Definition 3

Let $E$ and $F$ be two hilbertian spaces. A mapping $u \in \mathscr{L}(E; F)$ is said to be partially isometric if $\|u(x)\| = \|x\|$ for all $x$ belonging to the initial subspace of $u$.

Let $u \in \mathscr{L}(E; F)$ and let $N$ be its kernel and $I$ its image. To say that $u$ is partially isometric is the same as saying that the linear mapping $\tilde{u}: E/N \to I$ deduced from $u$ is isometric (V, p. 13). Then the subspace $I$ of $F$ is complete, hence closed, and is the final subspace of $u$. Consequently, $u$ induces a hilbertian space isomorphism from the initial subspace of $u$ onto its final subspace.