On a problem of Horváth concerning barrelled spaces of vector valued continuous functions vanishing at infinity

Abstract

Let $C_{0}\left( \Omega ,X\right)$ be the linear space of all continuous functions from a locally compact normal space $\Omega $ into a normed space $X$ vanishing at infinity, equipped with the supremum-norm topology. The main result of the paper says that if $X$ is barrelled, then the space $C_{0}\left( \Omega ,X\right) $ is always barrelled. This answers a question posed by J. Horváth.