Corollary 1

Let E and F be separated locally convex spaces and $u\in \mathscr{L}(E; F)$. In order that $u$ be a Fredholm mapping, it is necessary and sufficient that there exist a mapping $v\in \mathscr{L}(F; E)$ such that the linear mappings $1_E-v\circ u$ and $1_F-u\circ v$ be compact.

Since a continuous linear mapping of finite rank is compact, the stated condition is necessary.

Let $v$ be an element of $\mathscr{L}(F; E)$ such that the linear mappings $1_E-v\circ u$ and $1_F-u\circ v$ are compact. By Theorem 3$,v\circ u$ and $u\circ v$ are Fredholm mappings. Let $p$ and $q$ be quasi-inverses of $v\circ u$ and $u\circ v$ respectively. Put $w=p\circ v$ and $w'=v\circ q$. With the notations of No.$^o2$ of III, p. 40, one has $w\circ u\equiv 1_E$ and $u\circ w'\equiv 1_F$, whence $w\equiv w\circ u\circ w'\equiv w'$. It follows that $w$ is a quasi-inverse of $u$.