A revised closed graph theorem for quasi-Suslin spaces

Abstract

Some results about the continuity of special linear maps between F-spaces recently obtained by Drewnowski have motivated us to revise a closed graph theorem for quasi-Suslin spaces due to Valdivia. We extend Valdivia’s theorem by showing that a linear map with closed graph from a Baire tvs into a tvs admitting a relatively countably compact resolution is continuous. This also applies to extend a result of De Wilde and Sunyach. A topological space X is said to have a (relatively countably) compact resolution if X admits a covering {A α :α ∈ ℕℕ} consisting of (relatively countably) compact sets such that A α ⊆ A β for α ⩽ β. Some applications and two open questions are provided.