Exercise 7, § 3

a)** Let E be a (DF) space, U a convex, balanced and closed set in E, which absorbs the bounded subsets of E, and let $(x_n)$ be a sequence of points of $\complement U$. Show that there exists a neighbourhood V of 0 in E which does not contain any of the $x_n$. (Let $(A_n)$ be a countable base for the canonical bornology of E. Show that, by induction we can define a sequence $(\lambda_n)$ of numbers $> 0$ and a sequence $(V_n)$ of convex, balanced and closed neighbourhoods of 0 such that $\lambda_n A_n \subset \bigcap_{j=1}^\infty V_j, \lambda_n A_n \subset U$ and $x_n \in \complement V_n$ for all $n$. For this, if the $\lambda_j$ and $V_j$ have been constructed for $j \leq n$, 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 all $j \leq n$, 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 an everywhere dense countable set, then 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 for 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 infra-barrelled. d) Deduce from b) and from exerc. 6 that if every bounded subset of E is metrizable for the topology induced by that of E, then E is infrabarrelled.