Abstract SOME RESULTS IN REPRESENTABLE BANACH SPACES

Some results are presented, concerning aclass of Banach spaces introduced by G. Godefroy and M. Talagrand, the representable Banach spaces. The main aspects considered here are the stability in forming tensor products, and the topological properties of the weak * dual unitball. This note deals with some results on a class of Banach spaces introduced by G. Godefroy and M. Talagrand in [7], the representable Banach spaces. We begin by recalling some concepts. First, a Hausdorff topological space is said to be analytic if it is a continuous image of the irrationals. A measurable space is said to be a Souslin space if it is Borel isomorphic to some space (A, E), where A is a subset of the real line and E is the u-algebra of the Borel subsets of A. Let X be a normed space, and D a subset of its dual X *. D is said to be norming if x--> sup{1 x * (x) 1: x * E D} defines a norm equivalent to the given one. If X admits a countable norming set D C X*, we say that X is a space of countable type. In this case we can consider the topology uD of pointwise convergente on the points of D, which is a metrizable topology on X. We say that X is a representable normed space if there is a countable norming set D C X * such that (X, uD) is analytic. If (X, uD) is analytic for every such D, we say that X is universally representable. These spaces have been introduced in [7], where some questions about them are discussed. A Hausdorfftopological space E is said to be angelic when for every relatively countably compact subset A C E the following holds: a) A is relatively com-pact, and b) every closure point x E Á is the limit of a sequence contained in A. The fundamental facts about angelic spaces can be found in [5]. A remarkable class of angelic compact spaces, the Rosenthal compacts, has been studied by Godefroy in [6]. A compact Hausdorff space is said to be a Rosenthal compact when it is homeomorphic to a subset of Bl (I), the space of first class function