|
|
1 | (98) |
|
|
1 | (9) |
|
|
1 | (1) |
|
1.1.2 Ordered Vector Spaces |
|
|
2 | (4) |
|
1.1.3 Convex Sets and Convex Functions |
|
|
6 | (2) |
|
1.1.4 The Minkowski Functional, Norms and Seminorms |
|
|
8 | (1) |
|
1.1.5 Limit Inferior and Limit Superior of Sequences of Real Numbers |
|
|
9 | (1) |
|
|
10 | (9) |
|
1.2.1 The Notion of Topological Space |
|
|
10 | (2) |
|
|
12 | (1) |
|
|
13 | (1) |
|
1.2.4 Continuous Functions |
|
|
13 | (2) |
|
1.2.5 Semicontinuous Functions |
|
|
15 | (1) |
|
1.2.6 Sequences and Nets in Topological Spaces |
|
|
16 | (1) |
|
1.2.7 Products of Topological Spaces. Tihonov's Theorem |
|
|
17 | (2) |
|
|
19 | (18) |
|
1.3.1 The Notion of Metric Space |
|
|
19 | (2) |
|
1.3.2 Uniformly Continuous, Lipschitz and Holder Functions |
|
|
21 | (3) |
|
1.3.3 The Distance Function |
|
|
24 | (1) |
|
1.3.4 The Pompeiu-Hausdorff Metric |
|
|
25 | (2) |
|
1.3.5 Characterizations of Continuity in the Metric Case |
|
|
27 | (2) |
|
1.3.6 Completeness and Baire Category |
|
|
29 | (1) |
|
1.3.7 Compactness in Metric Spaces |
|
|
30 | (1) |
|
|
31 | (2) |
|
|
33 | (1) |
|
1.3.10 Paracompact Spaces |
|
|
34 | (2) |
|
1.3.11 Partitions of Unity |
|
|
36 | (1) |
|
1.3.12 Sandwich and Approximation Results for Semicontinuous Functions |
|
|
36 | (1) |
|
|
37 | (31) |
|
1.4.1 Topological Vector Spaces |
|
|
38 | (1) |
|
1.4.2 Locally Convex Spaces |
|
|
39 | (4) |
|
|
43 | (1) |
|
1.4.4 The Best Approximation Problem |
|
|
44 | (1) |
|
|
45 | (1) |
|
1.4.6 The Bidual and Reflexivity |
|
|
46 | (1) |
|
1.4.7 Series and Summable Families in Normed Spaces |
|
|
47 | (2) |
|
1.4.8 Inner Product Spaces |
|
|
49 | (3) |
|
1.4.9 Ordered Topological Vector Spaces |
|
|
52 | (1) |
|
1.4.10 Spaces of Continuous Functions |
|
|
53 | (2) |
|
1.4.11 The Stone-Weierstrass Theorem |
|
|
55 | (2) |
|
1.4.12 Compactness in Spaces of Continuous Functions |
|
|
57 | (1) |
|
1.4.13 Extreme Points of Convex Sets |
|
|
58 | (1) |
|
1.4.14 Differentiability of Vector Functions |
|
|
58 | (2) |
|
1.4.15 Some Geometric Properties of Normed Spaces |
|
|
60 | (3) |
|
1.4.16 Quasi-Normed Spaces |
|
|
63 | (5) |
|
1.5 Elements of Measure Theory and Integration |
|
|
68 | (17) |
|
1.5.1 Algebras and a-Algebras |
|
|
68 | (1) |
|
|
69 | (5) |
|
1.5.3 Measurable Functions and Integration |
|
|
74 | (5) |
|
1.5.4 The Radon-Nikodym Theorem |
|
|
79 | (1) |
|
|
80 | (2) |
|
1.5.6 Riesz' Representation Theorem |
|
|
82 | (2) |
|
|
84 | (1) |
|
|
85 | (14) |
|
1.6.1 The Integration of Vector Functions |
|
|
85 | (6) |
|
|
91 | (2) |
|
1.6.3 The Radon-Nikodym Property |
|
|
93 | (6) |
|
2 Basic Facts Concerning Lipschitz Functions |
|
|
99 | (44) |
|
2.1 Lipschitz and Locally Lipschitz Functions |
|
|
99 | (3) |
|
2.2 Lipschitz Properties of Differentiable Functions |
|
|
102 | (14) |
|
2.2.1 Differentiable Functions |
|
|
102 | (5) |
|
2.2.2 Characterizations in Terms of Dini Derivatives |
|
|
107 | (9) |
|
2.3 Algebraic Operations with Lipschitz Functions |
|
|
116 | (4) |
|
2.4 Sequences of Lipschitz Functions |
|
|
120 | (4) |
|
2.5 Gluing Lipschitz Functions Together |
|
|
124 | (4) |
|
2.6 Lipschitz Partitions of Unity |
|
|
128 | (9) |
|
2.6.1 The Locally Lipschitz Partition of Unity |
|
|
129 | (1) |
|
2.6.2 The Lipschitz Partition of Unity |
|
|
130 | (4) |
|
2.6.3 A Proof of Rudin's Lemma |
|
|
134 | (3) |
|
2.7 Applications of Lipschitz Partitions of Unity |
|
|
137 | (5) |
|
2.7.1 A Sandwich-Type Theorem |
|
|
137 | (1) |
|
2.7.2 Selections of Set-Valued Mappings |
|
|
138 | (2) |
|
2.7.3 The Lipschitz Separability of the Space C(T) |
|
|
140 | (2) |
|
2.8 Bibliographic Comments |
|
|
142 | (1) |
|
3 Relations with Other Classes of Functions |
|
|
143 | (68) |
|
3.1 Lipschitz Properties of Convex Functions |
|
|
143 | (35) |
|
|
143 | (1) |
|
3.1.2 Normal Cones in Locally Convex Spaces |
|
|
144 | (2) |
|
3.1.3 Some Properties of Convex Vector-Functions |
|
|
146 | (3) |
|
3.1.4 Continuity Properties of Convex Functions |
|
|
149 | (9) |
|
3.1.5 Further Properties of Convex Vector-Functions |
|
|
158 | (4) |
|
3.1.6 Lipschitz Properties of Convex Vector-Functions |
|
|
162 | (1) |
|
3.1.7 Convex Functions on Locally Convex Spaces |
|
|
162 | (5) |
|
3.1.8 The Order-Lipschitz Property |
|
|
167 | (1) |
|
3.1.9 C-Bounded Functions |
|
|
168 | (4) |
|
3.1.10 Equi-Lipschitz Properties of Families of Continuous Convex Mappings |
|
|
172 | (3) |
|
3.1.11 Convex Functions on Metrizable TVS |
|
|
175 | (3) |
|
3.2 Transforming Continuous Functions into Lipschitz Functions |
|
|
178 | (7) |
|
3.3 Lipschitz Versus Absolutely Continuous Functions |
|
|
185 | (11) |
|
3.3.1 Absolutely Continuous Functions |
|
|
185 | (7) |
|
3.3.2 Another Characterization of Lipschitz Functions |
|
|
192 | (4) |
|
3.4 Differentiability of Lipschitz Functions: Rademacher's Theorem |
|
|
196 | (14) |
|
3.4.1 Rademacher's Theorem and Some Extensions |
|
|
196 | (3) |
|
3.4.2 The Converse of Rademacher's Theorem |
|
|
199 | (2) |
|
3.4.3 Infinite Dimensional Extensions |
|
|
201 | (5) |
|
3.4.4 Metric Measure Spaces |
|
|
206 | (4) |
|
3.5 Bibliographic Comments and Miscellaneous Results |
|
|
210 | (1) |
|
4 Extension Results for Lipschitz Mappings |
|
|
211 | (42) |
|
4.1 McShane Type Theorems |
|
|
211 | (9) |
|
|
211 | (4) |
|
4.1.2 The Extension of Locally Lipschitz Functions |
|
|
215 | (5) |
|
4.2 Extension Results for Lipschitz Vector-Functions |
|
|
220 | (12) |
|
4.2.1 Kirszbraun and Valentine |
|
|
221 | (1) |
|
4.2.2 The Contraction Extension Property and the Intersection of Balls |
|
|
221 | (3) |
|
4.2.3 The Proof of Theorem 4.2.3 |
|
|
224 | (3) |
|
|
227 | (5) |
|
4.3 Semi-Lipschitz Functions on Quasi-Metric Spaces |
|
|
232 | (7) |
|
4.3.1 Quasi-Metric Spaces |
|
|
232 | (3) |
|
4.3.2 Semi-Lipschitz Functions |
|
|
235 | (4) |
|
4.4 Lipschitz Functions with Values in Quasi-Normed Spaces |
|
|
239 | (5) |
|
4.5 Bibliographic Comments and Miscellaneous Results |
|
|
244 | (9) |
|
5 Extension Results for Lipschitz Mappings in Geodesic Spaces |
|
|
253 | (64) |
|
5.1 Some Definitions and Facts in Geodesic Metric Spaces |
|
|
253 | (24) |
|
|
258 | (7) |
|
|
265 | (1) |
|
|
266 | (1) |
|
5.1.4 Convex Combinations |
|
|
267 | (10) |
|
5.2 Kirszbraun and McShane Type Extension Results |
|
|
277 | (9) |
|
5.3 Continuity of Extension Operators |
|
|
286 | (20) |
|
5.3.1 Continuous Selections in Alexandrov Spaces |
|
|
288 | (12) |
|
5.3.2 Nonexpansive Selections in Hyperconvex Metric Spaces |
|
|
300 | (6) |
|
5.4 Dugundji Type Extension Results |
|
|
306 | (8) |
|
5.4.1 Continuous Extensions |
|
|
306 | (1) |
|
5.4.2 Lipschitz Extensions |
|
|
307 | (7) |
|
5.5 Bibliographic Comments and Miscellaneous Results |
|
|
314 | (3) |
|
6 Approximations Involving Lipschitz Functions |
|
|
317 | (18) |
|
6.1 Uniform Approximation via the Stone--Weierstrass Theorem |
|
|
317 | (1) |
|
6.2 Approximation via Locally Lipschitz Partitions of Unity |
|
|
318 | (2) |
|
6.3 Approximation via Lipschitz Extensions |
|
|
320 | (2) |
|
6.4 Baire's Theorem on the Approximation of Semicontinuous Functions |
|
|
322 | (6) |
|
|
322 | (2) |
|
|
324 | (4) |
|
6.5 The Homotopy of Lipschitz Functions |
|
|
328 | (3) |
|
|
331 | (1) |
|
6.7 Bibliographic Comments and Miscellaneous Results |
|
|
332 | (3) |
|
7 Lipschitz Isomorphisms of Metric Spaces |
|
|
335 | (30) |
|
|
335 | (1) |
|
7.2 Schauder Bases in Banach Spaces |
|
|
336 | (8) |
|
7.3 Separable Metric Spaces Embed in c0 |
|
|
344 | (7) |
|
7.4 A Characterization of the Completeness of Normed Spaces in Terms of bi-Lipschitz Functions |
|
|
351 | (5) |
|
7.5 Bibliographic Comments and Miscellaneous Results |
|
|
356 | (9) |
|
8 Banach Spaces of Lipschitz Functions |
|
|
365 | (192) |
|
8.1 The Basic Metric and Lipschitz Spaces |
|
|
365 | (8) |
|
8.2 Lipschitz Free Banach Spaces |
|
|
373 | (24) |
|
8.2.1 The Arens-Eells Space |
|
|
374 | (4) |
|
8.2.2 Lipschitz Free Banach Spaces Generated by Evaluation Functionals |
|
|
378 | (11) |
|
|
389 | (1) |
|
8.2.4 A Result of Dixmier and Ng |
|
|
390 | (4) |
|
8.2.5 The Lipschitz Conjugate Operator |
|
|
394 | (3) |
|
8.3 Little Lipschitz Functions |
|
|
397 | (6) |
|
|
398 | (1) |
|
8.3.2 Properties of the Space lip0(X) |
|
|
399 | (4) |
|
8.4 The Kantorovich--Rubinstein Metric |
|
|
403 | (33) |
|
8.4.1 A Sesquilinear Integral |
|
|
403 | (5) |
|
8.4.2 Lipschitz Functions |
|
|
408 | (1) |
|
8.4.3 The Kantorovich--Rubinstein Norm |
|
|
409 | (5) |
|
8.4.4 The Weak* Topology on cabv(X, t) |
|
|
414 | (3) |
|
8.4.5 The Modified Kantorovich--Rubinstein Norm |
|
|
417 | (8) |
|
8.4.6 Infinite Dimensional Extensions Do Not Work |
|
|
425 | (2) |
|
8.4.7 The Mass Transfer Problem |
|
|
427 | (4) |
|
8.4.8 The Kantorovich--Rubinstein Duality |
|
|
431 | (5) |
|
8.5 Hanin's Norm and Applications |
|
|
436 | (22) |
|
8.5.1 Definition and First Properties |
|
|
437 | (5) |
|
8.5.2 The Density of Measures with Finite Support |
|
|
442 | (3) |
|
8.5.3 The Dual of (cabv(X), || · || H) |
|
|
445 | (2) |
|
8.5.4 The Weak* Convergence of Borel Measures |
|
|
447 | (1) |
|
|
448 | (7) |
|
8.5.6 Hanin's Norm in the Hilbert Case |
|
|
455 | (3) |
|
8.6 Compactness Properties of Lipschitz Operators |
|
|
458 | (9) |
|
8.6.1 Compact and Weakly Compact Linear Operators |
|
|
459 | (2) |
|
8.6.2 Lipschitz Compact and Weakly Compact Operators |
|
|
461 | (3) |
|
8.6.3 The Analogs of the Schauder and Gantmacher Theorems for Lipschitz Operators |
|
|
464 | (3) |
|
8.7 Composition Operators |
|
|
467 | (39) |
|
8.7.1 Definition and Basic Properties |
|
|
468 | (5) |
|
8.7.2 Compactness of the Composition Operators |
|
|
473 | (6) |
|
8.7.3 Weakly Compact Composition Operators |
|
|
479 | (9) |
|
8.7.4 Composition Operators on Spaces of Vector Lipschitz Functions |
|
|
488 | (15) |
|
|
503 | (2) |
|
8.7.6 The Nemytskii Superposition Operator |
|
|
505 | (1) |
|
8.8 The Bishop--Phelps--Bollobas Property |
|
|
506 | (35) |
|
8.8.1 The Bishop--Phelps--Bollobas Theorem in Banach Spaces |
|
|
507 | (4) |
|
8.8.2 The Bishop--Phelps Theorem for Weak*-Closed Convex Subsets of the Dual Space |
|
|
511 | (1) |
|
8.8.3 The Bishop--Phelps Theorem Fails in the Complex Case and in Locally Convex Spaces |
|
|
511 | (1) |
|
8.8.4 Norm-Attaining Operators |
|
|
512 | (6) |
|
8.8.5 Support Functionals in Spaces of Lipschitz Functions |
|
|
518 | (10) |
|
8.8.6 Norm-Attaining Seminorms |
|
|
528 | (3) |
|
8.8.7 The Lip-BPB Property |
|
|
531 | (6) |
|
8.8.8 Asymptotically Uniformly Smooth Banach Spaces and Norm-Attaining Lipschitz Operators |
|
|
537 | (4) |
|
8.9 Applications to Best Approximation in Metric Spaces |
|
|
541 | (16) |
|
8.9.1 Best Approximation in Arbitrary Metric Spaces |
|
|
541 | (7) |
|
8.9.2 Lipschitz Duals of Metric Linear Spaces and Best Approximation |
|
|
548 | (9) |
References |
|
557 | (28) |
Index |
|
585 | |