Proposition 32.1.9.label Let $G$ be a locally compact group, $(H, \pi)$ be a unitary representation of $G$, and $M \subset H$ be an invariant subspace, then:

1. (1)

   $M^{\perp}$ is also an invariant subspace.

2. (2)

   $(H, \pi)$ is the direct sum of $(M, \pi_{M})$ and $(M^{\perp}, \pi_{M^\perp})$.

Moreover,

1. (4)

   $(H, \pi)$ is a direct sum of cyclic representations.