A short course on Banach space theory PDF
Preview A short course on Banach space theory
A SHORT COURSE ON BANACH SPACE THEORY LONDON MATHEMATICAL SOCIETY STUDENT TEXTS Managingeditor:ProfessorJ.W.Bruce, DepartmentofMathematics,UniversityofHull,UK 3 Localfields,J.W.S.CASSELS 4 Anintroductiontotwistortheory:Secondedition,S.A.HUGGETT&K.P.TOD 5 Introductiontogeneralrelativity,L.P.HUGHSTON&K.P.TOD 7 Thetheoryofevolutionanddynamicalsystems,J.HOFBAUER&K.SIGMUND 8 SummingandnuclearnormsinBanachspacetheory,G.J.O.JAMESON 9 AutomorphismsofsurfacesafterNielsenandThurston,A.CASSON&S.BLEILER 11 Spacetimeandsingularities,G.NABER 12 Undergraduatealgebraicgeometry,MILESREID 13 AnintroductiontoHankeloperators,J.R.PARTINGTON 15 Presentationsofgroups:Secondedition,D.L.JOHNSON 17 Aspectsofquantumfieldtheoryincurvedspacetime,S.A.FULLING 18 Braidsandcoverings:selectedtopics,VAGNLUNDSGAARDHANSEN 19 Stepsincommutativealgebra,R.Y.SHARP 20 Communicationtheory,C.M.GOLDIE&R.G.E.PINCH 21 RepresentationsoffinitegroupsofLietype,FRANC¸OISDIGNE&JEANMICHEL 22 Designs,graphs,codes,andtheirlinks,P.J.CAMERON&J.H.VANLINT 23 Complexalgebraiccurves,FRANCESKIRWAN 24 Lecturesonellipticcurves,J.W.S.CASSELS 25 Hyperbolicgeometry,BIRGERIVERSEN 26 AnintroductiontothetheoryofL-functionsandEisensteinseries,H.HIDA 27 HilbertSpace:compactoperatorsandthetracetheorem,J.R.RETHERFORD 28 Potentialtheoryinthecomplexplane,T.RANSFORD 29 Undergraduatecommutativealgebra,M.REID 31 TheLaplacianonaRiemannianmanifold,S.ROSENBERG 32 LecturesonLiegroupsandLiealgebras,R.CARTER,G.SEGAL,& I.MACDONALD 33 AprimerofalgebraicD-modules,S.C.COUTINHO 34 Complexalgebraicsurfaces,A.BEAUVILLE 35 Youngtableaux,W.FULTON 37 Amathematicalintroductiontowavelets,P.WOJTASZCZYK 38 Harmonicmaps,loopgroups,andintegrablesystems,M.GUEST 39 Settheoryfortheworkingmathematician,K.CIESIELSKI 40 Ergodictheoryanddynamicalsystems,M.POLLICOTT&M.YURI 41 Thealgorithmicresolutionofdiophantineequations,N.P.SMART 42 Equilibriumstatesinergodictheory,G.KELLER 43 Fourieranalysisonfinitegroupsandapplications,AUDREYTERRAS 44 Classicalinvarianttheory,PETERJ.OLVER 45 Permutationgroups,P.J.CAMERON 46 Riemannsurfaces:APrimer,A.BEARDON 47 Intoductorylecturesonringsandmodules,J.BEACHY 48 Settheory,A.HAJNA´L,P.HAMBURGER,&A.MATE 49 K-theoryforC∗-algebras,M.RORDAM,F.LARSEN,&N.LAUSTSEN 50 Abriefguidetoalgebraicnumbertheory,H.P.F.SWINNERTON-DYER 51 Stepsincommutativealgebra,R.Y.SHARP 52 FiniteMarkovchainsandalgorithmicapplications,O.HAGGSTROM 53 Theprimenumbertheorem,G.J.O.JAMESON 54 Topicsingraphautomorphismsandreconstruction,J.LAURI&R.SCAPELLATO 55 Elementarynumbertheory,grouptheory,andRamanujangraphs,G.DAVIDOFF, P.SARNAK,&A.VALETTE A SHORT COURSE ON BANACH SPACE THEORY N. L. CAROTHERS BowlingGreenStateUniversity Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press TheEdinburghBuilding,Cambridge,UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridg e.org /9780521842839 ©N.L.Carothers2004 Thisbookisincopyright.Subjecttostatutoryexceptionandtotheprovisionof relevantcollectivelicensingagreements,noreproductionofanypartmaytakeplace withoutthewrittenpermissionofCambridgeUniversityPress. Firstpublishedinprintformat 2005 - ---- eBook (NetLibrary) - --- eBook (NetLibrary) - ---- hardback - --- hardback - ---- paperback - --- paperback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof sforexternalorthird-partyinternetwebsitesreferredtointhisbook,anddoesnot guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate. Contents Preface pagexi 1 ClassicalBanachSpaces 1 TheSequenceSpaces(cid:2) andc 1 p 0 Finite-DimensionalSpaces 2 The L Spaces 3 p TheC(K)Spaces 4 HilbertSpace 6 “Neoclassical”Spaces 7 TheBigQuestions 7 NotesandRemarks 9 Exercises 9 2 Preliminaries 11 ContinuousLinearOperators 11 Finite-DimensionalSpaces 12 ContinuousLinearFunctionals 13 Adjoints 15 Projections 16 Quotients 17 ACuriousApplication 20 NotesandRemarks 20 Exercises 20 3 BasesinBanachSpaces 24 Schauder’sBasisforC[0,1] 28 TheHaarSystem 30 NotesandRemarks 32 Exercises 33 4 BasesinBanachSpacesII 34 AWealthofBasicSequences 34 DisjointlySupportedSequencesin L and(cid:2) 35 p p vii viii Contents EquivalentBases 38 NotesandRemarks 41 Exercises 42 5 BasesinBanachSpacesIII 44 BlockBasicSequences 44 Subspacesof(cid:2) andc 47 p 0 ComplementedSubspacesof(cid:2) andc 49 p 0 NotesandRemarks 51 Exercises 54 6 SpecialPropertiesofc0,(cid:2)1,and(cid:2)∞ 55 TrueStoriesAbout(cid:2) 55 1 TheSecretLifeof(cid:2)∞ 60 Confessionsofc 63 0 NotesandRemarks 65 Exercises 65 7 BasesandDuality 67 NotesandRemarks 71 Exercises 72 8 L Spaces 73 p BasicInequalities 73 ConvexFunctionsandJensen’sInequality 74 ATestforDisjointness 77 ConditionalExpectation 78 NotesandRemarks 82 Exercises 83 9 L SpacesII 85 p TheRademacherFunctions 85 Khinchine’sInequality 87 TheKadec–Pe(cid:1)lczyn´skiTheorem 91 NotesandRemarks 97 Exercises 98 10 L SpacesIII 99 p UnconditionalConvergence 99 Orlicz’sTheorem 101 NotesandRemarks 106 Exercises 106 11 Convexity 107 StrictConvexity 108 NearestPoints 112 Smoothness 113 Contents ix UniformConvexity 114 Clarkson’sInequalities 117 AnElementaryProofThat L∗ = L 119 p q NotesandRemarks 122 Exercises 122 12 C(K)Spaces 124 TheCantorSet 124 CompletelyRegularSpaces 125 NotesandRemarks 134 Exercises 134 13 WeakCompactnessin L 136 1 NotesandRemarks 141 Exercises 141 14 TheDunford–PettisProperty 142 NotesandRemarks 146 Exercises 147 15 C(K) SpacesII 148 TheStone–CˇechCompactification 148 ReturntoC(K) 153 NotesandRemarks 155 Exercises 155 16 C(K) SpacesIII 156 TheStone–CˇechCompactificationofaDiscreteSpace 156 AFewFactsAboutβN 157 “Topological”MeasureTheory 158 TheDualof(cid:2)∞ 161 TheRieszRepresentationTheoremforC(βD) 162 NotesandRemarks 165 Exercises 165 Appendix:TopologyReview 166 Separation 166 LocallyCompactHausdorffSpaces 167 WeakTopologies 169 ProductSpaces 170 Nets 171 NotesandRemarks 172 Exercises 172 References 173 Index 181
The list of books you might like
