Proposition 2

Let E and F be separated locally convex spaces and let $u$ be an element of $\mathscr{L}(E; F)$. The following properties are equivalent:

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

(ii) The mapping $u$ is a strict morphism, its kernel is of finite dimension, its image is closed and of finite codimension in F ;

(iii) There exist closed vector subspaces of finite codimension $E_1$ of E and $F_1$ of F such that $u$ defines by passing to the subspaces an isomorphism of $E_1$ onto $F_1$;

(iv) There exist decompositions into topological direct sum $E = E_1\oplus E_2$ and $F = F_1\oplus F_2$ such that $E_2$ and $F_2$ are of finite dimension, that $u$ vanishes on $E_2$ and defines by passing to the subspaces an isomorphism of $E_1$ onto $F_1$.

(i) $=\Rightarrow$ (iii): Let $v$ be a quasi-inverse of $u, E_1$ the kernel of $1_E-v\circ u$ and $F_1$ that of $1_F-u\circ v$. Since the linear mappings $1_E-v\circ u$ and $1_F-u\circ v$ are continuous and of finite rank, $E_1$ and $F_1$ are closed vector subspaces of finite codimension of E and F respectively. Let $x\in E_1$. We have

whence $u(x)\in F_1$. Therefore $u(E_1)\subset F_1$; analogously, $v(F_1)\subset E_1$. The continuous linear mappings $u_1: E_1\rightarrow F_1$ and $v_1: F_1\rightarrow E_1$ induced by $u$ and $v$ are then isomorphisms inverse to one another, since $v\circ u$ and $1_E$ (resp. $u\circ v$ and $1_F$) coincide on $E_1$ (resp. on $F_1$).

(iii) $=\Rightarrow$ (ii): Let $E_1$ and $F_1$ satisfy the hypothesis of (iii). We have $E_1\cap$ Ker($u$) $=\{0\}$ and $F_1\subset$ Im($u$), hence Ker($u$) is finite-dimensional and Im($u$) is closed and of finite codimension in F. It follows from Prop. 1 of III, p. 39 that the mapping $u$ is a strict morphism.

(ii) $=\Rightarrow$ (iv): Suppose condition (ii) satisfied. The closed vector subspace $E_2=$ Ker($u$) of E is finite-dimensional, and there exists a vector subspace $E_1$ of E which is a topological supplement of $E_2$ (TVS, II, p. 27, Cor. 2). The vector subspace $F_1=$ Im($u$) of F is closed and of finite codimension, and admits a topological supplement $F_2$ in F. By Prop. 1 of III, p. 39, the mapping $u|E_1$ is a strict morphism, hence $u$ induces an isomorphism of $E_1$ onto $F_1$.

(iv) $=\Rightarrow$ (i): Under the hypotheses of (iv), the linear mapping of F into E which coincides with $u^{-1}_1$ on $F_1$ and is zero on $F_2$ is a quasi-inverse of $u$.