Introduction to Banach spaces analysis and probability. Vol.2

406 Pages·2018·1.951 MB·English

by  Li D., Queffelec H

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
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ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107162624
DOI:10.1017/9781316677391
OriginallypublishedinFrenchasIntroductionàl’étudedesespacesdeBanach
bySociétéMathématiquedeFrance,2004
©SociétéMathématiquedeFrance2004
FirstpublishedinEnglishbyCambridgeUniversityPress2018
Englishtranslation©CambridgeUniversityPress2018
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
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ISBN–2VolumeSet978-1-107-16263-1Hardback
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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