Exercise 14, § 4

Let E be the strict inductive limit of an increasing sequence of locally convex spaces $E_n$ (II, p. 33). Show that the topology of E is the finest of the topologies compatible with the vector space structure of E, whether locally convex or not, and inducing on $E_n$ a coarser topology than the given topology $\mathscr{T}_n$. (Let $V_0$ be a neighbourhood of 0 for such a topology $\mathscr{T}$ and $(V_n^n)_{n \geq 0}$, a sequence of neighbourhoods of 0 for $\mathscr{T}$ such that $V_{n+1}^{n+1} + V_{n+1}^{n+1} \subset V_n^n$ for all $n \geq 0$; for all $n \geq 1$, consider, in $E_n$, a convex neighbourhood $W_n^n$ of 0 that is contained in $E_n \cap V_n$, and take the convex envelope of the union of the $W_n^n$ in E.)