Definition 2

Let E and F be topological vector spaces over K (EVT, I, p. 1, def. 1). Let $u\in \mathscr{P}(E; F)$ be a partial operator from E into F.

One says that $u$ is an operator with dense domain if the domain of $u$ is dense in E.

One says that $u$ is closed if the graph of $u$ is closed in the topological vector space $E\times F$. One says that $u$ is closable if it possesses a closed extension.

Let E, F and G be topological vector spaces over K. Every extension of an operator $u\in \mathscr{P}(E; F)$ with dense domain has dense domain. Moreover, if $v\in \mathscr{L}(E; F)$, then $u+v$ is an operator with dense domain. If $v: F\rightarrow G$ (resp. $w: G\rightarrow E$) is an isomorphism of topological vector spaces, then $v\circ u$ (resp $u\circ w$) is an operator with dense domain.