Remark

Let E and F be topological vector spaces over K. Let $u$ be a closable operator of E into F. The domain of the closure of $u$ is contained in the closure of the domain of $u$ in E. In general, it is distinct from it (exercise 1 of IV, p. 344, b)).

If $u\in \mathscr{P}(E; F)$ is closable and dom($u$) $= E$, then $u=\overline{u}$ is closed, since then dom($\overline{u}$) $=$ dom($u$).