Handbook of the Geometry of Banach Spaces: Volume 1

Handbook of the Geometry of Banach Spaces 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. Content: Preface Pages v-vi William B. Johnson, Joram Lindenstrauss List of Contributors Page vii Chapter 1 Basic concepts in the geometry of Banach spaces Original Research Article Pages 1-84 William B. Johnson, Joram Lindenstrauss Chapter 2 Positive operators Original Research Article Pages 85-122 Y.A. Abramovich, C.D. Aliprantis Chapter 3 L p spaces Original Research Article Pages 123-159 Dale Alspach, Edward Odell Chapter 4 Convex geometry and functional analysis Original Research Article Pages 161-194 Keith Ball Chapter 5 Λ P -sets in analysis: Results, problems and related aspects Original Research Article Pages 195-232 Jean Bourgain Chapter 6 Martingales and singular integrals in Banach spaces Original Research Article Pages 233-269 Donald L. Burkholder Chapter 7 Approximation properties Original Research Article Pages 271-316 Peter G. Casazza Chapter 8 Local operator theory, random matrices and Banach spaces Original Research Article Pages 317-366 Kenneth R. Davidson, Stanislaw J. Szarek Chapter 9 Applications to Mathematical Finance Original Research Article Pages 367-391 Freddy Delbaen, Walter Schachermayer Chapter 10 Perturbed minimization principles and applications Original Research Article Pages 393-435 Robert Deville, Nassif Ghoussoub Chapter 11 Operator ideals Original Research Article Pages 437-496 Joe Diestel, Hans Jarchow, Albrecht Pietsch Chapter 12 Special Banach lattices and their applications Original Research Article Pages 497-532 S.J. Dilworth Chapter 13 Some aspects of the invariant subspace problem Original Research Article Pages 533-559 P. Enflo, V. Lomonosov Chapter 14 Special bases in function spaces Original Research Article Pages 561-597 T. Figiel, P. Wojtaszczyk Chapter 15 Infinite dimensional convexity Original Research Article Pages 599-670 V.P. Fonf, J. Lindenstrauss, R.R. Phelps Chapter 16 Uniform algebras as Banach spaces Original Research Article Pages 671-706 T.W. Gamelin, S.V. Kislyakov Chapter 17 Euclidean structure in finite dimensional normed spaces Original Research Article Pages 707-779 Apostolos A. Giannopoulos, Vitali D. Milman Chapter 18 Renormings of Banach spaces Original Research Article Pages 781-835 Gilles Godefroy Chapter 19 Finite dimensional subspaces of L p Original Research Article Pages 837-870 William B. Johnson, Gideon Schechtman Chapter 20 Banach spaces and classical harmonic analysis Original Research Article Pages 871-898 S.V. Kislyakov Chapter 21 Aspects of the isometric theory of Banach spaces Original Research Article Pages 899-939 Alexander Koldobsky, Hermann König Chapter 22 Eigenvalues of operators and applications Original Research Article Pages 941-974 Hermann König Author index Pages 975-991 Subject index Pages 993-1005

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.
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. Presents an overview of the various aspects of modern Banach space theory and its applications. This book also discusses 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 goal of this chapter is to reveal the power of descriptive set theory in penetrating the structure of Banach spaces.