 Strict inductive limit topology

In the standard closed-stage setting, let $E_1\subseteq E_2\subseteq \cdots$ be Hausdorff locally convex spaces, with each $E_n$ a closed topological linear subspace of $E_{n+1}$. The strict inductive limit topology on $E={\bigcup }_n{E}_n$ is the finest topology that makes $E$ a locally convex space and makes every inclusion $E_n\to E$ continuous. Each stage retains its original topology as a closed subspace of the limit.^[ref]

For a locally convex space $F$, a linear map $T:E\to F$ is continuous exactly when every restriction $T{|}_{E_n}$ is continuous. This universal property concerns locally convex targets; it does not identify the construction with an unrestricted final topology in the category of all topological spaces.

For example, the union of finite-dimensional coordinate spaces ${\mathbb{R}}^n$, included by appending zeros, is the space of finitely supported sequences. The functional $\ell (x)={\sum }_j{x}_j$ is continuous for the strict inductive limit topology, since each restriction is a finite sum. Hence the coordinate unit vectors do not converge to zero, although they do in the coordinatewise product topology on ${\mathbb{R}}^{\mathbb{N}}$. The choice of topology changes convergence on the same algebraic vectors.