Corollary

Let $E_1$ and $E_2$ be complex Hilbert spaces and let $u\in \mathscr{L}(E_1; E_2)$. Then Im($u$) $=$ Im($|u^*|$).

Let $(j,|u|)$ be the polar decomposition of $u$. One has $|u^*|=j|u|j^*=uj^*$ by the preceding proposition. By EVT, V, p. 41, Prop. 5, the mapping $j^*$ is partially isometric. Its final space is Ker($j$)$^{\circ}=$ Ker($u$)$^{\circ}$ (Prop. 10, c)). The assertion follows.