A theorem of Riesz type with Pettis integrals in topological vector spaces

Let X be a locally convex Hausdorff space and let C0(S,X) be the space of all continuous functions f : S →X, with compact support on the locally compact space S. In this paper we prove a Riesz representation theorem for a class of bounded operators T : C0(S,X)→X, where the representing integrals are X-valued Pettis integrals with respect to bounded signed measures on S. Under the additional assumption that X is a locally convex space, having the convex compactness property, or either, X is a locally convex space whose dual X is a barrelled space for an appropriate topology, we obtain a complete identification between all X-valued Pettis integrals on S and the bounded operators T : C0(S,X)→X they represent. Finally we give two illustrations of the representation theorem proved, in the particular case when X is the topological dual of a locally convex space. © 2007 Elsevier Inc. All rights reserved.