**Colloquium Publications**

Volume: 48;
2000;
488 pp;
Hardcover

MSC: Primary 46;
Secondary 22; 28; 47; 52; 54

Print ISBN: 978-0-8218-0835-1

Product Code: COLL/48

List Price: $79.00

Individual Member Price: $63.20

**Electronic ISBN: 978-1-4704-3194-5
Product Code: COLL/48.E**

List Price: $79.00

Individual Member Price: $63.20

# Geometric Nonlinear Functional Analysis: Volume 1

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*Yoav Benyamini; Joram Lindenstrauss*

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.

#### Table of Contents

# Table of Contents

## Geometric Nonlinear Functional Analysis: Volume 1

- Cover Cover11
- Title page v6
- Dedication vii8
- Contents ix10
- Introduction 114
- Retractions, extensions and selections 1124
- Retractions, extensions and selections (special topics) 3548
- Fixed points 6174
- Differentiation of convex functions 8396
- The Radon-Nikodým property 99112
- Negligible sets and Gâteaux differentiability 125138
- Lipschitz classification of Banach spaces 169182
- Uniform embeddings into Hilbert space 185198
- Uniform classification of spheres 197210
- Uniform classification of Banach spaces 219232
- Nonlinear quotient maps 261274
- Oscillation of uniformly continuous functions on unit spheres of finite-dimensional subspaces 281294
- Oscillation of uniformly continuous functions on unit spheres of infinite-dimensional subspaces 301314
- Perturbations of local isometries 341354
- Perturbations of global isometries 359372
- Twisted sums 373386
- Group structure on Banach spaces 391404
- Appendices 409422
- Bibliography 455468
- Index 481494
- Back Cover Back Cover1503

#### 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