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Format: 1 online resource (xxx, 374 pages) : , digital, PDF file(s).

ISBN: 1-108-30008-1 , 1-108-29816-8 , 1-316-67739-7

Series Statement: Cambridge studies in advanced mathematics ; 167

Uniform Title: Introduction à l’étude des espaces de Banach.

Content: 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.

Note: Originally published in French as Introduction à l'étude des espaces de Banach by Société Mathématique de France, 2004. , Title from publisher's bibliographic system (viewed on 10 Nov 2017). , Cover -- Half-title -- Seires information -- Title page -- Copyright information -- Dedication -- Contents of Volume 2 -- Contents of Volume 1 -- Preface -- 1 Euclidean Sections -- I Introduction -- II An Inequality of Concentration of Measure -- II.1 Asymptotic Behavior of Gaussian Variables -- II.2 The Maurey-Pisier Inequality -- III Comparison of Gaussian Vectors -- III.1 Statement of the Problem -- III.2 The Comparison Theorem. Applications -- III.3 The Slepian-Gordon Theorem -- IV Dvoretzky's Theorem -- IV.1 Preliminary Remarks -- IV.2 The Dvoretzky-Rogers and Lewis Lemmas -- IV.3 The Proof of Dvoretzky's Theorem -- IV.4 Examples -- IV.5 The Milman-Schechtman Theorem -- V The Lindenstrauss-Tzafriri Theorem -- VI Comments -- VII Exercises -- 2 Separable Banach Spaces without the Approximation Property -- I Introduction and Definitions -- II The Grothendieck Reductions -- III The Counterexamples of Enflo and Davie -- IV Comments -- V Exercises -- 3 Gaussian Processes -- I Introduction -- II Gaussian Processes -- II.1 Definitions and Notation -- II.2 The Marcus-Shepp Theorem -- III Brownian Motion -- III.1 Introduction -- III.2 The C[sup(as)] Model of Brownian Motion -- IV Dudley's Majoration Theorem -- IV.1 The Entropy Integral -- IV.2 The Dudley Majoration Theorem -- IV.3 Absence of a Converse to Dudley's Theorem -- V Fernique's Minoration Theorem for Stationary Processes -- V.1 Processes Indexed by a Cantor Tree -- V.2 Minoration of Stationary Gaussian Processes -- V.3 An Equivalent Form of the Entropy Integral -- VI The Elton-Pajor Theorem -- VI.1 Combinatorial Preliminaries -- VI.2 Volume Inequalities -- VI.3 Real Case -- VI.4 Complex Case -- VII Comments -- VIII Exercises -- 4 Reflexive Subspaces of L[sup(1)] -- I Introduction -- II Structure of Reflexive Subspaces of L[sup(1)] -- II.1 Reflexive Subspaces and Convergence in Measure. , II.2 Local Structure of Reflexive Subspaces of L[sup(1)] -- III Examples of Reflexive Subspaces of L[sup(1)] -- III.1 Stable Variables -- III.2 Omega(q) Sets -- IV Maurey's Factorization Theorem and Rosenthal's Theorem -- V Finite-Dimensional Subspaces of L[sup(1)] -- V.1 Statement of the Result -- V.2 K-Convexity -- V.3 An Auxiliary Result -- V.4 Proof of Talagrand's Theorem -- VI Comments -- VII Exercises -- 5 The Method of Selectors. Examples of Its Use -- I Introduction -- II Extraction of Quasi-Independent Sets -- II.1 Quasi-Independent sets -- II.2 Characterization of Sidon Sets -- II.3 Proof of the Sufficient Condition -- II.4 Applications -- III Sums of Sines and Vectorial Hilbert Transforms -- III.1 Introduction -- III.2 Sums of Sines -- IV Minoration of the K-Convexity Constant -- V Comments -- VI Exercises -- 6 The Pisier Space of Almost Surely Continuous Functions. Applications -- I Introduction -- II Complements on Banach-Valued Variables -- II.1 The Itô-Nisio Theorem -- II.2 An Almost Sure ``Tauberian'' Theorem -- III The C[sup(as)] Space -- III.1 Equivalent Definitions -- III.2 The Marcus-Pisier Theorem -- III.3 Duality Between C[sup(as)] and M[sub(2,[Psi[sub(2)]])] -- IV Applications of the Space C[sup(as)] -- IV.1 Characterization of Sidon Sets -- IV.2 The Katznelson Dichotomy Problem -- V The Bourgain-Milman Theorem -- V.1 Banach and Arithmetic Diameters -- V.2 Proof of Theorem V.4 -- V.3 Proof of Theorem V.5 -- VI Comments -- VII Exercises -- Appendix A News in the Theory of Infinite-Dimensional Banach Spaces in the Past 20 Years -- Appendix B An Update on Some Problems in High-Dimensional Convex Geometry and Related Probabilistic Results -- Appendix C A Few Updates and Pointers -- Appendix D On the Mesh Condition for Sidon Sets -- References -- Notation Index for Volume 2 -- Author Index for Volume 2 -- Subject Index for Volume 2. , Notation Index for Volume 1 -- Author Index for Volume 1 -- Subject Index for Volume 1.

Additional Edition: ISBN 1-107-16262-9

Language: English