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The conclusion of the part of the Peter-Weyl theorem stating that unitary representations of the group split as an orthogonal direct sum of finite dimensional ones is false if the group is not compact. There are non-compact groups which simply do not admit any finite dimensional representations, like the metaplectic group $Mp_{2p}$, which is the double cover of the symplectic group $\mathrm{Sp}_{2p}$, which has, on the other hand, irreducible infinite dimensional representations.