Peter–Weyl theorem - Wikipedia

Abstraction: Harmonic analysis decomposition theorem for compact topological groups

Key points:

• Three-part theorem (Peter & Weyl, 1927): (1) matrix coefficients of irreducible representations are dense in C(G); (2) every unitary representation of a compact group decomposes into an orthogonal direct sum of finite-dimensional irreducible unitary representations; (3) the regular representation on L2(G) decomposes as a direct sum of all irreducibles, with multiplicity equal to their degree
• Matrix coefficients of irreducible unitary representations, renormalized, form an orthonormal basis of L2(G)
• Generalizes Fourier series: for the circle group U(1), the theorem reduces to standard Fourier series theory
• Key corollary: every compact Lie group has a faithful finite-dimensional representation and is isomorphic to a closed subgroup of GL(n)
• Characters of irreducible representations form a Hilbert basis for square-integrable class functions; foundational for Weyl's classification of compact Lie group representations

Connections: Hermann Weyl · Fritz Peter · Representation Theory · Harmonic Analysis · Group Theory

Source: http://en.wikipedia.org/wiki/Peter%E2%80%93Weyl_theorem