Barrelledness conditions on S(Σ;E) and B(Σ;E).

Abstract

Let Ω be a nonempty set, and let Σ be a field of subsets of Ω. If E is a locally convex space we denote by S(Σ;E) the vector space of all Σ-simple functions defined on Ω with values in E, and by B(Σ;E) the vector space of all functions defined on Ω with values in E which are uniform limits of Σ-simple functions. We give some results characterizing when the spaces S(Σ;E) and B(Σ;E) endowed with the uniform convergence topology are barrelled or infrabarrelled.