Exercise 12, § 4

a) Let E be a locally convex space, $(A_n)$ an increasing sequence of convex balanced sets in E such that $A = \bigcup_n A_n$ is absorbent. Let $(W_n)$ be a decreasing sequence of convex balanced neighbourhoods of 0; then the convex balanced envelope V of the $W_n \cap A_n$ is absorbent; if E is barrelled, $\overline{V}$ is a neighbourhood of 0. b) Let $\mathfrak{F}$ be a filter on E; suppose that for every n, there exists a set $M_n \in \mathfrak{F}$ such that $(M_n + W_n) \cap A_{2n} = \varnothing$. Let $V_n$ be the convex balanced envelope of the $W_k \cap A_k$ for $k \leq n - 1$ and of $W_n$ in such a way that $V_n$ is a neighbourhood of 0 and that we have $\frac{1}{2} \overline{V} \subset V_n$ for all n. Show that $(M_n + V_n) \cap A_n = \varnothing$ for all n. c) Deduce from a) and b) that if E is barrelled and if $\mathfrak{F}$ is a Cauchy filter on E, then there exists an integer N such that, for all $M \in \mathfrak{F}$ and every neighbourhood W of 0 in E, $M + W$ meets $A_N$. (Argue by contradiction; with the notations of b), consider a set $M \in \mathfrak{F}$ with small order $\frac{1}{2} \overline{V}$.)