= Haar averaging produces invariant complements
{c}
{title2=$V=W\oplus W^\perp$}
For a finite-dimensional continuous complex representation $R$ of a <compact group>, normalize <Haar measure> to mass one and average any positive-definite <Hermitian inner product>:
$$
\langle v,w\rangle_K=\int_K\langle R(g)v,R(g)w\rangle_0\,d\mu(g).
$$
The resulting inner product is positive-definite and invariant. If $W$ is an <invariant subspace>, then $W^\perp$ is invariant as well, since $\langle R(g)v,w\rangle_K=\langle v,R(g^{-1})w\rangle_K$. Thus $V=W\oplus W^\perp$. Restriction to a <compact real form>, followed by <integration of a Lie-algebra representation>, proves the <Weyl complete reducibility theorem>.
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