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Peter-Weyl Theorem is concerned with expressing $L^2(G)$ as closure of direct sum of subspace generated by irreducible unitary representation. Every irreducible representation on a compact group is finite dimensional. Why $L^2(G)$ has exactly subspace with dimension number of copies of corresponding irreducible unitary representation? I have problem with investigating why there are no more copies !!

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If not, consider the restriction of regular representation to the complement of the direct sum of the subspaces corresponding to irreducible representations will have finite dimensional unitary subrepresentation which lead to contradiction.

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