On spaces of vector-valued continuous functions

Abstract

Let X be a Hausdorff completely regular space and E be a Hausdorff locally convex topological vector space. Then C(X;E) denotes the linear space of the continuous functions on X, with values in E. Previously [C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A596-A598; MR0271712 (42 #6593)], L. Nachbin introduced the topologies τω|A, τλ|A and τδ|A, where A is a subspace of C(X;E). In this paper, the author studies the case when A is a Cb(X)-submodule of C(X;E) (Cb(X) is the linear space of bounded continuous functions on X). He proves that in this case the topologies τω|A and τλ|A coincide with the compact-open topology, and that the topology τδ|A coincides with the compact-open topology coming from the repletion (= realcompactification) of X.