On the dual space C0*(S, X)

. 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 C0*(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.