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  "body": "\n## Summary\n\nA normal family is a family of holomorphic functions in which every sequence has a subsequence that converges uniformly on compact sets, see the [[definition|QWSD2JDRP0]].\nIt is the complex-analytic version of a relatively compact set of functions.\n[[Montel's theorem|ZVFVBVJMCB]] gives simple tests for normality.\nNormal families are a key tool in the proof of the [[Riemann mapping theorem|02WHMEG5ZR]] and in complex dynamics.\n\n## Explanations\n\n### L1 human\n\nIn a finite set, every sequence repeats some element infinitely often.\nA normal family is a family of functions with a similar property: from any sequence one can pick a subsequence that settles down to a limit.\nThe limit is reached uniformly on each closed and bounded piece of the domain, though maybe not on the whole domain.\nFor example, the powers $z, z^2, z^3, \\dots$ on the unit disc form a normal family, since they tend to $0$ on each smaller disc.\n",
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