Geometric Nonlinear Functional Analysis, Part 1This 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. |
Contents
| 1 | |
Fixed Points | 51 |
Differentiation of Convex Functions | 83 |
The RadonNikodým Property | 99 |
11 | 114 |
21 | 122 |
31 | 144 |
40 | 153 |
Nonlinear Quotient Maps | 261 |
Oscillation of Uniformly Continuous Functions on Unit | 281 |
Oscillation of Uniformly Continuous Functions on Unit | 301 |
Perturbations of Local Isometries | 341 |
Perturbations of Global Isometries | 359 |
Twisted Sums | 373 |
Group Structure on Banach Spaces | 391 |
Appendices | 409 |
48 | 159 |
Lipschitz Classification of Banach Spaces | 169 |
Uniform Embeddings into Hilbert Space | 185 |
Uniform Classification of Spheres | 197 |
Uniform Classification of Banach Spaces | 219 |
53 | 435 |
| 455 | |
| 477 | |
| 481 | |
Other editions - View all
Geometric Nonlinear Functional Analysis, Volume 48, Issue 1 Yoav Benyamini,Joram Lindenstrauss Limited preview - 2000 |
Geometric Nonlinear Functional Analysis, Part 1 Yoav Benyamini,Joram Lindenstrauss No preview available - 2000 |
Common terms and phrases
assume B₁ Banach lattice Banach space basic sequence bounded linear operator Chapter choose closed co-Lipschitz compact complemented subspace construction contain a subspace continuous function converges coordinates Corollary countable defined denote dense E₁ estimate example F₁ finite finite-dimensional subspace follows Fréchet differentiable function f Gâteaux differentiable gives Haar null hence Hilbert space implies inductively inequality integer isometry isomorphic l₁ Lemma Let f Lipschitz constant Lipschitz equivalent Lipschitz function Lipschitz map Lipschitz retract metric space modulus of continuity nonnegative null sets numbers obtain positive definite proof of Theorem Proposition proved quasi-norm quotient map reflexive satisfies scalars separable Banach space space F subspace F subspace isomorphic superreflexive surjective twisted sum ultrafilter ultrapower unconditional basis uniform uniformly continuous uniformly convex uniformly homeomorphic uniformly smooth unit ball unit sphere unit vector basis weakly
Popular passages
Page 9 - This research was supported by the Fund for the Promotion of Research at the Technion and by the Technion VPR Fund - L.
Page 464 - Petrie and JD Randall, Transformation Groups on Manifolds (1984) 83. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (1984) 84.