Proposition 3

Let E and F be locally convex spaces and $u\in \mathscr{L}(E; F)$. Let $E_1($resp. $F_1)$ be a closed subspace of finite codimension of E (resp. F). We suppose that $u$ maps $E_1$ into $F_1$ and denote by $u_1\in \mathscr{L}(E_1; F_1)$ the mapping which coincides with $u$ on $E_1$. For $u$ to be a Fredholm mapping, it is necessary and sufficient that $u_1$ be one. We then have

(5) ind($u$)$-$ ind($u_1$) $=$ codim$_F(F_1)-$ codim$_E(E_1)$.

Let $i: E_1\rightarrow E$ and $j: F_1\rightarrow F$ be the canonical injections. They are Fredholm mappings, and we have

ind($i$) $=$ codim$_E(E_1)$, ind($j$) $=$ codim$_F(F_1)$.

Since $j\circ u_1=u\circ i$, we see that $u$ is a Fredholm mapping if and only if $u_1$ is one (Remarks 3 and 4 of III, p. 41). If this is so, we have

ind($j$) $+$ ind($u_1$) $=$ ind($j\circ u_1$) $=$ ind($u\circ i$) $=$ ind($u$) $+$ ind($i$),

whence formula (5).