Analysis on Polish spaces and an introduction to optimal transportation

著者

    • Garling, D. J. H.

書誌事項

Analysis on Polish spaces and an introduction to optimal transportation

D.J.H. Garling

(London Mathematical Society student texts, 89)

Cambridge University Press, 2018

  • : hardback
  • : pbk

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注記

Includes bibliographical references (p. 339-341) and index

内容説明・目次

内容説明

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.

目次

  • 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. Cadlag 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.

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