Introduction to Banach spaces : analysis and probability. Volume 1 /.
Daniel Li, Hervé Queffélec ; translated from the French by Danièle Gibbons and Greg Gibbons.
Bok Engelsk 2017 · Electronic books.
| Originaltittel | [ Introduction à l’étude des espaces de Banach. .] English
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| Medvirkende | |
| Omfang | 1 online resource (ix, 431 pages) : : digital, PDF file(s).
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| Utgave | 1st ed.
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| Opplysninger | 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 -- Series information -- Title page -- Copyright information -- Dedication -- Contents of Volume 1 -- Contents of Volume 2 -- Preface -- Preliminary Chapter Weak and Weak* Topologies. Filters, Ultrafilters. Ordinals -- I Introduction -- II Weak and Weak* Topologies -- II.1 The Eberlein-Šmulian Theorem -- II.2 The Krein-Milman Theorem -- III Filters, Ultrafilters. Ordinals -- III.1 Filters and Ultrafilters -- III.2 Limit along a Filter -- III.3 Filtering Families -- III.4 Ordinals -- IV Exercises -- 1 Fundamental Notions of Probability -- I Introduction -- II Convergence -- II.1 Convergence in Probability and Convergence Almost Sure -- II.2 Convergence in Distribution in L[sup(0)] -- III Series of Independent Random Variables -- III.1 Inequalities of Kolmogorov and Paley-Zygmund -- III.2 The Paul Lévy Theorem -- IV Khintchine's Inequalities -- IV.1 Statement of the Inequalities -- IV.2 Applications of Khintchine's Inequalities -- V Martingales -- V.1 Conditional Expectation -- V.2 Martingales -- VI Comments -- VII Exercises -- 2 Bases in Banach Spaces -- I Introduction -- II Schauder Bases: Generalities -- II.1 Definition. Associated Projections -- II.2 Universality of C([0,1]) for Separable Spaces -- III Bases and the Structure of Banach Spaces -- III.1 Bases and Isomorphisms -- III.2 Block Bases -- III.3 Bases and Duality -- IV Comments -- V Exercises -- 3 Unconditional Convergence -- I Introduction -- II Unconditional Convergence -- III Unconditional Bases -- IV The Canonical Basis of c[sub(0)] -- V The James Theorems -- VI The Gowers Dichotomy Theorem -- VII Comments -- VIII Exercises -- 4 Banach Space Valued Random Variables -- I Introduction -- II Definitions. Convergence -- II.1 The Space L[sup(0)](E) -- II.2 Convergence in Probability in L[sup(0)](E) -- II.3 Almost Sure Convergence in L[sup(0)](E).. - 7 Some Properties of L[sup(p)]-Spaces -- I Introduction -- II The Space L[sup(1)] -- II.1 The Dunford-Pettis Theorem -- II.2 Unconditionality -- III The Trigonometric System -- III.1 The Riesz Projection -- III.2 The Marcinkiewicz Theorem and the Marcel Riesz Theorem -- III.3 Conditionality of the Trigonometric System -- IV The Haar Basis in L[sup(p)] -- IV.1 The Riesz-Thorin Interpolation Theorem -- IV.2 The Martingale Square Function -- IV.3 Applications to Unconditional Bases -- IV.4 The Extremal Character of the Haar Basis -- V Another Proof of Grothendieck's Theorem -- V.1 The Spaces H[sup(p)] -- V.2 An Alternative Proof of Grothendieck's Theorem -- VI Comments -- VII Exercises -- 8 The Space [ell[sub(1)]] -- I Introduction -- II Rosenthal's [ell[sub(1)]] Theorem -- II.1 Rosenthal-Bourgain-Fremlin-Talagrand -- II.2 Proof of Rosenthal's [ell[sub(1)] Theorem in the Real Case -- II.3 Proof of Rosenthal's [ell[sub(1)] Theorem in the Complex Case -- III Further Results on Spaces Containing [ell[sub(1)]] -- III.1 The Odell-Rosenthal Theorem -- III.2 Duals of Spaces That Do Not Contain [ell[sub(1)] -- IV Comments -- V Exercises -- Annex Banach Algebras. Compact Abelian Groups -- I Introduction -- II Banach Algebras -- II.1 Invertible Elements and Maximal Ideals -- II.2 Spectrum of an Element -- II.3 Spectral Radius -- II.4 Characters of a Commutative Algebra -- II.5 Involutive Banach Algebras -- III Compact Abelian Groups -- III.1 Haar Measure -- III.2 Convolution -- III.3 The Dual Group -- III.4 The Fourier Transform -- III.5 Approximate Identities -- References -- Notation Index for Volume 1 -- Author Index for Volume 1 -- Subject Index for Volume 1 -- Notation Index for Volume 2 -- Author Index for Volume 2 -- Subject Index for Volume 2.. - II.4 Convergence in Distribution in L[sup(0)](E) -- II.5 Spaces L[sup(p)](Omega,A,P -- E) -- III The Paul Lévy Symmetry Principle and Applications -- III.1 Symmetry Principle -- III.2 Applications -- IV The Contraction Principle -- IV.1 Qualitative Version -- IV.2 Quantitative Version -- V The Kahane Inequalities -- V.1 Differences with the Scalar Case -- V.2 The Kahane Inequalities -- V.3 Walsh Functions and Hypercontractivity -- VI Comments -- VII Exercises -- 5 Type and Cotype of Banach Spaces. Factorization through a Hilbert Space -- I Introduction -- II Complements of Probability -- II.1 Gaussian Vectors -- II.2 p-Stable Variables -- III Complements on Banach Spaces -- III.1 Local Reflexivity -- III.2 Ultraproducts -- IV Type and Cotype of Banach Spaces -- IV.1 Introduction -- IV.2 Definitions and First Properties -- IV.3 Examples -- IV.4 Complements -- V Factorization through a Hilbert Space and Kwapie[acute(n)]'s Theorem -- V.1 Definition and Local Character of the Factorization -- V.2 Subordination Criterion. Factorization Theorem -- V.3 Kwapie[acute(n)]'s Theorem -- VI Some Applications of the Notions of Type and Cotype -- VII Comments -- VIII Exercises -- 6 p-Summing Operators. Applications -- I Introduction -- II p-Summing Operators -- II.1 Definition -- II.2 The Pietsch Factorization Theorem -- III Grothendieck's Theorem -- III.1 Grothendieck's Inequality -- III.2 Grothendieck's Theorem -- III.3 The Dual Form of Grothendieck's Theorem -- IV Some Applications of p-Summing Operators -- IV.1 The Dvoretzky-Rogers Theorem -- IV.2 2-Summing Norm of the Identity in Finite-Dimensional Spaces -- IV.3 Unconditional Bases of [ell[sub(1)]] and c[sub(0)] -- V Sidon Sets -- V.1 Definitions -- V.2 Examples -- V.3 Smallness Properties of Sidon Sets -- V.4 Sidon Sets and Spaces of Cotype 2 -- VI Comments -- VII Exercises.. - 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.
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| Emner | |
| Sjanger | |
| Dewey | |
| ISBN | 1-108-29815-X. - 1-316-67576-9
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