Exercise 7, § 3

a) Let E be a (DF) space, U a convex, balanced and closed set in E, which absorbs the bounded parts of E, $(x_n)$ a sequence of points of $\mathbf{C} U$. Prove that there exists a neighbourhood V of 0 in E which contains none of the $x_n$. (Let $(A_n)$ be a countable basis of the canonical bornology of E. Prove that one can define by recurrence a sequence $(\lambda_n)$ of numbers $> 0$ and a sequence $(V_n)$ of neighbourhoods of 0, convex, balanced and closed, such that $\lambda_n A_n \subset \bigcap_{j=1}^\infty V_j$, $\lambda_n A_n \subset U$ and $x_n \in \mathbf{C} V_n$ for every n. For this, the $\lambda_j$ and $V_j$ being determined for $j \leq n$, first take $\lambda_{n+1}$ such that $\lambda_{n+1} A_{n+1} \subset U$ and $\lambda_{n+1} A_{n+1} \subset V_j$ for every $j \leq n$, then take $V_{n+1}$ containing the closure of the convex envelope of the union of the $\lambda_j A_j$ for $j \leq n + 1$.) b) Deduce from a) that if M is a subset of E containing a countable everywhere dense set, the topology induced on M by the strong topology of the bidual $E''$ of E is identical with the topology induced by that of E. In particular, the convergent sequences in E are the same for the topology of E and the topology induced by the strong topology of $E''$; for every metrizable subset M of E, the topology induced on M by the topology of E is identical with the topology induced by the strong topology of $E''$. c) Deduce from a) that if there exists a countable everywhere dense set in E, then E is infratonnelé. d) Deduce from b) and exerc. 6 that if every bounded subset of E is metrizable for the topology induced by that of E, then E is infratonnelé.