# Peter-Weyl theorem

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A theorem on the approximation of functions on a compact topological group by means of representation functions (cf. Representation function). Let $\pi$ run through a family $\Sigma$ of representatives of all equivalence classes for the irreducible continuous unitary representations of a compact group $G$( cf. Representation of a topological group). Let $\mathop{\rm dim} \pi$ be the dimension of a representation $\pi$ and let $u _ {ij} ^ {( \pi ) }$ be its matrix elements in some orthonormal basis. The Peter–Weyl theorem asserts that the functions

$$\sqrt { \mathop{\rm dim} \pi } u _ {ij} ^ {( \pi ) } \ ( \pi \in \Sigma )$$

form an orthonormal basis in the space $L _ {2} (G)$ of square-summable functions with respect to the Haar measure on $G$( the measure of the entire group is taken to be 1). The algebra of all complex-valued representation functions on $G$, which coincides with the set of finite linear combinations of the functions $u _ {ij} ^ {( \pi ) }$, $\pi \in \Sigma$, is uniformly dense in the space of all continuous complex-valued functions in $G$.

If $G=T$ 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 $G$ is dense in the algebra of all continuous functions on $G$, constant on classes of conjugate elements. Another consequence is that for any element $a \in G$, $a \neq e$, there is an irreducible continuous representation $\phi$ of $G$ such that $\phi (a) \neq e$; if, on the other hand, $G$ is a compact Lie group, then $G$ has a faithful linear representation.

The Peter–Weyl theorem implies also the following more general assertion , . Suppose one is given a continuous linear representation $\phi$ of a compact group $G$ in a Fréchet space $E$. Then the subspace of representation elements of $E$ is dense in $E$. Here an element $v \in E$ is called a representation, or spherical or almost-invariant, element if the orbit $\phi (G)v$ generates a finite-dimensional subspace in $E$. This is applicable in particular to the case where $E$ is the space of sections of a certain smoothness class of smooth vector $G$- 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 $G$.

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=44951
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