Analysis on Polish Spaces and an Introduction to Optimal Transportation (London Mathematical Society Student Texts)
Cambridge University Press, 12/21/2017
EAN 9781108431767, ISBN10: 1108431763
Paperback, 358 pages, 22.9 x 15.2 x 2.1 cm
Language: English
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.
Introduction
Part I. Topological Properties
1. General topology
2. Metric spaces
3. Polish spaces and compactness
4. Semi-continuous functions
5. Uniform spaces and topological groups
6. Càdlàg functions
7. Banach spaces
8. Hilbert space
9. The Hahn–Banach theorem
10. Convex functions
11. Subdifferentials and the legendre transform
12. Compact convex Polish spaces
13. Some fixed point theorems
Part II. Measures on Polish Spaces
14. Abstract measure theory
15. Further measure theory
16. Borel measures
17. Measures on Euclidean space
18. Convergence of measures
19. Introduction to Choquet theory
Part III. Introduction to Optimal Transportation
20. Optimal transportation
21. Wasserstein metrics
22. Some examples
Further reading
Index.