Lemma 4

Let E be a complex Banach space and let $u$ be a closed injective partial operator on E such that $u^{-1}\in \mathscr{L}(E)$. Let $v\in \mathscr{L}(E)$ be such that $\|v\|<\|u^{-1}\|^{-1}$. Then the partial operator $u+v$ is injective, one has $(u+v)^{-1}\in \mathscr{L}(E)$, and

where the series is absolutely convergent in $\mathscr{L}(E)$. Moreover, one has

Since $\|v\| \|u^{-1}\|<1$, the series with general term $(-vu^{-1})^k$ is absolutely convergent in $\mathscr{L}$ (E), and its sum is the inverse of the endomorphism $1_E+vu^{-1}$ (prop. 2 of I, p. 22). Consequently, the partial operator $(1_E+vu^{-1})\circ u=u+v($IV, p. 226, rem. vi) is injective and the reciprocal partial operator $(u+v)^{-1}=u^{-1}\circ (1_E+vu^{-1})^{-1}($IV, p. 226, rem. iii) belongs to $\mathscr{L}(E)$. Since

$\|(1_E+vu^{-1})^{-1}\|\leqslant \sum^{+\infty}(\|v\| \|u^{-1}\|)^k=$ 1 $_{-1}$,

$k=0$

the lemma is proved.