Pelczynski's property (V) on spaces of vector valued functions

Abstract

Let $E$ be a separable Banach space and $\Omega$ be a compact Hausdorff space. It is shown that the space $C(\Omega,E)$ has property (V) if and only if $E$ does. Similar result is also given for Bochner spaces $L^p(\mu,E)$ if $1<p<\infty$ and $\mu$ is a finite Borel measure on $\Omega$.