Corollary 2

Let T be a separated vector subspace of finite dimension of F. Let $\pi : F\rightarrow F/T$ be the canonical surjection. In order that $u$ be a strict morphism with closed image, it is necessary and sufficient that $\pi \circ u$ be one.

The identical mapping of T onto itself extends to a continuous mapping $q: F\rightarrow T$ (EVT, II, p. 26, remark). The kernel S of $q$ is a closed topological supplementary subspace of T. Let $p: F\rightarrow F$ be the projector with image S associated with the decomposition $F = T\oplus S$. For $\pi \circ u$ to be a strict morphism with closed image, it is necessary and sufficient that $p\circ u$ be one. Now $p\circ u$ and $u$ have the same restriction to $\overset{-1}{u}$(S), which is a closed vector subspace of finite codimension of E, and the assertion follows from the proposition.