# Unitary Representations and the Peter-Weyl Theorem

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.
• Look at $Gv = \overline{span}(\{ \rho(g)v, g \in G\})$. If $Gv$ is a non-trivial subspace, since $\rho(g)$ is unitary $\rho(g)Gv^\perp$ is orthogonal to $\rho(g)Gv$ and $\rho = (\rho|_{Gv},\rho|_{Gv^\perp})$ – reuns Mar 21 at 2:32
When it says "$$H$$ splits" it means $$H$$ as a Hilbert space with a unitary action of $$G$$. In other words, the direct sum decomposition is indeed a decomposition of the representation $$\rho$$, not just of the Hilbert space $$H$$.