A strong barrelledness property for space $C(X,E)$

Abstract

A locally convex space (lcs) E is called s-barrelled [DiK] if every sequentially closed linear map,i.e. with sequentially closed graph, of E into a Fréchet space,i.e. a metrizable and complete lcs, is continuous. Let E be a lcs, X a locally compact topological space, $? X$ its Stone-Cech compactification. If $C(? X, E)$ is s-barrelled, then $C(X, E)$ is s-barrelled iff X is realcompact, where all spaces of continuous functions are provided with the compact-open topology. Some remarks and corollaries are also included.