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The Peter-Weyl Theorem states that:

The '''Peter-Weyl Theorem''' states that:

Let G be a [[compact]] [[topological group]]. For any G-invariant subspace V in L2(G), where G acts on the left, we consider the image of G in GL(V). It is closed, since G is compact, and a subgroup of the [[Lie group]] GL(V). It follows by a basic theorem (of Élie Cartan) that the image of G is also a Lie group. From this follows that:

Let G be a [[compact]] [[topological group]]. For any G-invariant subspace V in L2(G), where G acts on the left, we consider the image of G in GL(V). It is closed, since G is compact, and a subgroup of the [[Lie group]] GL(V). It follows by a basic theorem (of Élie Cartan) that the image of G is also a Lie group. From this follows that: