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  "body": "\n## Statement\n\nLet $G$ be a [[compact group|EVHCE5VR6Y]] with normalised [[Haar measure|VY0HXD3GC4]].\nLet $\\widehat{G}$ be a set of representatives of the isomorphism classes of continuous [[irreducible|FX3NEFYFFN]] [[unitary representations|74BPYJNZTM]] of $G$.\nEach $\\pi \\in \\widehat{G}$ is finite-dimensional; let $d_\\pi$ be its dimension, and let $\\pi_{ij}(g)$ be its matrix entries in an orthonormal basis.\nA matrix coefficient of $\\pi$ is a function $g \\mapsto \\ip{\\pi(g)u}{v}$ for vectors $u, v$ of its space.\n\n1. The matrix coefficients of the finite-dimensional continuous unitary representations of $G$ span a dense subspace of $C(G)$, with the supremum norm.\n2. The functions $\\sqrt{d_\\pi}\\, \\pi_{ij}$, for $\\pi \\in \\widehat{G}$ and $1 \\le i, j \\le d_\\pi$, form an orthonormal basis of the [[Hilbert space|HM2HHE3J51]] $L^2(G)$.\n3. Every continuous unitary representation of $G$ on a Hilbert space is a Hilbert direct sum of finite-dimensional irreducible representations.\n\n## Remarks\n\nThe proof applies the spectral theorem to self-adjoint convolution operators $f \\mapsto k * f$ on $L^2(G)$ with continuous kernels $k$.\nSuch operators are compact and commute with right translations, so their eigenspaces for nonzero eigenvalues are finite-dimensional invariant subspaces.\nLetting $k$ run through an approximate identity shows that these subspaces span a dense subspace, which gives parts 1 and 2.\nPart 3 follows by applying the same idea to a general representation.\nOrthonormality in part 2 is the [[Schur orthogonality for compact groups|ETNDTECY97]].\nExample: for $G = \\mathbb{T}$ the irreducible representations are the characters $z \\mapsto z^n$, $n \\in \\Z$, and part 2 says that the functions $e^{in\\theta}$ form an orthonormal basis of $L^2(\\mathbb{T})$, the completeness of Fourier series.\nFor a finite group, the theorem reduces to the decomposition of the regular representation, with $\\sum_\\pi d_\\pi^2 = \\abs{G}$.\n\n## Explanations\n\n### L1 human\n\nOn the circle, every reasonable function is a sum of waves $e^{in\\theta}$.\nThe Peter–Weyl theorem says the same for every compact group, where the waves are replaced by the entries of the matrices of irreducible representations.\nThese matrix entries are like a complete set of perpendicular axes for the space of functions on the group.\n\n### L2 human\n\nPart 1 implies that every compact Lie group has a faithful finite-dimensional representation, so it is isomorphic to a closed subgroup of some unitary group $U(n)$.\nPart 3 reduces the study of unitary representations of compact groups to finite-dimensional ones, where [[complete reducibility|N6JC6KNXJJ]] and characters are available.\nIn terms of the left and right regular actions, part 2 says that $L^2(G)$ is the Hilbert direct sum over $\\pi \\in \\widehat{G}$ of the spaces $\\End(V_\\pi)$, each of dimension $d_\\pi^2$.\n",
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