Proposition 10

Let E and F be normed spaces. The set $\mathscr{M}(E; F)$ is open in $\mathscr{L}(E; F)$. It is the interior of the set of mappings in $\mathscr{L}(E; F)$ which are injective.

Let A be the set of injective continuous linear mappings of E into F. For a mapping $u\in A$ to be a strict morphism, it is necessary and sufficient that its conorm $((u))$ be strictly positive (III, p. 62, remark). Now $((u))$ is the distance from $u$ to the complement of A in $\mathscr{L}(E; F)$(III, p. 65, prop. 9). The proposition follows.