This two-volume text provides a complete overview of the theory of Banach spaces, emphasising its interplay with classical and harmonic analysis (particularly Sidon sets) and probability. The authors give a full exposition of all results, as well as numerous exercises and comments to complement the text and aid graduate students in functional analysis. The book will also be an invaluable reference volume for researchers in analysis. Volume 1 covers the basics of Banach space theory, operatory theory in Banach spaces, harmonic analysis and probability. The authors also provide an annex devoted to compact Abelian groups. Volume 2 focuses on applications of the tools presented in the first volume, including Dvoretzky's theorem, spaces without the approximation property, Gaussian processes, and more. In volume 2, four leading experts also provide surveys outlining major developments in the field since the publication of the original French edition.
Author(s): Daniel Li, Hervé Queffélec
Series: Cambridge Studies in Advanced Mathematics 166
Publisher: Cambridge University Press
Year: 2017
Language: English
Pages: 462
Contents......Page 8
Contents of Volume 2......Page 11
Preface......Page 14
II Weak and Weak* Topologies......Page 32
III Filters, Ultrafilters. Ordinals......Page 38
IV Exercises......Page 43
I Introduction......Page 44
II Convergence......Page 46
III Series of Independent Random Variables......Page 52
IV Khintchine’s Inequalities......Page 61
V Martingales......Page 66
VI Comments......Page 73
VII Exercises......Page 74
II Schauder Bases: Generalities......Page 77
III Bases and the Structure of Banach Spaces......Page 90
IV Comments......Page 105
V Exercises......Page 107
II Unconditional Convergence......Page 114
III Unconditional Bases......Page 121
IV The Canonical Basis of c0......Page 125
V The James Theorems......Page 127
VI The Gowers Dichotomy Theorem......Page 132
VII Comments......Page 141
VIII Exercises......Page 142
II Definitions. Convergence......Page 148
III The Paul Lévy Symmetry Principle and Applications......Page 160
IV The Contraction Principle......Page 164
V The Kahane Inequalities......Page 169
VI Comments......Page 182
VII Exercises......Page 183
II Complements of Probability......Page 190
III Complements on Banach Spaces......Page 203
IV Type and Cotype of Banach Spaces......Page 208
V Factorization through a Hilbert Space and Kwapie´ n’s Theorem......Page 224
VI Some Applications of the Notions of Type and Cotype......Page 231
VII Comments......Page 234
VIII Exercises......Page 236
I Introduction......Page 241
II p-Summing Operators......Page 242
III Grothendieck’s Theorem......Page 248
IV Some Applications of p-Summing Operators......Page 258
V Sidon Sets......Page 262
VI Comments......Page 289
VII Exercises......Page 291
I Introduction......Page 297
II The Space L1......Page 298
III The Trigonometric System......Page 308
IV The Haar Basis in Lp......Page 315
V Another Proof of Grothendieck’s Theorem......Page 327
VI Comments......Page 336
VII Exercises......Page 346
II Rosenthal’s ℓ1 Theorem......Page 357
III Further Results on Spaces Containing ℓ1......Page 372
IV Comments......Page 381
V Exercises......Page 384
II Banach Algebras......Page 388
III Compact Abelian Groups......Page 395
References......Page 413
Notation Index for Volume 1......Page 444
Author Index for Volume 1......Page 446
Subject Index for Volume 1......Page 450
Notation Index for Volume 2......Page 456
Author Index for Volume 2......Page 457
Subject Index for Volume 2......Page 460
