Exercise 3, § 4

Let E be a Hausdorff locally convex space with a countably infinite base $(a_n)$. a) Show that E admits a countable, topologically independent base $(e_n)$ (using the fact that every line in E has a topological complement, define the $e_n$ by induction). b) Show that, for E to be barrelled, it is necessary and sufficient that the topology $\mathscr{T}$ of E is identical with the finest locally convex topology on E (observe that the convex balanced envelope of every sequence $(\lambda_n e_n)$ is closed in E). In particular, if $\mathscr{T}$ is metrizable, E is not barrelled (cf. exerc. 2).