolvedbynow.
  Inthepresentvolumewealsostatesomeopenproblems.Itisreasonabletoexpectthatmanyofthesewillbesolvedinthenottoofarfuture.Wefeel,however,thatmostofthetopicsdiscussedherehavereachedarelativelyfinalform,andthattheirpresentationwillnotberadicallyaffectedbythesolutionoftheopenproblems.Amongthetopicsdiscussedindetailinthisvolume,theonewhichseemstoustobetheleastwellunderstoodandwhichmightchangethemostinthefuture,isthatoftheapproximationproperty.

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