Lemma 2

Let E and F be Fréchet spaces and $u\in \mathscr{L}(E; F)$. The following conditions are equivalent:

(i) The cokernel of $u$ is of finite dimension ;

(ii) The mapping $^tu: F'_c\rightarrow E'_c$ is a strict morphism with closed image whose kernel is of finite dimension.

(i) $=\Rightarrow$ (ii) : Suppose that the cokernel of $u$ is of finite dimension. The mapping $u$ is then a strict morphism (lemma 6 of III, p. 52). According to prop. 2 of EVT, IV, p. 27, the image of $^tu$ is closed in $E'$ endowed with the weak topology, and fortiori in $E'_c$. The kernel of $^tu$ is the orthogonal of the image of $u($loc. cit.) ; it is therefore of finite dimension. Finally, $^tu$ is a strict morphism from $F'_c$ into $E'_c$ according to theorem 1 of EVT, IV, p. 28.

(ii) $=\Rightarrow$ (i) : Suppose that $^tu: F'_c\rightarrow E'_c$ is a strict morphism. According to EVT, IV, p. 28, th. 1, the image of $u$ is closed. The kernel of $^tu$ is the orthogonal of Im($u$) (EVT, IV, p. 27, prop. 2) ; if this kernel is of finite dimension, the image of $u$ is of finite codimension in F.