Lemma 1

Let E and F be locally convex spaces and let $u$ be a mapping of Fredholm from E into F. The kernel and the cokernel of $u$ are of finite dimension.

Let $v: F\rightarrow E$ be a quasi-inverse of $u$. The kernel of $u$ is contained in the image of the finite rank linear mapping $1_E-v\circ u$, hence is of finite dimension. The image of $u$ contains the kernel of the finite rank linear mapping $1_F-u\circ v$, hence is of finite codimension in F.