Theorem 2

Let E and F be Fréchet spaces, $u: E\rightarrow F$ a continuous linear mapping whose cokernel is of finite dimension, and $h: E\rightarrow F$ a compact linear mapping. The linear mapping $u+h$ is a strict morphism, its cokernel is of finite dimension and its image is closed.

According to lemma 6 of III, p. 52, it suffices to show that the cokernel of $u+h$ is of finite dimension. Now, according to prop. 9 of III, p. 9, the mapping $^th$ from $F'_c$ into $E'_c$ is compact ; theorem 2 results from theorem 1 and lemma 2.