I am following Folland's A Course in Abstract Harmonic Analysis, currently looking at unitary representations of compact groups (Chapter 5).
Let $\rho$ be a unitary representation of compact $G$ on $\mathcal{H}_{\rho}$, and $[\pi],[\pi']\in\hat{G}$.
From Proposition 5.3, we know that for an invariant subspace $M_{\pi}$ of representation space $\mathcal{H}_{\rho}, M_\pi'$ is the orthogonal complement of $M_\pi$ if $\pi$ and $\pi$ are not unitarily equivalent.
Then a note after this proposition says we can write the space $H_\rho$ as a direct sum of $M_\pi$ over the representatives of the equivalence classes $[\pi]$.
But, since $M_\pi'$ is the orthogonal complement of $M_\pi$, do we not have $H_\rho$ as a direct sum of $M_\pi$ and $M_\pi'$ only?
While $\rho$ might have more than two non-unitarily equivalent subspaces, it does not seem possible (or sensible) to write $\rho$ as a direct sum over all of them given that the entire representation space of it can be expressed using only one of these invariant substances, via a direct sum of the subspace and its orthogonal complement?
I hope my question is clear. Thanks.