Handbook of the geometry of Banach spaces

J. Lindenstrauss, W.B. Johnson0444828427, 9780444828422, 9780080532806, 0444513051

The Handbook presents an overview of most aspects of modern Banach space theory and its applications. The up-to-date surveys, authored by leading research workers in the area, are written to be accessible to a wide audience. In addition to presenting the state of the art of Banach space theory, the surveys discuss the relation of the subject with such areas as harmonic analysis, complex analysis, classical convexity, probability theory, operator theory, combinatorics, logic, geometric measure theory, and partial differential equations.
The Handbook begins with a chapter on basic concepts in Banach space theory which contains all the background needed for reading any other chapter in the Handbook. Each of the twenty one articles in this volume after the basic concepts chapter is devoted to one specific direction of Banach space theory or its applications. Each article contains a motivated introduction as well as an exposition of the main results, methods, and open problems in its specific direction. Most have an extensive bibliography. Many articles contain new proofs of known results as well as expositions of proofs which are hard to locate in the literature or are only outlined in the original research papers.
As well as being valuable to experienced researchers in Banach space theory, the Handbook should be an outstanding source for inspiration and information to graduate students and beginning researchers. The Handbook will be useful for mathematicians who want to get an idea of the various developments in Banach space theory.

---

Table of contents :
Contents of Volume 1……Page 1
Contents of Volume 2……Page 3
Preface……Page 5
List of Contributors……Page 7
1. Basic concepts in the geometry of Banach spaces (W.B. Johnson and J. Lindenstrauss) ……Page 8
2. Positive operators (Y.A. Abramovitch and C.D. Aliprantis) ……Page 92
3. $L_p$ spaces (D. Alspach and E. Odell) ……Page 130
4. Convex geometry and functional analysis (K. Ball) ……Page 168
5. $Lambda_P$-sets in analysis: Results, problems and related aspects (J. Bourgain) ……Page 202
6. Martingales and singular integrals in Banach spaces (D.L. Burkholder) ……Page 240
7. Approximation properties (P.G. Casazza) ……Page 278
8. Local operator theory, random matrices and Banach spaces (K.R. Davidson and S.J. Szarek) ……Page 324
9. Applications to mathematical finance (F. Delbaen and W. Schachermayer) ……Page 374
10. Perturbed minimization principles and applications (R. Deville and N. Ghoussoub) ……Page 400
11. Operator ideals (J. Diestel, H. Jarchow and A. Pietsch) ……Page 444
12. Special Banach lattices and their applications (S.J. Dilworth) ……Page 504
13. Some aspects of the invariant subspace problem (P. Enflo and V. Lomonosov) ……Page 540
14. Special bases in function spaces (T. Figiel and P. Wojtaszczyk) ……Page 568
15. Infinite dimensional convexity (V.P. Fonf, J. Lindenstrauss and R.R. Phelps) ……Page 606
16. Uniform algebras as Banach spaces (T.W. Gamelin and S.V. Kislyakov) ……Page 678
17. Euclidean structure in finite dimensional normal spaces (A.A. Giannopoulos and V.D. Milman) ……Page 714
18. Renormings of Banach spaces (G. Godefroy) ……Page 788
19. Finite dimensional subspaces of $L_p$ (W.B. Johnson and G. Schechtman) ……Page 844
20. Banach spaces and classical harmonic analysis (S.V. Kislyakov) ……Page 878
21. Aspects of the isometric theory of Banach spaces (A. Koldobsky and H. K”onig) ……Page 906
22. Eigenvalues of operators and applications (H. K”onig) ……Page 948
Author Index ……Page 982
Subject Index ……Page 1000