# Difference between revisions of "Peter-Weyl theorem"

A theorem on the approximation of functions on a compact topological group by means of representation functions (cf. Representation function). Let run through a family of representatives of all equivalence classes for the irreducible continuous unitary representations of a compact group (cf. Representation of a topological group). Let be the dimension of a representation and let be its matrix elements in some orthonormal basis. The Peter–Weyl theorem asserts that the functions form an orthonormal basis in the space of square-summable functions with respect to the Haar measure on (the measure of the entire group is taken to be 1). The algebra of all complex-valued representation functions on , which coincides with the set of finite linear combinations of the functions , , is uniformly dense in the space of all continuous complex-valued functions in .

If is the rotation group for the plane, this assertion coincides with an elementary theorem on approximating periodic continuous functions by trigonometric polynomials.

A consequence of the Peter–Weyl theorem is that the set of linear combinations of characters of the irreducible representations of is dense in the algebra of all continuous functions on , constant on classes of conjugate elements. Another consequence is that for any element , , there is an irreducible continuous representation of such that ; if, on the other hand, is a compact Lie group, then has a faithful linear representation.

The Peter–Weyl theorem implies also the following more general assertion , . Suppose one is given a continuous linear representation of a compact group in a Fréchet space . Then the subspace of representation elements of is dense in . Here an element is called a representation, or spherical or almost-invariant, element if the orbit generates a finite-dimensional subspace in . This is applicable in particular to the case where is the space of sections of a certain smoothness class of smooth vector -fibrations, for example, the space of tensor fields of a certain type or given smoothness class on a smooth manifold with a smooth action of a compact Lie group .

The Peter–Weyl theorem was proved in 1927 by F. Peter and H. Weyl .

How to Cite This Entry:
Peter-Weyl theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Peter-Weyl_theorem&oldid=17032
This article was adapted from an original article by A.L. OnishchikA.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article