Proposition 1

Let E and F be topological vector spaces over K. A partial operator $u$ of E into F is closable if and only if the closure of the graph $\Gamma_u$ of $u$ in $E\times F$ is a functional graph. There then exists a unique partial operator $v$ of E into F whose graph is $\overline{\Gamma}_u$, and it is the smallest closed extension of $u$.

If the closure of the graph of $u$ in $E\times F$ is a functional graph, it is the graph of a partial operator, and this is a closed extension of $u$, so that $u$ is closable. Conversely, suppose that $u\subset w$ with $w$ closed. The closure $\overline{\Gamma}_u$ of the graph of $u$ in $E\times F$ is contained in $\Gamma_w$, hence $\Gamma_u$ is a functional graph.

The last assertion follows from the fact that if $w$ is a closed extension of $u$, then the graph of $w$ contains $\overline{\Gamma}_u$.