Topological Modules/Barrelled spaces

Proposition (pointwise limit of continuous linear functions from a barrelled LCTVS into a Hausdorff TVS is continuous and linear):

Let $X$ be a barrelled LCTVS over a field $\mathbb {K}$, let $Y$ be a locally closed Hausdorff TVS over the same field, and suppose that a net $T_{\lambda }:X\to Y$ ($\lambda \in \Lambda$) of linear and continuous functions is given. Suppose further that $T:X\to Y$ is a function such that

$\lim _{\lambda \in \Lambda }T_{\lambda }(x)=T(x)$.

Then $T$ is itself a linear and continuous functional.

Proof: First note that $T$ is linear, since whenever $\alpha \in \mathbb {K}$ and $v,w\in X$, we have

$T(v+\alpha w)=\lim _{\lambda \in \Lambda }(T_{\lambda }(v)+\alpha T_{n}(w))=T(v)+\alpha T(w)$

since $Y$ is a Hausdorff space, where limits are well-defined, and by continuity of addition. Then note that $T$ is continuous, since for all $v\in X$ the set $\{T_{\lambda }|\lambda \in \Lambda \}$ is bounded, so that the Banach—Steinhaus theorem applies and the family $\{T_{\lambda }|\lambda \in \Lambda \}$ is uniformly bounded. Hence, suppose that $V\subseteq Y$ is a closed neighbourhood of the origin. By uniform boundedness, select $U\subseteq X$ to be an open neighbourhood of the origin so that

$\forall \lambda \in \Lambda :T_{n}(U)\subseteq V$.

We conclude that $T(U)\subseteq V$, since closed sets contain their net limits. We conclude since $Y$ is locally closed, so that $V$ represents a generic neighbourhood. $\Box$