A Banach–Dieudonné theorem for the space of bounded continuous functions on a separable metric space with the strict topology

Let X be a separable metric space and let β be the strict topology on the space of bounded continuous functions on X, which has the space of τ-additive Borel measures as a continuous dual space. We prove a Banach–Dieudonné type result for the space of bounded continuous functions equipped with β: the finest locally convex topology on the dual space that coincides with the weak topology on all weakly compact sets is a k-space. As a consequence, the space of bounded continuous functions with the strict topology is hypercomplete and a Pták space. Additionally, the closed graph, inverse mapping and open mapping theorems holds for linear maps between space of this type.