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. 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 167
Publisher: Cambridge University Press
Year: 2017
Language: English
Pages: 406
Contents......Page 8
Contents of Volume 1......Page 10
Preface......Page 14
II An Inequality of Concentration of Measure......Page 32
III Comparison of Gaussian Vectors......Page 39
IV Dvoretzky’s Theorem......Page 49
V The Lindenstrauss–Tzafriri Theorem......Page 71
VI Comments......Page 76
VII Exercises......Page 77
I Introduction and Definitions......Page 82
II The Grothendieck Reductions......Page 84
III The Counterexamples of Enflo and Davie......Page 90
IV Comments......Page 99
V Exercises......Page 101
II Gaussian Processes......Page 103
III Brownian Motion......Page 107
IV Dudley’s Majoration Theorem......Page 110
V Fernique’s Minoration Theorem for Stationary Processes......Page 116
VI The Elton–Pajor Theorem......Page 126
VII Comments......Page 153
VIII Exercises......Page 154
I Introduction......Page 158
II Structure of Reflexive Subspaces of L1......Page 159
III Examples of Reflexive Subspaces of L1......Page 173
IV Maurey’s Factorization Theorem and Rosenthal’s Theorem......Page 181
V Finite-Dimensional Subspaces of L1......Page 188
VI Comments......Page 207
VII Exercises......Page 211
II Extraction of Quasi-Independent Sets......Page 224
III Sums of Sines and Vectorial Hilbert Transforms......Page 248
IV Minoration of the K-Convexity Constant......Page 254
V Comments......Page 259
VI Exercises......Page 261
I Introduction......Page 265
II Complements on Banach-Valued Variables......Page 266
III The Cas Space......Page 274
IV Applications of the Space Cas......Page 292
V The Bourgain–Milman Theorem......Page 299
VI Comments......Page 313
VII Exercises......Page 318
Appendix A News in the Theory of Infinite-Dimensional Banach Spaces in the Past 20 Years......Page 321
Appendix B An Update on Some Problems in High-Dimensional Convex Geometry and Related Probabilistic Results......Page 328
Appendix C A Few Updates and Pointers......Page 338
Appendix D On the Mesh Condition for Sidon Sets......Page 347
References......Page 355
Notation Index for Volume 2......Page 386
Author Index for Volume 2......Page 387
Subject Index for Volume 2......Page 390
Notation Index for Volume 1......Page 394
Author Index for Volume 1......Page 396
Subject Index for Volume 1......Page 400
