Example

Let E and F be topological vector spaces over K.

1) Suppose the space F separated. Let $u$ be a linear mapping from E into F. If $u$ is continuous, then the partial operator $u$ is closed (TG, I, p. 53, Corollary 2). Suppose moreover that K is a non-discrete valued field and that E and F are metrisable complete topological vector spaces over K. By the closed graph theorem (EVT, I, p. 19, Corollary 5), the partial operator defined by $u$ is then closed if and only if $u$ is continuous.

2) Suppose the space E separated. Let $v$ be an injective continuous linear mapping from F into E. The partial operator $v^{-1}\in \mathscr{P}(E; F) ($cf. IV, p. 226) is then closed, since its graph is the image of the graph of $v$, which is closed, under the isomorphism of topological vector spaces from $F\times E$ onto $E\times F$ defined by $(y, x)\mapsto (x, y)$.