Grothendieck locally convex spaces of continuous vector valued functions

Abstract

Let ^{X, E) be the space of continuous functions from the completely regular Hausdorff space X into the Hausdorff locally convex space E, endowed with the compact-open topology. Our aim is to characterize the ^(X, E) spaces which have the following property: weak-star and weak sequential convergences coincide in the equicontinuous subsets of ^(X, E)'. These spaces are here called Grothendieck spaces. It is shown that in the equicontinuous subsets of E' the σ(E', E)- and β(E', ^-sequential convergences coincide, if ^(X, E) is a Grothendieck space and X contains an infinite compact subset. Conversely, if X is a G-space and E is a strict inductive limit of Frechet-Montel spaces ^(X, E) is a Grothendieck space. Therefore, it is proved that if £ is a separable Frechet space, then E is a Montel space if and only if there is an infinite compact Hausdorff X such that , E) is a Grothendieck space.