Proposition 4

Let E and F be Banach spaces, let $u_0: E\rightarrow F$ be a Fredholm mapping and $v_0$ a quasi-inverse of $u_0$. There exist an open neighbourhood U of $u_0$ in $\mathscr{L}(E; F)$ and an analytic mapping $\varphi : U\rightarrow \mathscr{L}(F; E)$ such that

(i) One has $\varphi (u_0) =v_0$;

(ii) For every $u$ in U, the mapping $\varphi (u)$ is a quasi-inverse of $u$, and in particular $u$ is a Fredholm mapping;

(iii) For every $u$ in U, one has ind($u$) $=$ ind($u_0$).

By prop. 2 of III, p. 42, (ii), there exist topological direct sum decompositions $E = E_1\oplus E_2$ and $F = F_1\oplus F_2$, and there exists $\alpha_0\in \mathscr{I}(E_1; F_1)$, such that $E_2$ and $F_2$ are finite-dimensional and that $u_0$ is represented by the matrix $\begin{pmatrix} \alpha_0 & 0 \\ 0 & 0 \end{pmatrix}$ relative to these decompositions.

Let U be the set of elements $u$ of $\mathscr{L}(E; F)$ such that, in the matrix representation $\begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix}$ of $u$ with respect to these decompositions, one has $\alpha \in \mathscr{I}(E_1; F_1)$. Since $\mathscr{I}(E_1; F_1)$ is open in $\mathscr{L}(E_1; F_1)$ (prop. 3 of III, p. 57), U is an open neighbourhood of $u_0$ in $\mathscr{L}(E; F)$. For $u=\begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix}$ in U, put

One has $\varphi (u_0) =v_0$ and the mapping $\varphi$ is analytic (loc. cit.). Modulo continuous linear mappings of finite rank, one has the congruences

Consequently, $\varphi (u)$ is a quasi-inverse of $u$. Every element $u$ of U defines by restriction an isomorphism of $E_1$ onto a topological supplement of $F_2$ in F. By prop. 3 of III, p. 44, one therefore has

ind($u$) $=$ codim$_F(u(E_1))-$ codim$_E(E_1)$

= dim(F$_2$)$-$ dim(E$_2$) $=$ ind($u_0$),

which concludes the proof.