Exercise 6, § 3

(a) Let $E$ be a (DF) space (IV, p. 58, exerc. 2), $U$ 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 $E$. Now prove that $U$ is a neighbourhood of 0 in $A$. (Let $(A_n)$ be a countable basis of the canonical bornology of $E$ (III, p. 3, déf. 5). Now prove that one can define by recurrence a sequence $(\lambda_n)$ of numbers $> 0$ and a sequence $(V_n)$ of neighbourhoods of 0 in $E$, convex, balanced and closed, such that $\lambda_n A_n \subset \frac{1}{3} U$, $\lambda_n A_n \subset \bigcap_{j=1}^\infty V_j$, $V_n \cap A_n \subset U$ whatever n may be. Now prove first that if the $\lambda_j$ and $V_j$ are determined for $j \leq n$, one 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 $j \leq n$. Prove then that one 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 purpose, denoting by A the convex hull of the $\lambda_j A_j$ for $j \leq n + 1$, show that one can take $V_{n+1} = A + \overline{V}$ for a suitable convex balanced neighbourhood V of 0; one will note for this purpose that it suffices to show that, if $B = A_{n+1} \cap \mathbf{C} U$, 0 is not adherent to 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).