Handbook of the Geometry of Banach Spaces: Vol 1

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:
Basic concepts in the geometry of Banach spaces (W.B. Johnson, J. Lindenstrauss). Positive operators (Y.A. Abramovitch, C.D. Aliprantis). Lp spaces (D. Alspach, E. Odell). Convex geometry and functional analysis (K. Ball). A p-sets in analysis: Results, problems and related aspects (J. Bourgain). Martingales and singular integrals in Banach spaces (D.L. Burkholder). Approximation properties (P.G. Casazza). Local operator theory, random matrices and Banach spaces (K.R. Davidson, S.J. Szarek). Applications to mathematical finance (F. Delbaen). Perturbed minimization principles and applications (R. Deville, N. Ghoussoub). Operator ideals (J. Diestel, H. Jarchow, A. Pietsch). Special Banach lattices and their applications(S.J. Dilworth). Some aspects of the invariant subspace problem (P. Enflo,V. Lomonosov). Special bases in function spaces (T. Figel, P. Wojtaszczyk). Infinite dimensional convexity (V. Fonf, J. Lindenstrauss, R.R. Phelps). Uniform algebras as Banach spaces (T.W. Gamelin, S.V. Kisliakov). Euclidean structure in finite dimensional normed spaces (A.A. Giannopoulos, V.D. Milman). Renormings of Banach spaces (G. Godefroy). Finite dimensional subspaces of Lp (W.B. Johnson, G. Schechtman). Banach spaces and classical harmonic analysis (S.V. Kisliakov). Aspects of the isometric theory of Banach spaces (A. Koldobsky, H. Konig). Eigenvalues of operators and applications (H. Konig).