Let $(E,\|\cdot\|)$ be a Banach space and let $A\colon E\to E$ be a bounded linear isomorphism with bounded inverse (i.e. a topological linear automorphism); let $B(r)$ be the ball with center 0 and radius $r$ (we allow $r=\infty$ ). Then for any Lipschitz map $\phi\colon B(r)\to E$ such that $\lip \phi < \|A^{-1}\|^{-1}$ and $\phi(0)=0$ , there are open sets $U\subset E$ and $V\subset B(r)$ and a map $T \colon U\to V$ such that $T(A+\phi) = I|_V$ and $(A+\phi)T = I|_U$ . In other words, there is a local inverse of $A+\phi$ near zero. Furthermore, the inverse $T$ is Lipschitz with $\lip T \leq (\|A\|+\lip \phi)^{-1}$ and $$B\left(r(\|A^{-1}\|^{-1} - \lip \phi)\right)\subset U.$$

Remark. The inclusion above implies that $A+\phi\colon E\to E$ is invertible if $r=\infty$ .

Remark. $\lip \phi$ denotes the smallest Lipschitz constant of $\phi$ .