Strict topologies on spaces of vector-valued functions

Abstract

Let X be a completely regular Hausdorff space, and {E_x : x ∈ X} a collection of non-trivial locally convex topological vector spaces indexed by X. Let E be their disjoint union. We investigate a species of strict topology on a vector space F of choice functions σ : X → E (σ(x) ∈ E_x), and obtain Stone–Weierstrass and spectral synthesis analogues. We also obtain completeness results in some special cases.