Exercise 6, § 3

a)** Let E be a (DF) space (IV, p. 57, exerc. 2), and let U be a convex set such that for every bounded, convex and balanced subset A of E, $U \cap A$ is a neighbourhood of 0 for the topology induced on A by that of E. Show that U is a neighbourhood of 0 in A. (Let $(A_n)$ be a countable base for the canonical bornology of E (III, p. 3, def. 5). Show that, by induction we can define a sequence $(\lambda_n)$ of numbers $> 0$ and a sequence $(V_n)$ of closed, convex and balanced neighbourhoods of 0 in E such that $\lambda_n A_n \subset \frac{1}{3} U, \lambda_n A_n \subset \bigcap_{j=1}^\infty V_j, V_n \cap A_n \subset U$ for every $n$.

First show that if $\lambda_j$ and $V_j$ have been constructed for $j \leq n$, then we can find $\lambda_{n+1}$ such that $\lambda_{n+1} A_{n+1} \subset \frac{1}{3} U$ and $\lambda_{n+1} A_{n+1} \subset V_j$ for $j \leq n$. Next prove that we can find $V_{n+1}$ such that $\lambda_j A_j \subset V_{n+1}$ for $j \leq n + 1$ and $V_{n+1} \cap A_{n+1} \subset U$; for this, letting A denote the convex envelope of the $\lambda_j A_j$ for $j \leq n + 1$, show that we can take $V_{n+1} = \overline{A + V}$ for a suitable convex balanced neighbourhood V of 0; we remark that for this it is enough to show that, if $B = A_{n+1} \cap \complement U$, then 0 is not in the closure of the set $B + 2A$. b) Deduce from a) that if $u$ is a linear mapping from E into a locally convex space F, such that the restriction of $u$ to every bounded subset of E is continuous, then $u$ is continuous ($cf.$ IV, p. 50, exerc. 15).