Editing Peter–Weyl theorem (section)

==Decomposition of square-integrable functions==
To state the third and final part of the theorem, there is a natural Hilbert space over ''G'' consisting of [[square-integrable function]]s, [[Lp space|<math>L^2(G)</math>]]; this makes sense because the [[Haar measure]] exists on ''G''. The group ''G'' has a [[unitary representation]] ρ on <math>L^2(G)</math> given by [[Group action (mathematics)|acting]] on the left, via

:<math>\rho(h)f(g) = f(h^{-1}g).</math>

The final statement of the Peter–Weyl theorem {{harv|Knapp|1986|loc=Theorem 1.12}} gives an explicit [[orthonormal basis]] of <math>L^2(G)</math>.  Roughly it asserts that the matrix coefficients for ''G'', suitably renormalized, are an [[orthonormal basis]] of ''L''<sup>2</sup>(''G'').  In particular, <math>L^2(G)</math> decomposes into an orthogonal direct sum of all the irreducible unitary representations, in which the multiplicity of each irreducible representation is equal to its degree (that is, the dimension of the underlying space of the representation).  Thus,

:<math>L^2(G) = \underset{\pi\in\Sigma}{\widehat{\bigoplus}} E_\pi^{\oplus\dim E_\pi}</math>

where Σ denotes the set of (isomorphism classes of) irreducible unitary representations of ''G'', and the summation denotes the [[closure (topology)|closure]] of the direct sum of the total spaces ''E''<sub>π</sub> of the representations π.

We may also regard <math>L^2(G)</math> as a representation of the direct product group <math>G\times G</math>, with the two factors acting by translation on the left and the right, respectively. Fix a representation <math>(\pi,E_\pi)</math> of <math>G</math>. The space of matrix coefficients for the representation may be identified with <math>\operatorname{End}(E_\pi)</math>, the space of linear maps of <math>E_\pi</math> to itself. The natural left and right action of <math>G\times G</math> on the matrix coefficients corresponds to the action on <math>\operatorname{End}(E_\pi)</math> given by
:<math>(g,h)\cdot A=\pi(g)A\pi(h)^{-1}.</math>
Then we may decompose <math>L^2(G)</math> as unitary representation of <math>G\times G</math> in the form

:<math>L^2(G) = \underset{\pi\in\Sigma}{\widehat{\bigoplus}} E_\pi\otimes E_\pi^*.</math>

Finally, we may form an orthonormal basis for <math>L^2(G)</math> as follows. Suppose that a representative π is chosen for each isomorphism class of irreducible unitary representation, and denote the collection of all such π by Σ. Let <math>\scriptstyle{u_{ij}^{(\pi)}}</math> be the matrix coefficients of π in an orthonormal basis, in other words

:<math>u^{(\pi)}_{ij}(g) = \langle \pi(g)e_j, e_i\rangle.</math>

for each ''g''&nbsp;∈&nbsp;''G''.  Finally, let ''d''<sup>(π)</sup> be the degree of the representation π. The theorem now asserts that the set of functions

:<math>\left\{\sqrt{d^{(\pi)}}u^{(\pi)}_{ij}\mid\, \pi\in\Sigma,\,\, 1\le i,j\le d^{(\pi)}\right\}</math>

is an orthonormal basis of <math>L^2(G).</math>

===Restriction to class functions===
A function <math>f</math> on ''G'' is called a ''class function'' if <math>f(hgh^{-1})=f(g)</math> for all <math>g</math> and <math>h</math> in ''G''. The space of square-integrable class functions forms a closed subspace of <math>L^2(G)</math>, and therefore a Hilbert space in its own right. Within the space of matrix coefficients for a fixed representation <math>\pi</math> is the [[Character (mathematics)|character]] <math>\chi_\pi</math> of <math>\pi</math>, defined by
:<math>\chi_\pi(g)=\operatorname{trace}(\pi(g)).</math>
In the notation above, the character is the sum of the diagonal matrix coefficients:
:<math>\chi_\pi=\sum_{i=1}^{d^{(\pi)}}u_{ii}^{(\pi)}.</math>
An important consequence of the preceding result is the following:
:'''Theorem''': The characters of the irreducible representations of ''G'' form a Hilbert basis for the space of square-integrable class functions on ''G''.
This result plays an important part in Weyl's classification of the [[Compact group#Representation theory of a connected compact Lie group|representations of a connected compact Lie group]].<ref>{{harvnb|Hall|2015}} Chapter 12</ref>

===An example: U(1)===
A simple but helpful example is the case of the group of complex numbers of magnitude 1, <math>G=S^1</math>. In this case, the irreducible representations are one-dimensional and given by
:<math>\pi_n(e^{i\theta})=e^{in\theta}.</math>
There is then a single matrix coefficient for each representation, the function
:<math>u_n(e^{i\theta})=e^{in\theta}.</math>
The last part of the Peter–Weyl theorem then asserts in this case that these functions form an orthonormal basis for <math>L^2(S^1)</math>. In this case, the theorem is simply a standard result from the theory of Fourier series.

For any compact group ''G'', we can regard the decomposition of <math>L^2(G)</math> in terms of matrix coefficients as a generalization of the theory of Fourier series. Indeed, this decomposition is often referred to as a Fourier series.

===An example: SU(2)===
We use the standard representation of the group [[Special unitary group#The group SU(2)|SU(2)]] as
:<math> \operatorname{SU}(2) = \left \{ \begin{pmatrix} \alpha&-\overline{\beta}\\ \beta & \overline{\alpha} \end{pmatrix}: \ \ \alpha,\beta\in\mathbb{C},\, |\alpha|^2 + |\beta|^2 = 1\right \}  ~,</math>
Thus, SU(2) is represented as the [[3-sphere]] <math>S^3</math> sitting inside <math>\mathbb{C}^2=\mathbb{R}^4</math>.
The irreducible representations of SU(2), meanwhile, are labeled by a non-negative integer <math>m</math> and can be realized as the natural action of SU(2) on the space of [[homogeneous polynomials]] of degree <math>m</math> in two complex variables.<ref>{{harvnb|Hall|2015}} Example 4.10</ref> The matrix coefficients of the <math>m</math>th representation are [[Spherical harmonics#Higher dimensions|hyperspherical harmonics]] of degree <math>m</math>, that is, the restrictions to <math>S^3</math> of homogeneous harmonic polynomials of degree <math>m</math> in <math>\alpha</math> and <math>\beta</math>. The key to verifying this claim is to compute that for any two complex numbers <math>z_1</math> and <math>z_2</math>, the function 
:<math>(\alpha,\beta)\mapsto (z_1\alpha+z_2\beta)^m</math> 
is harmonic as a function of <math>(\alpha,\beta)\in\mathbb{C}^2=\mathbb{R}^4</math>.

In this case, finding an orthonormal basis for <math>L^2(\operatorname{SU}(2))=L^2(S^3)</math> consisting of matrix coefficients amounts to finding an orthonormal basis consisting of hyperspherical harmonics, which is a standard construction in analysis on spheres.