Remark

— 1) Suppose that $E$ is the union of an increasing directed, non-enumerable family of subspaces $(E_\alpha)_{\alpha \in I}$, each $E_\alpha$ having a locally convex topology $\mathscr{T}_\alpha$, such that, for $E_\alpha \subset E_\beta$, the topology induced on $E_\alpha$ by $\mathscr{T}_\beta$ is identical with $\mathscr{T}_\alpha$. It may be the case that the topology induced on each $E_\alpha$ by the topology $\mathscr{T}$ is equal to $\mathscr{T}_\alpha$ and that the $E_\alpha$ are Hausdorff and complete, but that $E$ is not complete for $\mathscr{T}$ (INT, III, 2nd ed., § 1, exerc. 2).

2) Let $F$ be a locally convex space, which is the union of an increasing sequence of vector subspaces $(F_n)$, and for each index $n$, let $\mathscr{T}_n$ be the topology induced on $F_n$ by the topology $\mathscr{T}$ of $F$. One should beware that in general $\mathscr{T}$ is not the inductive limit of the $\mathscr{T}_n$.

3) Suppose that $E$ is the strict inductive limit of the sequence $(E_n)$; if $F$ is a closed (in $\mathscr{T}$) vector subspace of $E$, it may be the case that the strict inductive limit of the topologies induced by the $\mathscr{T}_n$ on $F \cap E_n$ is strictly finer than the topology induced by $\mathscr{T}$ (IV, p. 63, exerc. 10).