Spaces of weakly continuous functions.

Abstract

This paper is very much in the spirit of a paper by H. Corson [Trans. Amer. Math. Soc. 101 (1961), 1–15; MR0132375 (24 2220)]. Let E be a real Banach space. The bw-topology on E is the finest topology which agrees with the weak topology on all bounded subsets of E. Cwb(E) [Cwbu(E)] is the set of real functions which are weakly continuous [weakly uniformly continuous] on all bounded sets in E. Cwb(E) is always barrelled; a sufficient condition is given for Cwb(E) to be bornological (under the compact-open topology). As a main result, the following are shown to be equivalent: (1) E is reflexive; (ii) Cwbu(E) is a Fr´echet space; (iii) Cwbu(E) is a Pt´ak space; (iv) Cwbu(E) is complete; (v) Cwbu(E) is barrelled; (vi) Cwbu(E) = Cwb(E).