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A theorem on the approximation of functions on a compact [[Topological group|topological group]] by means of representation functions (cf. [[Representation function|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|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 [[#References|[5]]], [[#References|[6]]]. 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 .

====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  F. Peter,   H. Weyl,   "Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe"  ''Math. Ann.'' , '''97'''  (1927)  pp. 737–755</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  F. Peter,   H. Weyl,   "Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe" , ''Gesammelte Abhandlungen H. Weyl'' , '''III''' :  73 , Springer  (1968)  pp. 58–75</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Pontryagin,   "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Hewitt,   K.A. Ross,   "Abstract harmonic analysis" , '''1–2''' , Springer  (1979)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  C. Chevalley,   "Theory of Lie groups" , '''1''' , Princeton Univ. Press  (1946)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  R.S. Palais,   T.E. Stewart,   "The cohomology of differentiable transformation groups"  ''Amer. J. Math.'' , '''83''' :  4  (1961)  pp. 623–644</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  G.D. Mostow,   "Cohomology of topological groups and solvmanifolds"  ''Ann. of Math.'' , '''73''' :  1  (1961)  pp. 20–48</TD></TR></table>

====Comments====
A representation element is now usually called a  $  G $-
finite element.

The statement that the algebra of complex-valued representation functions is uniformly dense in the algebra of continuous functions on  $  G $
is known as the Weyl approximation theorem. The Peter–Weyl theorem gives a complete description of the (left or right) regular representation in terms of its irreducible components. In particular, each irreducible component occurs with a multiplicity equal to its dimension, cf. [[#References|[a1]]], Chapt. 7, §2. There exists a generalized Peter–Weyl theorem for unimodular Lie groups, cf. [[#References|[a1]]], Chapt. 14, §2. The description of  $  L _ {2} (G) $(
and the other unitary representations) in terms of the irreducible representations, including the fact that the irreducible unitary representations are finite dimensional, is known as Peter–Weyl theory, cf. e.g. [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.O. Barut,   R. Raçzka,   "Theory of group representations and applications" , '''1–2''' , PWN  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Wawrzyńczyk,   "Group representations and special functions" , Reidel  (1984)  pp. Sect. 4.4</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.W. Knapp,   "Representation theory of semisimple groups" , Princeton Univ. Press  (1988)  pp. 17</TD></TR></table>