Lemma 6

Let E and F be Fréchet spaces and $u\in \mathscr{L}(E; F)$. Suppose that the image of $u$ is of finite codimension in F. Then the image of $u$ is closed in F and $u$ is a strict morphism.

Let G be a vector subspace of F supplementary to Im($u$); the subspace G is closed (TVS, I, p. 14, cor. 1), hence the quotient space $F/G$ is a Fréchet space (TG, IX, p. 25, prop. 4). Let $\pi : F\rightarrow F/G$ be the canonical surjection. The mapping $\pi \circ u$ is surjective, hence strict (TVS, I, p. 19, cor. 3). The assertion then results from cor. 2 of III, p. 40.