Proposition 13

Let E and F be Banach spaces. The sets $\mathscr{E}(E; F)$ and $\mathscr{Q}\mathscr{E}(E; F)$ are open in $\mathscr{L}(E; F)$. More precisely, if $u$ is an element of $\mathscr{Q}\mathscr{E}(E; F)$ and $v$ an element of $\mathscr{L}(E; F)$ such that $\|v-u\|<((u))$, then $v\in \mathscr{Q}\mathscr{E}(E; F)$ and

dim Ker($v$)$\leqslant$ dim Ker($u$), dim Coker($v$)$\leqslant$ dim Coker($u$),

ind($v$) $=$ ind($u$).

We have seen above that $^tu\in \mathscr{Q}\mathscr{M}(F'; E')$. Moreover, for every element $v$ of $\mathscr{L}(E; F)$, one has $\|^tv-^tu\|=\|v-u\|$ and $((^tu)) = ((u))$ (EVT, IV, p. 7, prop. 8 and III, p. 63, prop. 8). According to prop. 12, it follows from these relations that if $\|v-u\|<((u))$, then $^tv$ belongs to $\mathscr{Q}\mathscr{M}(F'; E')$ and that one has the inequalities

dim Ker($^tv$)$\leqslant$ dim Ker($^tu$), dim Coker($^tv$)$\leqslant$ dim Coker($^tu$)

as well as the equality ind($^tv$) $=$ ind($^tu$). The proposition then follows from lemma 4.