Peter–Weyl theorem

Remarks

The proof applies the spectral theorem to self-adjoint convolution operators $f \mapsto k * f$ on $L^2(G)$ with continuous kernels $k$. Such operators are compact and commute with right translations, so their eigenspaces for nonzero eigenvalues are finite-dimensional invariant subspaces. Letting $k$ run through an approximate identity shows that these subspaces span a dense subspace, which gives parts 1 and 2. Part 3 follows by applying the same idea to a general representation. Orthonormality in part 2 is the Schur orthogonality for compact groups. Example: for $G = \mathbb{T}$ the irreducible representations are the characters $z \mapsto z^n$, $n \in \Z$, and part 2 says that the functions $e^{in\theta}$ form an orthonormal basis of $L^2(\mathbb{T})$, the completeness of Fourier series. For a finite group, the theorem reduces to the decomposition of the regular representation, with $\sum_\pi d_\pi^2 = \abs{G}$.

Going deeper

For a student

Part 1 implies that every compact Lie group has a faithful finite-dimensional representation, so it is isomorphic to a closed subgroup of some unitary group $U(n)$. Part 3 reduces the study of unitary representations of compact groups to finite-dimensional ones, where complete reducibility and characters are available. In terms of the left and right regular actions, part 2 says that $L^2(G)$ is the Hilbert direct sum over $\pi \in \widehat{G}$ of the spaces $\End(V_\pi)$, each of dimension $d_\pi^2$.