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Geometric Nonlinear Functional Analysis: Volume 1
Yoav Benyamini, Technion--Israel Institute of Technology, Haifa, Israel, and Joram Lindenstrauss, Hebrew University, Jerusalem, Israel
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Colloquium Publications
2000; 488 pp; hardcover
Volume: 48
ISBN-10: 0-8218-0835-4
ISBN-13: 978-0-8218-0835-1
List Price: US$75
Member Price: US$60
Order Code: COLL/48
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The book presents a systematic and unified study of geometric nonlinear functional analysis. This area has its classical roots in the beginning of the twentieth century and is now a very active research area, having close connections to geometric measure theory, probability, classical analysis, combinatorics, and Banach space theory.

The main theme of the book is the study of uniformly continuous and Lipschitz functions between Banach spaces (e.g., differentiability, stability, approximation, existence of extensions, fixed points, etc.). This study leads naturally also to the classification of Banach spaces and of their important subsets (mainly spheres) in the uniform and Lipschitz categories.

Many recent rather deep theorems and delicate examples are included with complete and detailed proofs. Challenging open problems are described and explained, and promising new research directions are indicated.

Readership

Graduate students and research mathematicians interested in functional analysis; theoretical computer scientists.

Reviews

"Important monograph, written by leading specialists in the field ... notes and remarks ... contain interesting historical notes and information about a vast amount of related results ... Challenging open problems are described and explained. The vast majority of the material appears for the first time in book form, and many quite recent deep results are proved."

-- European Mathematical Society Newsletter

"Much ... is explained in this book in a splendid and fascinating way. The considerable amount of mathematics is divided into seventeen chapters which are essentially independent. Three or four of them chosen ad-libitum would provide an excellent basis for a graduate course ... researchers will enjoy this invitation to the deepest parts of functional analysis."

-- Mathematical Reviews

"Without any doubt, this is one of the great books on nonlinear analysis which will certainly become a standard reference. It is not only a must for every math library all over the world, but also for all researchers interested in functional analysis, operator theory, geometry of Banach spaces, and nonlinear problems."

-- Zentralblatt MATH

Table of Contents

  • Introduction
  • Retractions, extensions and selections
  • Retractions, extensions and selections (special topics)
  • Fixed points
  • Differentiation of convex functions
  • The Radon-Nikodým property
  • Negligible sets and Gâteaux differentiability
  • Lipschitz classification of Banach spaces
  • Uniform embeddings into Hilbert space
  • Uniform classification of spheres
  • Uniform classification of Banach spaces
  • Nonlinear quotient maps
  • Oscillation of uniformly continuous functions on unit spheres of finite-dimensional subspaces
  • Oscillation of uniformly continuous functions on unit spheres of infinite-dimensional subspaces
  • Perturbations of local isometries
  • Perturbations of global isometries
  • Twisted sums
  • Group structure on Banach spaces
  • Appendices
  • Bibliography
  • Index
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