Consider part II of the Peter-Weyl Theorem (see this wikipedia page for more information):
Let $\rho$ be a unitary representation of a compact $G$ on a complex Hilbert space $H$. Then $H$ splits into an orthogonal direct sum of irreducible finite-dimensional unitary representations of $G$.
What does the conclusion have to with $\rho$? The conclusion seems to have nothing to do with the given unitary representation $\rho$. I always thought Peter-Weyl said that $\rho$ splits into a orthogonal direct sum of finite dimensional unitary representations, not the Hilbert space.