Independent functions and the geometry of Banach spaces

The main objective of this survey is to present the 'state of the art' of those parts of the theory of independent functions which are related to the geometry of function spaces. The 'size' of a sum of inde- pendent functions is estimated in terms of classical moments and also in terms of general symmetric function norms. The exposition is centred on the Rosenthal inequalities and their various generalizations and sharp con- ditions under which the latter hold. The crucial tool here is the recently developed construction of the Kruglov operator. The survey also provides a number of applications to the geometry of Banach spaces. In particu- lar, variants of the classical Khintchine-Maurey inequalities, isomorphisms between symmetric spaces on a finite interval and on the semi-axis, and a description of the class of symmetric spaces with any sequence of symmet- rically and identically distributed independent random variables spanning a Hilbert subspace are considered. Bibliography: 87 titles.