Editing Peter–Weyl theorem (section)

===Structure of compact topological groups===
From the Peter–Weyl theorem, one can deduce a significant general structure theorem. Let ''G'' be a compact topological group, which we assume [[Hausdorff space|Hausdorff]]. For any finite-dimensional ''G''-invariant subspace ''V'' in ''L''<sup>2</sup>(''G''), where ''G'' [[Group action (mathematics)|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 [[Closed subgroup theorem|theorem]] of [[Élie Cartan]] that the image of ''G'' is a Lie group also.

If we now take the [[Limit (category theory)|limit]] (in the sense of [[category theory]]) over all such spaces ''V'', we get a result about ''G'': Because ''G'' acts faithfully on ''L''<sup>2</sup>(''G''), ''G'' is an ''inverse limit of Lie groups''. It may of course not itself be a Lie group: it may for example be a [[profinite group]].