| Description |
1 online resource (ix, 553 pages) : illustrations |
| Series |
Encyclopedia of mathematics and its applications ; v. 70 |
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Encyclopedia of mathematics and its applications ; v. 70.
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| Contents |
Cover; Half-title; Title; Copyright; Contents; Preface; Introduction; 0 Preliminaries; 0.1 Banach spaces and operators; 0.2 Finite dimensional spaces and operators; 0.3 Classical sequence spaces; 0.4 Classical function spaces; 0.5 Lorentz spaces; 0.6 Interpolation methods; 0.7 Summation operators; 0.8 Finite representability and ultrapowers; 0.9 Extreme points; 0.10 Various tools; 1 Ideal norms and operator ideals; 1.1 Ideal norms; 1.2 Operator ideals; 1.3 Classes of Banach spaces; 2 Ideal norms associated with matrices; 2.1 Matrices; 2.2 Parseval ideal norms and 2-summing operators |
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2.3 Kwapien ideal norms and Hilbertian operators2.4 Ideal norms associated with Hilbert matrices; 3 Ideal norms associated with orthonormal systems; 3.1 Orthonormal systems; 3.2 Khintchine constants; 3.3 Riemann ideal norms; 3.4 Dirichlet ideal norms; 3.5 Orthonormal systems with special properties; 3.6 Tensor products of orthonormal systems; 3.7 Type and cotype ideal norms; 3.8 Characters on compact Abelian groups; 3.9 Discrete orthonormal systems; 3.10 Some universal ideal norms; 3.11 Parseval ideal norms; 4 Rademacher and Gauss ideal norms; 4.1 Rademacher functions |
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4.2 Rademacher type and cotype ideal norms4.3 Operators of Rademacher type; 4.4 B-convexity; 4.5 Operators of Rademacher cotype; 4.6 MP-convexity; 4.7 Gaussian random variables; 4.8 Gauss versus Rademacher; 4.9 Gauss type and cotype ideal norms; 4.10 Operators of Gauss type and cotype; 4.11 Sidon constants; 4.12 The Dirichlet ideal norms 6(#n, ftn) and 6(Sn, Sn); 4.13 Inequalities between 6(Rn, Rn) and g(R,n,J n); 4.14 The vector-valued Rademacher projection; 4.15 Parseval ideal norms and 7-summing operators; 4.16 The Maurey-Pisier theorem; 5 Trigonometric ideal norms |
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5.1 Trigonometric functions5.2 The Dirichlet ideal norms 6(£n, £n); 5.3 Hilbert matrices and trigonometric systems; 5.4 The vector-valued Hilbert transform; 5.5 Fourier type and cotype ideal norms; 5.6 Operators of Fourier type; 5.7 Operators of Fourier cotype; 5.8 The vector-valued Fourier transform; 5.9 Fourier versus Gauss and Rademacher; 6 Walsh ideal norms; 6.1 Walsh functions; 6.2 Walsh type and cotype ideal norms; 6.3 Operators of Walsh type; 6.4 Walsh versus Rademacher; 6.5 Walsh versus Fourier; 7 Haar ideal norms; 7.1 Martingales; 7.2 Dyadic martingales; 7.3 Haar functions |
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7.4 Haar type and cotype ideal norms7.5 Operators of Haar type; 7.6 Super weakly compact operators; 7.7 Martingale type ideal norms; 7.8 J-convexity; 7.9 Uniform g-convexity and uniform p-smoothness; 7.10 Uniform convexity and uniform smoothness; 8 Unconditionality; 8.1 Unconditional Riemann ideal norms; 8.2 Unconditional Dirichlet ideal norms; 8.3 Random unconditionality; 8.4 Fourier unconditionality; 8.5 Haar unconditionality/UMD; 8.6 Random Haar unconditionality; 8.7 The Dirichlet ideal norms (Wn, Wn); 8.8 The Burkholder-Bourgain theorem; 9 Miscellaneous; 9.1 Interpolation |
| Summary |
Orthonormal Systems and Banach Space Geometry describes the interplay between orthonormal expansions and Banach space geometry. Using harmonic analysis as a starting platform, classical inequalities and special functions are used to study orthonormal systems leading to an understanding of the advantages of systems consisting of characters on compact Abelian groups. Probabilistic concepts such as random variables and martingales are employed and Ramsey's theorem is used to study the theory of super-reflexivity. The text yields a detailed insight into concepts including type and co-type of Banach spaces, B-convexity, super-reflexivity, the vector-valued Fourier transform, the vector-valued Hilbert transform and the unconditionality property for martingale differences (UMD). A long list of unsolved problems is included as a starting point for research. This book should be accessible to graduate students and researchers with some basic knowledge of Banach space theory, real analysis, probability and algebra |
| Bibliography |
Includes bibliographical references (pages 523-545) and index |
| Notes |
Print version record |
| Subject |
Banach spaces.
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Mathematical analysis.
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MATHEMATICS -- Calculus.
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MATHEMATICS -- Mathematical Analysis.
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Banach spaces
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Mathematical analysis
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| Form |
Electronic book
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| Author |
Wenzel, Jörg.
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| LC no. |
98227912 |
| ISBN |
9781107089105 |
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1107089107 |
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