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  "body": "\n## Definition\n\nLet $G$ be a [[topological group|92700QAB5Y]] and let $H$ be a complex [[Hilbert space|HM2HHE3J51]].\nLet $U(H)$ be the group of [[unitary operators|MM0EJ2GDE4]] on $H$.\nA *unitary representation* of $G$ on $H$ is a map $\\pi \\colon G \\to U(H)$ such that:\n\n1. $\\pi$ is a group homomorphism: $\\pi(gh) = \\pi(g)\\pi(h)$ for all $g, h \\in G$;\n2. $\\pi$ is strongly continuous: for every $v \\in H$, the map $G \\to H$, $g \\mapsto \\pi(g)v$, is continuous.\n\n## Remarks\n\nClause 2 says that $\\pi$ is continuous for the [[strong operator topology|KHJ2G2R5G7]] on $U(H)$.\nIt is equivalent to weak continuity, the continuity of $g \\mapsto \\ip{\\pi(g)u}{v}$ for all $u, v \\in H$.\n\nExample: $G = (\\R, +)$ on $H = L^2(\\R)$ by $(\\pi(t)f)(x) = f(x - t)$.\nTranslation preserves $\\int \\abs{f}^2$ and has inverse $\\pi(-t)$, so each $\\pi(t)$ is unitary, and $\\pi(s + t) = \\pi(s)\\pi(t)$.\nFor continuity, $\\norm{\\pi(t)f - f}_2 \\to 0$ as $t \\to 0$ holds for compactly supported continuous $f$ by uniform continuity, and then for all $f \\in L^2(\\R)$ since these are dense and each $\\pi(t)$ has norm $1$.\n\nNon-example: let $\\ell^2(\\R)$ be the Hilbert space of square-summable functions on the set $\\R$, with orthonormal basis $(\\delta_s)_{s \\in \\R}$, and let $\\pi(t)\\delta_s = \\delta_{s + t}$.\nEach $\\pi(t)$ is unitary and $\\pi$ is a homomorphism, but clause 2 fails for the usual topology of $\\R$.\nIndeed $\\norm{\\pi(t)\\delta_0 - \\delta_0} = \\sqrt{2}$ for every $t \\ne 0$, so $\\pi(t)\\delta_0$ does not tend to $\\delta_0$ as $t \\to 0$.\n\nTwo unitary representations $\\pi_1$ on $H_1$ and $\\pi_2$ on $H_2$ are *unitarily equivalent* if there is a unitary $T \\colon H_1 \\to H_2$ with $T\\pi_1(g) = \\pi_2(g)T$ for all $g \\in G$.\nA unitary representation is *irreducible* if $H \\ne 0$ and the only closed subspaces invariant under all $\\pi(g)$ are $0$ and $H$.\n",
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  "name": "Unitary representation",
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