Lie Groups Beyond an Introduction
《Lie Groups Beyond an Introduction》 · 第 245 页 · Chapter IV，Section 3「Peter-Weyl Theorem」，Theorem 4.20
Theorem 4.20 (Peter-Weyl Theorem). If $G$ is a compact group, then the linear span of all matrix coefficients for all finite-dimensional irreducible unitary representations of $G$ is dense in $L^2(G)$.