Pointwise Compactness in Spaces of Continuous Functions

Abstract

In this paper we describe a class of topological spaces X such that Cp(X), the space of continuous functions on Xendowed with the topology of pointwise convergence, is an angelic space. This class contains the topological spaces with a dense and countably determined subspace; in particular the topological spaces which are K-analytic in the sense of G. Choquet. Our results include previous ones of A. Grothendieck, L. Kelley and I. Namioka, J. D. Pryce, R. Haydon, M. De Wilde, K. Floret and M. Talagrand. As a consequence we obtain an improvement of the Eberlein-Šmulian theorem in the theory of locally convex spaces. This result allows us to deduce, for instance, that (LF)-spaces and dual metric spaces, in particular DF)-spaces of Grothendieck, are weakly angelic. In this way the answer to a question posed by K. Floret about the weak angelic character of (LF)-spaces is given.