Book reviews: “Daniel Li and Herv´e Queff´elec, Introduction to Banach spaces: analysis and probability” (2 volumes, translated from the French by Dani`ele Gibbons and Greg Gibbons), Cambridge Studies in Advanced Mathematics: no. 166: vol. 1, xxx+431 p., ISBN: 978-1-107-16051-4/hbk ($ 94.99), 978-1-316-67576-2/ebook; no. 167: vol. 2, xxx+374 p., ISBN: 978-1-107-16262-4/hbk ($ 99.99), 978-1-316-67739- 1/ebook, ISBN 978-1-107-16263-1/set ($ 165.00), Cambridge University Press, Cambridge, 2017.

Abstract

The subject of the book is the interplay between the Banach space theory and probability theory, which has a long and fruitful history. Although the study of Banach space valued random variables started in the 1950s, their importance in the study of Banach spaces became clear only after the introduction, at the beginning of 1970s, of the notions of type and cotype (presented in volume I, Chapter 5), proving their intimately connections with Banach spaces. On the other hand, probability methods allow the proof of some deep results in Banach space theory as, for instance, Dvoretzky’s theorem on the Euclidean sections of convex bodies in Banach spaces, the relevance of martingales in the study of Radon-Nikody´m property, Davie’s proof on the existence of Banach spaces without approximation property, Gowers’ dichotomy theorem.