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El. knyga: Lipschitz Functions

  • Formatas: EPUB+DRM
  • Serija: Lecture Notes in Mathematics 2241
  • Išleidimo metai: 23-May-2019
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030164898
  • Formatas: EPUB+DRM
  • Serija: Lecture Notes in Mathematics 2241
  • Išleidimo metai: 23-May-2019
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030164898

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The aim of this book is to present various facets of the theory and applications of Lipschitz functions, starting with classical and culminating with some recent results. Among the included topics we mention: characterizations of Lipschitz functions and relations with other classes of functions, extension results for Lipschitz functions and Lipschitz partitions of unity, Lipschitz free Banach spaces and their applications, compactness properties of Lipschitz operators, Bishop-Phelps type results for Lipschitz functionals, applications to best approximation in metric and in metric linear spaces, Kantorovich-Rubinstein norm and applications to duality in the optimal transport problem, Lipschitz mappings on geodesic spaces.

The prerequisites are basic results in real analysis, functional analysis, measure theory (including vector measures) and topology, which, for reader's convenience, are surveyed in the first chapter of the book.

Recenzijos

"This is an excellent source for all aspects of the notion of Lipschitz continuity and will undoubtedly become a standard reference. (M. Kunzinger, Monatshefte für Mathematik, Vol. 196 (1), 2021)

The book is accessible to graduate students, but it also contains recent results of interest to researchers in various domains as metric geometry, mathematical analysis, and functional analysis. My opinion this book will be of interest to everyone whose domain of interest is mathematical analysis and its applications. (Andrey Zahariev, zbMATH 1431.26002, 2020) This book provides a very large amount of interesting information. it is very useful that they are brought to readers' attention. (Mikhail Ostrovskii, Mathematical Reviews, December, 2019)

1 Prerequisites
1(98)
1.1 Ordered Sets
1(9)
1.1.1 Preorder and Order
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)
1.2 Topological Spaces
10(9)
1.2.1 The Notion of Topological Space
10(2)
1.2.2 Separation Axioms
12(1)
1.2.3 Compactness
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)
1.3 Metric Spaces
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)
1.3.8 Equivalent Metrics
31(2)
1.3.9 Ultrametric Spaces
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)
1.4 Functional Analysis
37(31)
1.4.1 Topological Vector Spaces
38(1)
1.4.2 Locally Convex Spaces
39(4)
1.4.3 Normed Spaces
43(1)
1.4.4 The Best Approximation Problem
44(1)
1.4.5 Weak Topologies
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)
1.5.2 Measures
69(5)
1.5.3 Measurable Functions and Integration
74(5)
1.5.4 The Radon-Nikodym Theorem
79(1)
1.5.5 Borel Measures
80(2)
1.5.6 Riesz' Representation Theorem
82(2)
1.5.7 Radon Measures
84(1)
1.6 Vector Measures
85(14)
1.6.1 The Integration of Vector Functions
85(6)
1.6.2 Vector Measures
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)
3.1.1 Overview
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)
4.1.1 McShane's Theorem
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)
4.2.4 Flett's Theorem
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)
5.1.1 Alexandrov Spaces
258(7)
5.1.2 Hyperconvex Spaces
265(1)
5.1.3 Hyperbolic Spaces
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)
6.4.1 Baire's Function
322(2)
6.4.2 Baire's Theorem
324(4)
6.5 The Homotopy of Lipschitz Functions
328(3)
6.6 Lipschitz Manifolds
331(1)
6.7 Bibliographic Comments and Miscellaneous Results
332(3)
7 Lipschitz Isomorphisms of Metric Spaces
335(30)
7.1 Introduction
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)
8.2.3 Pestov's Approach
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)
8.3.1 De Leeuw's Map
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)
8.5.5 Double Duality
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)
8.7.5 The Arens Product
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
Stefan Cobzas is Emeritus Professor at the Babes-Bolyai University, Department of Mathematics, Cluj-Napoca, Romania. He graduated from the same university in 1968 and obtained a Ph.D. in 1979. His scientific interests concern mainly applied functional analysis -- optimization and best approximation in Banach spaces. In the last years he worked on some problems in asymmetric functional analysis and published several papers and a book (in the series Frontiers in Mathematics, Birkhauser-Springer, 2013) on this topic. Radu Miculescu is Professor at Transilvania University of Brasov, Romania. He graduated from Bucharest University, Romania, in 1992 and obtained his Ph.D in 1999 from the same university with a thesis concerning Lipschitz functions. In the last period his scientific interest includes Hutchinson-Barnsley fractals. Adriana Nicolae is Associate Professor at the Babes-Bolyai University, Department of Mathematics, Cluj-Napoca, Romania. She graduated in 2007 from the same university and focused during her Ph.D. on various aspects in metric fixed point and best approximation theory mainly in the setting of geodesic metric spaces. In the last years she also addressed problems in areas such as geometry and analysis in metric spaces, optimization, or proof mining.