Peter-Weyl Theorem

The Peter-Weyl Theorem is a foundational result of harmonic analysis for compact topological groups, not necessarily abelian, first proved by Hermann Weyl together with his student Fritz Peter in 1927. It has three parts: the matrix coefficients of the group's irreducible representations are dense in the space of continuous functions on the group and in the space of square-integrable functions on it; every unitary representation of the group decomposes completely into irreducible pieces; and the group's regular representation on its own square-integrable functions decomposes as the direct sum of all its irreducible unitary representations, with the matrix coefficients of those representations forming an orthonormal basis. When the group is the circle group of unit complex numbers, this last statement reduces to the ordinary theory of Fourier series.