Exercise 13, § 4

a) Let E be a barrelled space, $(C_n)$ an increasing sequence of convex, balanced sets such that $E = \bigcup_n C_n$. Let U be a convex, balanced and absorbent set such that for every n, $U \cap C_n$ is closed in $C_n$. Show that U is a neighbourhood of 0 in E. (Show that $\overline{U} \subset 2U$, by considering a filter $\mathfrak{F}$ on U converging to a point $x \in E$ and applying exerc. 12, c)). b) Let E be a barrelled space, $(E_n)$ an increasing sequence of subspaces of E such that $E = \bigcup_n E_n$.

Show that if U is a subset of E such that $U \cap E_n$ is a barrel in $E_n$ for every n, then U is a neighbourhood of 0 in E. In particular, E is the strict inductive limit (II, p. 33) of the sequence $(E_n)$.