A simple proof of the Grothendieck theorem on the Dieudonné property of $C\textunderscore 0(T)$

Abstract

Abstract. Let T be a locally compact Hausdorff space and let C 0 (T ) be the Banach space of all complex valued continuous functions vanishing at infinity in T , provided with the supremum norm. Let X be a locally convex Hausdorff space (briefly, an lcHs) which is quasicomplete. A simple proof of the Grothendieck theorem on the Dieudonné property of C 0 (T ) is given. The present proof is much simpler than that given in an earlier work of the author (Characterizations of weakly compact operators on C 0 (T ), Trans. Amer. Math. Soc. 350 (1998), 4849-4867).