Editing Peter–Weyl theorem (section)

==Decomposition of a unitary representation==
The second part of the theorem gives the existence of a decomposition of a [[unitary representation]] of ''G'' into finite-dimensional representations. Now, intuitively groups were conceived as rotations on geometric objects, so it is only natural to study representations which essentially arise from continuous '''[[Group action (mathematics)|actions]]''' on Hilbert spaces. (For those who were first introduced to dual groups consisting of characters which are the continuous homomorphisms into the [[circle group]], this approach is similar except that the circle group is (ultimately) generalised to the group of unitary operators on a given Hilbert space.)

Let ''G'' be a topological group and ''H'' a complex Hilbert space.

A continuous linear action ∗ : ''G'' × ''H'' → ''H'', gives rise to a continuous map ρ<sub>∗</sub> : ''G'' → ''H''<sup>''H''</sup> (functions from ''H'' to ''H'' with the [[Strong operator topology|strong topology]]) defined by: ρ<sub>∗</sub>(''g'')(''v'') = ''∗(g,v)''. This map is clearly a homomorphism from ''G'' into GL(''H''), the bounded linear operators on ''H''. Conversely, given such a map, we can uniquely recover the action in the obvious way.

Thus we define the '''representations of ''G'' on a Hilbert space ''H''''' to be those [[group homomorphisms]], ρ, which arise from continuous actions of ''G'' on ''H''. We say that a representation ρ is '''unitary''' if ρ(''g'') is a [[unitary operator]] for all ''g''&nbsp;∈&nbsp;''G''; i.e., <math>\langle \rho(g)v,\rho(g)w\rangle = \langle v,w\rangle</math> for all ''v'', ''w''&nbsp;∈&nbsp;''H''. (I.e. it is unitary if ρ : ''G'' → U(''H''). Notice how this generalises the special case of the one-dimensional Hilbert space, where U('''C''') is just the circle group.)

Given these definitions, we can state the second part of the Peter–Weyl theorem {{harv|Knapp|1986|loc=Theorem 1.12}}:

<blockquote>'''Peter–Weyl Theorem (Part II).''' Let ρ be a unitary representation of a compact group ''G'' on a complex Hilbert space ''H''.  Then ''H'' splits into an orthogonal [[direct sum of representations|direct sum]] of irreducible finite-dimensional unitary representations of ''G''.</blockquote>