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  "body": "\n## Definition\n\nLet $G$ be a [[Lie group|7JBRGCVJRB]] and let $V$ be a finite-dimensional vector space over $\\R$ or $\\C$.\nA *representation* of $G$ on $V$ is a continuous group homomorphism $\\pi \\colon G \\to \\mathrm{GL}(V)$, where $\\mathrm{GL}(V)$ is the [[general linear group|0FCHE902D7]] of $V$.\n\nLet $V$ be an infinite-dimensional complex Hilbert or Banach space.\nA representation of $G$ on $V$ is a homomorphism $\\pi$ from $G$ to the group of bounded invertible operators on $V$.\nIt must be strongly continuous: $g \\mapsto \\pi(g) v$ is continuous from $G$ to $V$ for each $v \\in V$.\n\n## Remarks\n\nExample: $G = \\mathrm{SO}(3)$, $V = \\R^3$ and $\\pi(g) = g$.\nThe map is the inclusion of a closed subgroup of $\\mathrm{GL}_3(\\R)$, so it is a continuous homomorphism.\n\nNon-example: let $f \\colon \\R \\to \\R$ be an additive map that is not linear, which exists by choosing a basis of $\\R$ over $\\Q$.\nThen $\\pi(t) = e^{f(t)}$ is a homomorphism $\\R \\to \\mathrm{GL}_1(\\R)$, but it is not continuous, so it is not a representation of the Lie group $\\R$.\n\nA continuous homomorphism of Lie groups is automatically smooth.\nSo a finite-dimensional representation $\\pi$ has a differential $d\\pi$ at the identity, which is a representation of the Lie algebra of $G$.\nIf $G$ is connected, $d\\pi$ determines $\\pi$.\nA unitary representation is one where each $\\pi(g)$ preserves a fixed inner product on $V$.\n",
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