Definition 2

Let $E$ and $F$ be two hilbertian spaces and $u \in \mathscr{L}(E; F)$. The orthogonal of the kernel of $u$ in $E$ is said to be the initial subspace of $u$ and the closure of the image of $u$ in $F$ is called the final subspace of $u$. The orthoprojector from $E$ (resp. $F$) onto the initial (resp. final) subspace of $u$ is called the initial (resp. final) orthoprojector of $u$.

Let $P$ be the initial subspace of $u$. Since $E$ is the direct sum of $P$ and of the kernel of $u$, we have $u(P) = u(E)$.