Barrelled space

Remarks

Some authors define barrelled spaces among all topological vector spaces, with the same condition. Example: every Banach space $E$ is barrelled. If $T$ is a barrel, then $E = \bigcup_{n \ge 1} nT$ since $T$ is absorbing. By the Baire category theorem some $nT$, hence $T$, has an interior point. Convexity and balance then give that $T$ contains a ball around $0$. The same argument shows that every Fréchet space is barrelled. Non-example: let $c_{00}$ be the space of finitely supported real sequences with the norm $\norm{x} = \sup_n \abs{x_n}$, and let $T = \set{x : \abs{x_n} \le 1/n \text{ for all } n}$. $T$ is a barrel, but it is not a neighbourhood of $0$: for each $\varepsilon > 0$ the vector $\varepsilon e_n$ has norm $\varepsilon$ and lies outside $T$ once $n > 1 / \varepsilon$. In a barrelled space, every family of continuous linear maps into a locally convex space that is bounded at each point is equicontinuous.