LipschitzInverseMappingTheorem 2004-06-17 07:09:51 2008-07-19 14:45:32 Theorem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{mathrsfs} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\lip}{\operatorname{Lip}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Per}{\operatorname{Per}} 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 $$B\left(r(\|A^{-1}\|^{-1} - \lip \phi)\right)\subset U.$$ \emph{Remark.} The inclusion above implies that $A+\phi\colon E\to E$ is invertible if $r=\infty$. \emph{Remark.} $\lip \phi$ denotes the smallest Lipschitz constant of $\phi$.