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).