ON THE DUAL SPACE C∗0 (S,X)

Abstract. Let S be a locally compact Hausdorff space and let us consider the space C0(S,X) of continuous functions vanishing at infinity, from S into the Banach space X. A theorem of I. Singer, settled for S compact, states that the topological dual C∗0 (S,X) is isometrically isomorphic to the Banach space rσbv(S,X ∗) of all regular vector measures of bounded variation on S with values in the strong dual X∗. Using the Riesz-Kakutani theorem and some routine topological arguments, we propose a constructive detailed proof which is, as far as we know, different from that supplied elsewhere. Preliminaries Let S be a locally compact Hausdorff space equipped with its Borel σ-field BS, and letX be a Banach space. We denote by C0(S,X) the Banach space (uniform norm) of all continuous functions f: S → X, vanishing at infinity. If X = R, we put C0(S,X) = C0(S). According to the Riesz-Kakutani theorem [7, Theorem 6.19], the dual C∗0 (S) is isometric to the Banach space of all scalar regular measures on S with the variation norm. All the measures we will deal with here are supposed to be defined on the σ-field BS. We denote by X ∗ the strong dual of X. If λ: BS → Y is an additive set function from BS into the Banach space Y, then the variation of λ is usually defined by the extented positive set function |λ|(•) given by: |λ|(E) = sup