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

Abstract

Let X be a locally convex Hausdorff space and let C 0 ( 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 : C 0 ( 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 : C 0 ( 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.