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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $E$ be a Hausdorff [[locally convex space|B6GY2KJMWW]] over $\\mathbb{K}$.\nA *barrel* in $E$ is a subset $T \\subseteq E$ that is [[closed|0ZCK8DQYN1]], [[convex|02AS45T3H9]], [[balanced|2TDSHJJYS4]] and [[absorbing|NM1BCK3ZX9]].\n$E$ is *barrelled* if every barrel in $E$ is a [[neighbourhood|0ZCY0Z80B6]] of $0$.\n\n## Remarks\n\nSome authors define barrelled spaces among all topological vector spaces, with the same condition.\nExample: every Banach space $E$ is barrelled.\nIf $T$ is a barrel, then $E = \\bigcup_{n \\ge 1} nT$ since $T$ is absorbing.\nBy the Baire category theorem some $nT$, hence $T$, has an interior point.\nConvexity and balance then give that $T$ contains a ball around $0$.\nThe same argument shows that every Fréchet space is barrelled.\nNon-example: let $c_{00}$ be the space of finitely supported real sequences with the norm $\\norm{x} = \\sup_n \\abs{x_n}$, and let $T = \\set{x : \\abs{x_n} \\le 1/n \\text{ for all } n}$.\n$T$ is a barrel, but it is not a neighbourhood of $0$: for each $\\varepsilon \u003e 0$ the vector $\\varepsilon e_n$ has norm $\\varepsilon$ and lies outside $T$ once $n \u003e 1 / \\varepsilon$.\nIn a barrelled space, every family of continuous linear maps into a locally convex space that is bounded at each point is equicontinuous.\n",
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