Lemma 4

Let E and F be Banach spaces. In order that an element $u$ of $\mathscr{L}(E; F)$ be a strict morphism of E into F, it is necessary and sufficient that $^tu$ be a strict morphism of $F'$ into $E'$. In this case, one has dim Coker($^tu$) $=$ dim Ker($u$) and dim Ker($^tu$) $=$ dim Coker($u$).

For $u$ to be a strict morphism, it is necessary and sufficient that $^tu$ be one (EVT, IV, p. 29, cor. 3); in this case, the image of $u$ is closed (EVT, IV, p. 28, th. 1) and the vector space Ker($^tu$) (resp. Coker($^tu$)) is canonically isomorphic to the dual of the normed space Coker($u$) (resp. Ker($u$)) by EVT, IV, p. 27, prop. 2. The lemma follows.

Let E and F be Banach spaces. Denote by $\mathscr{E}(E; F)$ the set of continuous linear surjective mappings from E into F and by $\mathscr{Q}\mathscr{E}(E; F)$ the set of continuous linear mappings from E into F whose image is of finite codimension. Every element of $\mathscr{Q}\mathscr{E}(E; F)$ is a strict morphism with closed image (III, p. 52, lemma 6). It follows from lemma 4 that $\mathscr{E}(E; F)$ and $\mathscr{Q}\mathscr{E}(E; F)$ are the inverse images respectively of $\mathscr{M}(F'; E')$ and $\mathscr{Q}\mathscr{M}(F'; E')$ under the continuous mapping $u\mapsto^tu$ from $\mathscr{L}(E; F)$ into $\mathscr{L}(F'; E')$.