Let S be a locally compact Hausdorff space and let us consider the space C 0 ( 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 r σ bv ( 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.