On the semi-reflexivity of biprojective tensor product spaces

Abstract

Let E and F be Banach spaces. If both E and F contain a Schauder basis, it is proved by Holub [5] that the Banach space Yb(E, F) of continuous linear maps from E into F, carried the topology of bounded convergence in E, is reflexive if, and only if, all the maps of it are compact. This result is later extended by Ruckle [lo] to the case both E and F have the approximation property and by Holub [6] to that either E or F has the approximation property. Heinrich [4] has also proved it under a weaker assumption than the approximation property of E or F. In all the preceding considerations. tensor product techniques are essentially applied. On the other hand, by a conditional weak compactness criterium of Lewis [9], it follows that the reflexivity of certain tensor products of reflexive Banach Spaces is equivalent to their weak sequential completeness (ibid.). In [15] an extension of the above criterium to a fairly large class of locally convex spaces is obtained, which yields Theorem 3.1 below, so that the techniques and the results of [16] actually supply the present work. Thus, let E and F be complete locally convex spaces, E BE F the respective completed (biprojective) e-tensor product and let T be the (vector) subspace of (E’ OF’)*, consisting of the o((E’ OF’)*, E’ @ F’)-limits of all weakly Cauchy sequences in E BE F. First, for certain semi-reflexive spaces E and F, it is therein verified that E Gi. F is semi-reflexive if, and only if, it is weakly sequentially complete (Theorem 3.1). On the other hand, under some mild restrictions on E and F, it is also proved that E a, F is semi-reflexive if, and only if, both E and F are semi-reflexive and moreover, every linear map u E T transfers equicontinuous subsets of E’ into relatively compact subsets of F (Theorem 3.2). From these results and certain results of [16] a number of corollaries, referred to the semireflexivity of several (locally convex) spaces of vector-valued maps are derived. In particular, they have, among otherthings, a special bearing, into the case under consideration, on the reflexivity criterium for the Banach space &(E, F), stated above.