= Conjugation averaging on a compact group
{title2=$\mathcal Af(g)=\int_G f(hgh^{-1})\,dh$}
Normalized <Haar measure> makes this averaging a norm-one projection onto continuous <class functions>. Averaging a <trace> coefficient replaces its coefficient <endomorphism> by $\int\theta(h)^{-1}\alpha\theta(h)\,dh$. On an irreducible summand this average is $(\operatorname{tr}\alpha/\dim V)I$ by <Schur lemma>. It follows that averaging finite-dimensional <matrix coefficients> yields finite linear combinations of <irreducible characters>. Together with the <Peter-Weyl theorem>, this proves uniform density of those characters among continuous <class functions>.
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