Corollary 2

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

(i) The mapping $u$ is a Fredholm mapping of index zero;

(ii) There exists an isomorphism $v$ of E onto F such that $u-v$ is of finite rank ;

(iii) There exists an isomorphism $v$ of E onto F such that $u-v$ is compact.

(i) $=\Rightarrow$ (ii) : Suppose that $u$ is a Fredholm mapping of index zero. There exist decompositions into topological direct sums $E = E_1\oplus E_2$ and $F = F_1\oplus F_2$ with $E_2$ and $F_2$ finite-dimensional such that $u$ vanishes on $E_2$ and defines by restriction an isomorphism $u_1$ of $E_1$ onto $F_1($III, p. 42, prop. 2). If ind($u$) $= 0$, then the dimensions of $E_2$ and $F_2$ are equal and there exists an isomorphism $v$ of E onto F extending $u_1$. The kernel of $u-v$ contains $E_1$, hence $u-v$ is of finite rank.

We have (ii) $=\Rightarrow$ (iii) since every continuous linear mapping of finite rank from E into F is compact, and (iii) $=\Rightarrow$ (i) according to th. 3.