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  "body": "\n## Statement\n\nLet $G$ and $H$ be real [[Lie groups|7JBRGCVJRB]].\nLet $\\varphi \\colon G \\to H$ be a [[continuous group homomorphism|9J5PQMG3HJ]].\nThen $\\varphi$ is smooth; in fact it is real-analytic.\n\n## Remarks\n\nThe graph $\\Gamma = \\set{(g, \\varphi(g)) : g \\in G}$ is a subgroup of $G \\times H$, and it is closed since $\\varphi$ is continuous.\nBy the [[closed subgroup theorem|3NJGY8QSTE]], $\\Gamma$ is an embedded Lie subgroup.\nThe projection $\\Gamma \\to G$ is a bijective smooth homomorphism; such a map has constant rank, so it is a diffeomorphism, and $\\varphi$ is its inverse followed by the smooth projection $\\Gamma \\to H$.\nExample: a continuous homomorphism $\\varphi \\colon \\R \\to \\R$ satisfies $\\varphi(x + y) = \\varphi(x) + \\varphi(y)$ and is therefore $\\varphi(x) = cx$, which is smooth.\nContinuity is necessary: using a basis of $\\R$ as a vector space over $\\Q$, there are additive maps $\\R \\to \\R$ that are not continuous, hence not smooth.\n\n## Explanations\n\n### L1 human\n\nFor a map between Lie groups that respects the group laws, mere continuity already forces smoothness.\nSo the smooth structure of a Lie group adds nothing beyond its topology and group law.\n\n### L2 human\n\nA direct consequence is that a topological group carries at most one smooth structure making it a Lie group.\nIndeed, the identity map between two such structures is a continuous homomorphism, so it is a diffeomorphism.\nThis is why one can speak of \"the\" Lie group structure of a group such as $SO(n)$.\nIt also means continuous representations of Lie groups on finite-dimensional spaces are automatically smooth, so their differentials give representations of the Lie algebra.\nThe question of which topological groups admit a Lie group structure at all is Hilbert's fifth problem.\n",
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  "name": "Cartan's theorem on continuous homomorphisms",
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