Proposition 3

Let E and F be topological vector spaces over K, F being separated. Let $u$ be a closed partial operator from E into F. The kernel of $u$ is a closed subspace of E.

In fact, the kernel of $u$ is the inverse image of the closed subspace $\Gamma_u\cap (E\times \{0\})$ of $E\times F$ by the continuous linear mapping $x\mapsto (x,0)$ of E into $E\times F$.

In the rest of this No., it is assumed that K is a non-discrete valued field.