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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $E$ be a Hausdorff [[barrelled space|ZE6DXEJB8H]] over $\\mathbb{K}$.\n$E$ is a *Montel space* if every [[closed|0ZCK8DQYN1]] and [[bounded|3BAJ8ZTVPX]] subset of $E$ is [[compact|0R6EWFKRCF]].\n\n## Remarks\n\nA locally convex space in which closed bounded sets are compact, without the barrelled condition, is called *semi-Montel*.\nExample: for an open set $\\Omega \\subseteq \\C$, the space $H(\\Omega)$ of holomorphic functions with the seminorms $p_K(f) = \\sup_{z \\in K} \\abs{f(z)}$, $K \\subseteq \\Omega$ compact, is a Montel space.\nIt is a Fréchet space, hence barrelled.\nA bounded set is a family that is uniformly bounded on each compact set, so [[Montel's theorem|ZVFVBVJMCB]] gives the compactness of closed bounded sets.\nNon-example: an infinite-dimensional Banach space is barrelled, but its closed unit ball is closed and bounded and not compact, by Riesz's lemma.\nEvery Montel space is reflexive, and its strong dual is again a Montel space.\n",
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