Given a unitary representation of a group on the vector space $V$, in which a scalar product $\langle\ldots \mid \ldotsangle$ is defined, and given a submodule $W$, and its orthogonal complement $W_{\perp}=\{\vec{v} \in V \mid\langle\vec{v} \mid \vec{w}angle=0 \forall \vec{w} \in W\}$, show that $V \cong W \oplus W_{\perp}$.