On the closed graph theorem

Abstract

Our main purpose is to describe those separated locally convex spaces which can serve as domain spaces for a closed graph theorem in which the range space is an arbitrary Banach space of (linear) dimension at most c, the cardinal number of the real line R. These are the δ-barrelled spaces which are considered in §4. Many of the standard elementary Banach spaces, including in particular all separable ones, have dimension at most c. Also it is known that an infinite dimensional Banach space has dimension at least c (see e.g. [8]). Thus if we classify Banach spaces by dimension we are dealing, in a natural sense, with the first class which contains infinite dimensional spaces.