# Lipschitz inverse mapping theorem

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 $\operatorname{Lip}\phi<\|A^{-1}\|^{-1}$ 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 $\operatorname{Lip}T\leq(\|A\|+\operatorname{Lip}\phi)^{-1}$ and

 $B\left(r(\|A^{-1}\|^{-1}-\operatorname{Lip}\phi)\right)\subset U.$

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

Remark. $\operatorname{Lip}\phi$ denotes the smallest Lipschitz constant of $\phi$.

Title Lipschitz inverse mapping theorem LipschitzInverseMappingTheorem 2013-03-22 14:25:13 2013-03-22 14:25:13 Koro (127) Koro (127) 7 Koro (127) Theorem msc 46B07 msc 47J07