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Sonja Žunar presented joint work on topological aspects of smooth automorphic forms. Her 2022 slides explain the use of locally convex spaces and the strict inductive limit topology in organizing spaces of such functions. This supplies a topology compatible with an increasing sequence of function spaces, rather than treating automorphic forms only as an unstructured collection of functions.^[ref]

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.