Proposition 4

Let E and F be Banach spaces (resp. Hilbert spaces), and let $u$ be a partial operator from E into F. Then $u$ is closed if and only if the normed space $E_u$ is a Banach space (resp. a Hilbert space).

It is enough to treat the case of Banach spaces. The norm of $E_u$ is obtained, by transport of structure by means of the bijective linear mapping $(x, y)\mapsto x$ of $\Gamma_u$ onto dom($u$), from the norm obtained by restriction to the subspace $\Gamma_u$ of the norm $(x, y)\mapsto (\|x\|^2_E+\|y\|^2_F)^{1/2}$ on the Banach space $E\oplus F$. Thus the space $E_u$ is a Banach space if and only if the subspace $\Gamma_u$ of $E\oplus F$ is closed.

If $u$ and $v$ are partial operators from E into F and from F into G, respectively, and if $u$ is closed, then the partial operator $v\circ u$ is not closed in general, even if $v$ is continuous (exercise 1 of IV, p. 344, c)). One nevertheless has the following sufficient condition: