A characterization of when C(K,E) is a Grothendieck space, for reflexive spaces E.

Abstract

Let C(K,E) be the vector space of all continuous functions on a compact Hausdorff space K with values in a reflexive Banach space E, endowed with the usual uniform norm. We prove in this paper that the Banach space C(K,E) is a Grothendieck space if and only if C(K,E) does not contain a complemented subspace isomorphic to c0