Proposition 11.4.4.label Let $E$ be a locally convex space over $K \in \RC$. If $E$ is a Baire space, then $E$ is barrelled.
Proof, [II.7.1, SW99]. Let $D \subset E$ be a Barrel, then $E = \bigcup_{n \in \natp}nD$ is a countable union of closed sets. Since $E$ is Baire, there exists $n \in \natp$, $U \in \cn_{E}(0)$ circled, and $x \in E$ such that $x + U \in nB$. In which case,
\[U \subset (x + U) - (x + U) \subset nB - nB = 2nB\]
so $2nB$ and thus $B$ is a neighbourhood of $0$.$\square$
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