Proposition 11.4.5.label Let $\seqi{E}$ be locally convex spaces over $K \in \RC$, $E$ be a vector space over $K$, and $\seqi{T}$ such that $T_{i} \in \hom(E_{i}; E)$ for all $i \in I$, then the inductive locally convex topology on $E$ induced by $\seqi{T}$ is barrelled.
Proof, [II.7.2, SW99]. Let $D \subset E$ be a barrel, then for each $i \in I$, $T_{i}^{-1}(D) \subset E_{i}$ is also a barrel, and thus a neighbourhood of $0$ in $E_{i}$. By (5) of Definition 11.8.1, $D$ is a neighbourhood of $0$ in $E$, so $E$ is barrelled.$\square$
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