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E-raamat: Analysis on Polish Spaces and an Introduction to Optimal Transportation

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Analysis on Polish Spaces and an Introduction to Optimal TransportationSuurem pilt
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This detailed account of analysis on Polish spaces contains results that apply to probability theory and a gentle introduction to optimal transportation. Containing more than 200 elementary exercises, it is a useful resource for advanced mathematical students and also for researchers in mathematical analysis.

A large part of mathematical analysis, both pure and applied, takes place on Polish spaces: topological spaces whose topology can be given by a complete metric. This analysis is not only simpler than in the general case, but, more crucially, contains many important special results. This book provides a detailed account of analysis and measure theory on Polish spaces, including results about spaces of probability measures. Containing more than 200 elementary exercises, it will be a useful resource for advanced mathematical students and also for researchers in mathematical analysis. The book also includes a straightforward and gentle introduction to the theory of optimal transportation, illustrating just how many of the results established earlier in the book play an essential role in the theory.

Arvustused

'This book provides a detailed and concise account of analysis and measure theory on Polish spaces, including results about probability measures. Containing more than 200 elementary exercises, it will be a useful resource for advanced mathematical students and also for researchers in analysis.' Luca Granieri, Mathematical Reviews

Muu info

Detailed account of analysis on Polish spaces with a straightforward introduction to optimal transportation.
Introduction 1(6)
PART ONE TOPOLOGICAL PROPERTIES
7(170)
1 General Topology
9(9)
1.1 Topological Spaces
9(6)
1.2 Compactness
15(3)
2 Metric Spaces
18(20)
2.1 Metric Spaces
18(3)
2.2 The Topology of Metric Spaces
21(3)
2.3 Completeness: Tietze's Extension Theorem
24(3)
2.4 More on Completeness
27(2)
2.5 The Completion of a Metric Space
29(2)
2.6 Topologically Complete Spaces
31(2)
2.7 Baire's Category Theorem
33(2)
2.8 Lipschitz Functions
35(3)
3 Polish Spaces and Compactness
38(12)
3.1 Polish Spaces
38(1)
3.2 Totally Bounded Metric Spaces
39(2)
3.3 Compact Metrizable Spaces
41(6)
3.4 Locally Compact Polish Spaces
47(3)
4 Semi-continuous Functions
50(6)
4.1 The Effective Domain and Proper Functions
50(1)
4.2 Semi-continuity
50(3)
4.3 The Brezis--Browder Lemma
53(1)
4.4 Ekeland's Variational Principle
54(2)
5 Uniform Spaces and Topological Groups
56(15)
5.1 Uniform Spaces
56(3)
5.2 The Uniformity of a Compact Hausdorff Space
59(2)
5.3 Topological Groups
61(3)
5.4 The Uniformities of a Topological Group
64(2)
5.5 Group Actions
66(1)
5.6 Metrizable Topological Groups
67(4)
6 Cadlag Functions
71(8)
6.1 Cadlag Functions
71(1)
6.2 The Space (D[ 0, 1], d∞)
72(1)
6.3 The Skorohod Topology
73(2)
6.4 The Metric dB
75(4)
7 Banach Spaces
79(18)
7.1 Normed Spaces and Banach Spaces
79(3)
7.2 The Space BL(X) of Bounded Lipschitz Functions
82(1)
7.3 Introduction to Convexity
83(3)
7.4 Convex Sets in a Normed Space
86(2)
7.5 Linear Operators
88(3)
7.6 Five Fundamental Theorems
91(4)
7.7 The Petal Theorem and Danes's Drop Theorem
95(2)
8 Hilbert Spaces
97(15)
8.1 Inner-product Spaces
97(4)
8.2 Hilbert Space; Nearest Points
101(3)
8.3 Orthonormal Sequences; Gram--Schmidt Orthonormalization
104(3)
8.4 Orthonormal Bases
107(1)
8.5 The Frechet--Riesz Representation Theorem; Adjoints
108(4)
9 The Hahn--Banach Theorem
112(16)
9.1 The Hahn--Banach Extension Theorem
112(4)
9.2 The Separation Theorem
116(2)
9.3 Weak Topologies
118(1)
9.4 Polarity
119(1)
9.5 Weak and Weak* Topologies for Normed Spaces
120(4)
9.6 Banach's Theorem and the Banach--Alaoglu Theorem
124(1)
9.7 The Complex Hahn--Banach Theorem
125(3)
10 Convex Functions
128(5)
10.1 Convex Envelopes
128(2)
10.2 Continuous Convex Functions
130(3)
11 Subdifferentials and the Legendre Transform
133(22)
11.1 Differentials and Subdifferentials
133(1)
11.2 The Legendre Transform
134(3)
11.3 Some Examples of Legendre Transforms
137(2)
11.4 The Episum
139(1)
11.5 The Subdifferential of a Very Regular Convex Function
140(3)
11.6 Smoothness
143(5)
11.7 The Fenchel--Rockafeller Duality Theorem
148(1)
11.8 The Bishop--Phelps Theorem
149(2)
11.9 Monotone and Cyclically Monotone Sets
151(4)
12 Compact Convex Polish Spaces
155(7)
12.1 Compact Polish Subsets of a Dual Pair
155(2)
12.2 Extreme Points
157(3)
12.3 Dentability
160(2)
13 Some Fixed Point Theorems
162(15)
13.1 The Contraction Mapping Theorem
162(3)
13.2 Fixed Point Theorems of Caristi and Clarke
165(2)
13.3 Simplices
167(1)
13.4 Sperner's Lemma
168(2)
13.5 Brouwer's Fixed Point Theorem
170(1)
13.6 Schauder's Fixed Point Theorem
171(2)
13.7 Fixed Point Theorems of Markov and Kakutani
173(2)
13.8 The Ryll--Nardzewski Fixed Point Theorem
175(2)
PART TWO MEASURES ON POLISH SPACES
177(120)
14 Abstract Measure Theory
179(12)
14.1 Measurable Sets and Functions
179(3)
14.2 Measure Spaces
182(2)
14.3 Convergence of Measurable Functions
184(3)
14.4 Integration
187(1)
14.5 Integrable Functions
188(3)
15 Further Measure Theory
191(19)
15.1 Riesz Spaces
191(3)
15.2 Signed Measures
194(2)
15.3 M(X), L1 and L∞
196(3)
15.4 The Radon--Nikodym Theorem
199(4)
15.5 Orlicz Spaces and LP Spaces
203(7)
16 Borel Measures
210(33)
16.1 Borel Measures, Regularity and Tightness
210(4)
16.2 Radon Measures
214(1)
16.3 Borel Measures on Polish Spaces
215(1)
16.4 Lusin's Theorem
216(2)
16.5 Measures on the Bernoulli Sequence Space Ω(N)
218(4)
16.6 The Riesz Representation Theorem
222(3)
16.7 The Locally Compact Riesz Representation Theorem
225(1)
16.8 The Stone--Weierstrass Theorem
226(2)
16.9 Product Measures
228(3)
16.10 Disintegration of Measures
231(3)
16.11 The Gluing Lemma
234(2)
16.12 Haar Measure on Compact Metrizable Groups
236(2)
16.13 Haar Measure on Locally Compact Polish Topological Groups
238(5)
17 Measures on Euclidean Space
243(14)
17.1 Borel Measures on R and Rd
243(2)
17.2 Functions of Bounded Variation
245(2)
17.3 Spherical Derivatives
247(2)
17.4 The Lebesgue Differentiation Theorem
249(1)
17.5 Differentiating Singular Measures
250(1)
17.6 Differentiating Functions in bv0
251(3)
17.7 Rademacher's Theorem
254(3)
18 Convergence of Measures
257(23)
18.1 The Norm ||·||TV
257(1)
18.2 The Weak Topology w
258(2)
18.3 The Portmanteau Theorem
260(4)
18.4 Uniform Tightness
264(2)
18.5 The β Metric
266(3)
18.6 The Prokhorov Metric
269(2)
18.7 The Fourier Transform and the Central Limit Theorem
271(5)
18.8 Uniform Integrability
276(2)
18.9 Uniform Integrability in Orlicz Spaces
278(2)
19 Introduction to Choquet Theory
280(17)
19.1 Barycentres
280(2)
19.2 The Lower Convex Envelope Revisited
282(2)
19.3 Choquet's Theorem
284(1)
19.4 Boundaries
285(4)
19.5 Peak Points
289(2)
19.6 The Choquet Ordering
291(2)
19.7 Dilations
293(4)
PART THREE INTRODUCTION TO OPTIMAL TRANSPORTATION
297(42)
20 Optimal Transportation
299(16)
20.1 The Monge Problem
299(1)
20.2 The Kantorovich Problem
300(3)
20.3 The Kantorovich--Rubinstein Theorem
303(2)
20.4 c-concavity
305(3)
20.5 c-cyclical Monotonicity
308(2)
20.6 Optimal Transport Plans Revisited
310(3)
20.7 Approximation
313(2)
21 Wasserstein Metrics
315(10)
21.1 The Wasserstein Metrics Wp
315(2)
21.2 The Wasserstein Metric W1
317(1)
21.3 W1 Compactness
318(2)
21.4 Wp Compactness
320(2)
21.5 Wp-Completeness
322(1)
21.6 The Mallows Distances
323(2)
22 Some Examples
325(14)
22.1 Strictly Subadditive Metric Cost Functions
325(1)
22.2 The Real Line
326(1)
22.3 The Quadratic Cost Function
327(2)
22.4 The Monge Problem on Rd
329(2)
22.5 Strictly Convex Translation Invariant Costs on Rd
331(5)
22.6 Some Strictly Concave Translation-Invariant Costs on Rd
336(3)
Further Reading 339(3)
Index 342
D. J. H. Garling is a Fellow of St John's College, Cambridge, and Emeritus Reader in Mathematical Analysis at the University of Cambridge. He has written several books on mathematics, including Inequalities: A Journey into Linear Algebra (Cambridge, 2007) and A Course in Mathematical Analysis (Cambridge, 2013).
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