Suppose $(E,\la.,.\ra)$ is an inner product space and $u\in\Hom(E)$, such that $\im u$ is non-degenerated. Then there is an orthogonal decomposition: $E=\im u\oplus\ker u^*$. Moreover $\im u$ is non-degenerated iff $\ker u^*$ is non-degenerated.
By exam: $(\im u)^\perp=u^{*-1}(0)=\ker u^*$ and since $\im u$ is non-degenerate we conclude by lemma that $E=\im u\oplus\ker u^*$ is an orthogonal decomposition.