Exercise 7, § 4

A locally convex space E is said to be infra-barrelled if every barrel of E which is borniverous (III, p. 39, exerc. 11) is a neighbourhood of 0 in E. Every bornological space is infra-barrelled; every barrelled space is infrabarrelled. Show that the completion of a Hausdorff infrabarrelled space is barrelled (use the fact that in a Hausdorff locally convex space E, each barrel absorbs every convex, balanced, bounded and semi-complete subset of E).