A Choquet theorem for general subspaces of vector-valued functions

Abstract

Let X be a compact Hausdorff space, let E be a (real or complex) Banach space, and let C(X, E) stand for the Banach space of all continuous E-valued functions defined on X under the supremum norm. If A is an arbitrary linear subspace of C(X, E), then it is shown that each bounded linear functional l on A can be represented by a boundary E*-valued vector measure μ on X that has the same norm as l. This result constitutes an extension to vector-valued functions of the so-called analytic version of Choquet's integral representation theorem.