The Geometric Theory of the Classical Banach Spaces

In the theory of Banach spaces a rather small class of spaces has always played a central role (actually even before the formulation of the general theory). This class —the class of classical Banach spaces— contains the Lp (p) spaces (p a measure, 1 < p < °°) and the C(K) spaces (K compact Hausdorff) and some related spaces. These spaces are very important in various applications of Banach spaces in mathematical analysis. They are, however, also of major importance in the abstract theory of Banach spaces. Among the questions studied in the theory of classical Banach spaces are (i) classification of the classical spaces, (ii) special properties of the classical spaces, in particular properties which characterize these spaces, and to a lesser extent (iii) the relation between the classical spaces and general Banach spaces. The purpose of this talk is to give a condensed survey of recent developments in some aspects of this theory. For the sake of simplicity we shall concentrate our attention on the separable classical spaces.