Irreducible unitary representations of compact groups are finite dimensional

Given: AC, a compact Hausdorff group $K$, and an irreducible strongly continuous unitary representation $\pi$ of $K$ on a nonzero complex Hilbert space $H$.

Under AC there is a normalized Haar probability measure on $K$, and for every $\xi \in H$ with $\xi \mathrm{\ne }0$ the rank-one average $Q_{\xi }:={\int }_K\pi (k)R_{\xi }\pi (k{)}^{-1}d\mu (k)$, where $R_{\xi }x=⟨x,\xi ⟩\xi$, is a well-defined bounded operator on $H$ that is self-adjoint, nonzero, compact (Compact linear operator), and satisfies $\pi (g)Q_{\xi }=Q_{\xi }\pi (g)$ for every $g\in K$ (A positive rank-one Haar average is a nonzero compact intertwiner, Normalized Haar probability on a compact group, A bounded linear operator between normed spaces).

Schur's lemma: for an irreducible strongly continuous unitary representation of a topological group on a nonzero complex Hilbert space, every bounded operator commuting with the representation is a scalar multiple of the identity (Schur lemma for complex unitary representations).

If a nonzero scalar multiple $c{I}_H$ of the identity of a real or complex Hilbert space is a compact operator, then $H$ admits an ordered basis of finite length; this implication is choice free (A nonzero compact scalar identity forces finite dimension).

$\pi$ is a group homomorphism, and $I_H\mathrm{\ne }0$ because $H\mathrm{\ne }\{0\}$ (Strongly continuous unitary representations, invariant linear subspaces and intertwiners).