Introduction to Banach spaces analysis and probability. Vol.2 PDF
Preview Introduction to Banach spaces analysis and probability. Vol.2
CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 167 EditorialBoard B. BOLLOBÁS, W. FULTON, F. KIRWAN, P. SARNAK, B. SIMON, B. TOTARO INTRODUCTION TO BANACH SPACES: ANALYSIS AND PROBABILITY This two-volume text provides a complete overview of the theory of Banach spaces, emphasisingitsinterplaywithclassicalandharmonicanalysis(particularlySidonsets) andprobability.Theauthorsgiveafullexpositionofallresults,aswellasnumerous exercisesandcommentstocomplementthetextandaidgraduatestudentsinfunctional analysis. The book will also be an invaluable reference volume for researchers in analysis. Volume 1 covers the basics of Banach space theory, operator theory in Banach spaces,harmonicanalysisandprobability.Theauthorsalsoprovideanannexdevoted tocompactAbeliangroups. Volume2focusesonapplicationsofthetoolspresentedinthefirstvolume,including Dvoretzky’stheorem,spaceswithouttheapproximationproperty,Gaussianprocesses andmore.Fourleadingexpertsalsoprovidesurveysoutliningmajordevelopmentsin thefieldsincethepublicationoftheoriginalFrenchedition. Daniel Li is Emeritus Professor at Artois University, France. He has published over 40papersandtwotextbooks. Hervé Queffélec is Emeritus Professor at Lille 1 University. He has published over 60 papers, two research books and four textbooks, including Twelve Landmarks of Twentieth-CenturyAnalysis(2015). CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS EditorialBoard: B.Bollobás,W.Fulton,F.Kirwan,P.Sarnak,B.Simon,B.Totaro AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridgeUniversityPress.Fora completeserieslistingvisit:www.cambridge.org/mathematics. Alreadypublished 131 D.A.CravenThetheoryoffusionsystems 132 J.VäänänenModelsandgames 133 G.Malle&D.TestermanLinearalgebraicgroupsandfinitegroupsofLietype 134 P.LiGeometricanalysis 135 F.MaggiSetsoffiniteperimeterandgeometricvariationalproblems 136 M.Brodmann&R.Y.SharpLocalcohomology(2ndEdition) 137 C.Muscalu&W.SchlagClassicalandmultilinearharmonicanalysis,I 138 C.Muscalu&W.SchlagClassicalandmultilinearharmonicanalysis,II 139 B.HelfferSpectraltheoryanditsapplications 140 R.Pemantle&M.C.WilsonAnalyticcombinatoricsinseveralvariables 141 B.Branner&N.FagellaQuasiconformalsurgeryinholomorphicdynamics 142 R.M.DudleyUniformcentrallimittheorems(2ndEdition) 143 T.LeinsterBasiccategorytheory 144 I.Arzhantsev,U.Derenthal,J.Hausen&A.LafaceCoxrings 145 M.VianaLecturesonLyapunovexponents 146 J.-H.Evertse&K.Gyo˝ryUnitequationsinDiophantinenumbertheory 147 A.PrasadRepresentationtheory 148 S.R.Garcia,J.Mashreghi&W.T.RossIntroductiontomodelspacesandtheiroperators 149 C.Godsil&K.MeagherErdo˝s–Ko–Radotheorems:Algebraicapproaches 150 P.MattilaFourieranalysisandHausdorffdimension 151 M.Viana&K.OliveiraFoundationsofergodictheory 152 V.I.Paulsen&M.RaghupathiAnintroductiontothetheoryofreproducingkernelHilbertspaces 153 R.Beals&R.WongSpecialfunctionsandorthogonalpolynomials 154 V.JurdjevicOptimalcontrolandgeometry:Integrablesystems 155 G.PisierMartingalesinBanachspaces 156 C.T.C.WallDifferentialtopology 157 J.C.Robinson,J.L.Rodrigo&W.SadowskiThethree-dimensionalNavier–Stokesequations 158 D.HuybrechtsLecturesonK3surfaces 159 H.Matsumoto&S.TaniguchiStochasticanalysis 160 A.Borodin&G.OlshanskiRepresentationsoftheinfinitesymmetricgroup 161 P.WebbFinitegrouprepresentationsforthepuremathematician 162 C.J.Bishop&Y.PeresFractalsinprobabilityandanalysis 163 A.BovierGaussianprocessesontrees 164 P.SchneiderGaloisrepresentationsand (ϕ,(cid:3))-modules 165 P.Gille&T.SzamuelyCentralsimplealgebrasandGaloiscohomology(2ndEdition) 166 D.Li&H.QueffelecIntroductiontoBanachspaces,I 167 D.Li&H.QueffelecIntroductiontoBanachspaces,II 168 J.Carlson,S.Müller-Stach&C.PetersPeriodmappingsandperioddomains(2ndEdition) 169 J.M.LandsbergGeometryandcomplexitytheory 170 J.S.MilneAlgebraicgroups Introduction to Banach Spaces: Analysis and Probability Volume2 DANIEL LI Universitéd’Artois,France HERVÉ QUEFFÉLEC UniversitédeLilleI,France TranslatedfromtheFrenchby DANIÈLE GIBBONS and GREG GIBBONS UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 4843/24,2ndFloor,AnsariRoad,Daryaganj,Delhi–110002,India 79AnsonRoad,#06–04/06,Singapore079906 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781107162624 DOI:10.1017/9781316677391 OriginallypublishedinFrenchasIntroductionàl’étudedesespacesdeBanach bySociétéMathématiquedeFrance,2004 ©SociétéMathématiquedeFrance2004 FirstpublishedinEnglishbyCambridgeUniversityPress2018 Englishtranslation©CambridgeUniversityPress2018 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. PrintedintheUnitedStatesofAmericabySheridanBooks,Inc. AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. ISBN–2VolumeSet978-1-107-16263-1Hardback ISBN–Volume1978-1-107-16051-4Hardback ISBN–Volume2978-1-107-16262-4Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. Dedicatedtothememoryof Jean-PierreKahane Contents Volume2 ContentsofVolume1 pageix Preface xiii 1 EuclideanSections 1 I Introduction 1 II AnInequalityofConcentrationofMeasure 1 III ComparisonofGaussianVectors 8 IV Dvoretzky’sTheorem 18 V TheLindenstrauss–TzafririTheorem 40 VI Comments 45 VII Exercises 46 2 SeparableBanachSpaceswithouttheApproximation Property 51 I IntroductionandDefinitions 51 II TheGrothendieckReductions 53 III TheCounterexamplesofEnfloandDavie 59 IV Comments 68 V Exercises 70 3 GaussianProcesses 72 I Introduction 72 II GaussianProcesses 72 III BrownianMotion 76 IV Dudley’sMajorationTheorem 79 V Fernique’sMinorationTheoremforStationaryProcesses 85 VI TheElton–PajorTheorem 95 VII Comments 122 VIII Exercises 123 vii viii Contents 4 ReflexiveSubspacesofL1 127 I Introduction 127 II StructureofReflexiveSubspacesofL1 128 III ExamplesofReflexiveSubspacesofL1 142 IV Maurey’sFactorizationTheoremandRosenthal’sTheorem 150 V Finite-DimensionalSubspacesofL1 157 VI Comments 176 VII Exercises 180 5 TheMethodofSelectors.ExamplesofItsUse 193 I Introduction 193 II ExtractionofQuasi-IndependentSets 193 III SumsofSinesandVectorialHilbertTransforms 217 IV MinorationoftheK-ConvexityConstant 223 V Comments 228 VI Exercises 230 6 ThePisierSpaceofAlmostSurelyContinuousFunctions. Applications 234 I Introduction 234 II ComplementsonBanach-ValuedVariables 235 III TheCasSpace 243 IV ApplicationsoftheSpaceCas 261 V TheBourgain–MilmanTheorem 268 VI Comments 282 VII Exercises 287 AppendixA NewsintheTheoryofInfinite-DimensionalBanach SpacesinthePast20Years 290 AppendixB AnUpdateonSomeProblemsinHigh-Dimensional ConvexGeometryandRelatedProbabilisticResults 297 AppendixC AFewUpdatesandPointers 307 AppendixD OntheMeshConditionforSidonSets 316 References 324 NotationIndexforVolume2 355 AuthorIndexforVolume2 356 SubjectIndexforVolume2 359 NotationIndexforVolume1 363 AuthorIndexforVolume1 365 SubjectIndexforVolume1 369 Contents Volume1 ContentsofVolume2 pagex Preface xiii ∗ PreliminaryChapter WeakandWeak Topologies.Filters, Ultrafilters.Ordinals 1 I Introduction 1 ∗ II WeakandWeak Topologies 1 III Filters,Ultrafilters.Ordinals 7 IV Exercises 12 1 FundamentalNotionsofProbability 13 I Introduction 13 II Convergence 15 III SeriesofIndependentRandomVariables 21 IV Khintchine’sInequalities 30 V Martingales 35 VI Comments 42 VII Exercises 43 2 BasesinBanachSpaces 46 I Introduction 46 II SchauderBases:Generalities 46 III BasesandtheStructureofBanachSpaces 59 IV Comments 74 V Exercises 76 3 UnconditionalConvergence 83 I Introduction 83 II UnconditionalConvergence 83 ix
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