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  "body": "\n## Summary\n\nA Montel space is a [[barrelled space|ZE6DXEJB8H]] in which every closed [[bounded set|3BAJ8ZTVPX]] is compact.\nThis is a Heine–Borel property, which fails in every infinite-dimensional normed space.\nThe name comes from [[Montel's theorem|ZVFVBVJMCB]]: the space of holomorphic functions on an open subset of $\\C$ is a Montel space.\nSpaces of smooth functions and of test functions used in distribution theory are Montel spaces too, and Montel spaces are reflexive.\n\n## Explanations\n\n### L1 human\n\nIn $\\R^n$, a closed and bounded set is compact: every sequence in it has a convergent subsequence.\nIn an infinite-dimensional Banach space this fails, already for the closed unit ball.\nMontel spaces are infinite-dimensional spaces where it holds again.\nFor holomorphic functions, a family that is bounded on each compact set has a subsequence that converges uniformly on compact sets.\nThat is the content of Montel's theorem, and it is why these spaces carry Montel's name.\n",
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