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  "body": "\n## Statement\n\nLet $G$ be a [[compact group|EVHCE5VR6Y]], let $V$ be a finite-dimensional complex vector space, and let $\\rho \\colon G \\to GL(V)$ be a continuous [[representation|WYGRXCG2RN]].\nThen:\n\n1. there is an [[inner product|0NB0AWGJBW]] on $V$ such that $\\ip{\\rho(g)u}{\\rho(g)v} = \\ip{u}{v}$ for all $g \\in G$ and $u, v \\in V$;\n2. $V$ is a direct sum $V = V_1 \\oplus \\cdots \\oplus V_k$ of [[irreducible|FX3NEFYFFN]] subrepresentations.\n\n## Remarks\n\nFor part 1, take any inner product $\\ip{\\cdot}{\\cdot}_0$ on $V$ and average it with the normalised [[Haar measure|VY0HXD3GC4]]: $\\ip{u}{v} = \\int_G \\ip{\\rho(g)u}{\\rho(g)v}_0 \\, dg$.\nInvariance of the Haar measure makes this inner product $G$-invariant, and it is positive definite because the integrand is positive when $u = v \\ne 0$.\nFor part 2, the orthogonal complement of an [[invariant subspace|0ACX5QT8TE]] for an invariant inner product is again invariant, and induction on $\\dim V$ finishes the proof.\nFor a finite group the Haar integral is the average over the group, and the theorem is Maschke's theorem over $\\C$.\nCompactness is needed: the representation of $\\R$ on $\\C^2$ given by $t \\mapsto \\begin{pmatrix} 1 \u0026 t \\\\ 0 \u0026 1 \\end{pmatrix}$ has the invariant line $\\C e_1$.\nIt has no invariant complement, since for $t \\ne 0$ the only eigenvectors of the matrix are the multiples of $e_1$.\nThe same proof works for real representations, with a real inner product.\n\n## Explanations\n\n### L1 human\n\nA representation of a compact group always breaks into irreducible pieces that do not interact.\nThe reason is that one can average over the group, which is possible because the group is compact.\nAveraging gives a way of measuring lengths and angles that the group does not change, and then perpendicular pieces stay perpendicular.\n\n### L2 human\n\nThe theorem reduces the study of finite-dimensional representations of compact groups to irreducible ones, which are then classified by characters through the [[Schur orthogonality relations|ETNDTECY97]].\nFor compact connected Lie groups, the irreducible representations are described by highest weights and the [[Weyl character formula|230F5FXPMQ]].\n[[Weyl's unitary trick|DDQ2WBQ9GB]] transfers the theorem to complex semisimple Lie groups and Lie algebras, which are not compact.\nThe [[Peter–Weyl theorem|6VC9QE39XH]] extends the result to unitary representations on infinite-dimensional Hilbert spaces.\n",
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