Vector-valued continuous functions with strict topologies and angelic topological spaces

Abstract

Abstract. It is proved that if -V is a metric space, E a Banach space containing a a-weakly-compact dense subset, then the space (Mr(X, E'), o(MT(X, £"), Cb(X, E))) is angehe, Cb(X, E) being all bounded continuous functions from X into E and MT(X, £") the dual of Cb(X, E) with the strict topology ß.A Hausdorff topological space Y is called angelic if (i) every relatively countably compact subset of Y is relatively compact, and (ii) for any point x in the closure of a relatively compact subset A of Y, there exists a sequence, in A, converging to x [4, p. 534]. In this paper X will stand for a completely regular Hausdorff space, E a Banach space over K, the field of real or complex numbers, C(X, E) (C(X)) all £-valued (AT-valued) continuous functions on X, and Cb(X, E) (Cb(X)) all bounded £-valued (/T-valued) continuous functions on X. We shall use the notations of [5] for locally convex spaces. Also the notations and results from [2] will be used. The topologies ß, ß0, ßx, ße (also denoted by ßx) are defined on Cb(X) in [9], [7], [8]-these topologies are defined for K = R, the reals, but naturally extend to K = C, the complex field, ß, ß0, ßx, ßx are defined on Cb(X, E) in [1], [2]. If %°° = %X(X, E) = [H c Cb(X, E): H pointwise equicontinuous and uniformly bounded}, the topology ßx is the finest locally convex topology on Cb(X, E) agreeing with pointwise topology on each H E %°°. It is well known that MT(X, E') = (Cb(X, E), ß)', M,(X, E') = (Cb(X, E), ß0)' and MX(X, E') = (Cb(X, E), ßj (see [2]). If J is a metric, MX(X, E') = Mr(X, £") and ß and ßx are both Mackey [2]; from this it follows that ß = ßx in this case.