# Topological Modules/Barrelled spaces

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

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

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

Then ${\displaystyle T}$ is itself a linear and continuous functional.

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

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

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

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

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