= 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 $E=\bigcup_nE_n$ 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.<On topological aspects of smooth-automorphic forms by Sonja Žunar>[Slide 7, strict inductive limits and closed-stage properties]{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 $\ell(x)=\sum_jx_j$ 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.
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