Exercise 17, § 4

a) Let E be a Hausdorff locally convex space, L a vector subspace of E of finite codimension, and T a borniverous barrel in L. Show that there exists a borniverous barrel T' in E such that $T' \cap L = T$. (Reduce to the case where L is a hyperplane in E. Let $E_0$ be the bornological space associated with E (III, p. 40, exerc. 1), $L_0$ the hyperplane L with the topology induced by that of $E_0$; observe that T is a neighbourhood of 0 in $L_0$ and consider the following two cases : that $L_0$ is dense in $E_0$, or is closed in $E_0$; show that for T' we can take the closure $\overline{T}$ of T in E or the sum of T and a compact convex set of dimension 1). b) Let E be an infrabarrelled space, L a vector subspace of E of finite codimension. Deduce from a) that L is infrabarrelled (cf. IV, p. 64, exerc. 11).