= Compact averaging of a rank-one operator
{title2=$K_\xi=\int_G U_gR_{\xi,\xi}U_g^*\,dg$}
For a strongly continuous <unitary representation> of a <compact group>, this <Bochner integral> exists in the <Hilbert-Schmidt norm> and is a positive <compact operator> commuting with the representation. It is nonzero if $\xi\ne0$, because $\langle K_\xi\xi,\xi\rangle=\int_G|\langle U_g\xi,\xi\rangle|^2\,dg>0$. A nonzero <eigenspace> is finite-dimensional and invariant. Applying this on <orthogonal complements> yields an <orthogonal direct sum> of finite-dimensional <irreducible representations>, with no separability assumption on the represented <Hilbert space>.
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