Theorem 1 (Peter–Weyl)

Let G be a compact topological group. The biregular representation $\boldsymbol{\varrho }_G$ of G is the Hilbert sum of the subrepresentations $\boldsymbol{\varrho }_G(\pi )$ for $\pi \in \widehat{G}$.

The spaces $\boldsymbol{\varrho }_G(\pi )$ are pairwise orthogonal; they are irreducible subrepresentations of the biregular representation of G (prop. 5) which are pairwise non-isomorphic (prop. 3 of V, p. 461). The theorem then results from the fact that the sum Θ(G) of the spaces $\boldsymbol{\varrho }_G(\pi )$ for $\pi$ ranging over $\widehat{G}$ is dense in $L^2(G)$ (prop. 4).