# Lipschitz inverse mapping theorem

Let $(E,\parallel \cdot \parallel )$ be a Banach space^{} and let $A:E\to E$ 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 $r=\mathrm{\infty}$). Then for any Lipschitz map
$\varphi :B(r)\to E$
such that $$ and $\varphi (0)=0$, there are open sets
$U\subset E$ and $V\subset B(r)$ and a map $T:U\to V$ such that $T(A+\varphi )={I|}_{V}$ and $(A+\varphi )T={I|}_{U}$.
In other words, there is a local inverse of $A+\varphi $ near zero. Furthermore, the inverse $T$ is Lipschitz with $\mathrm{Lip}T\le {(\parallel A\parallel +\mathrm{Lip}\varphi )}^{-1}$ and

$$B\left(r({\parallel {A}^{-1}\parallel}^{-1}-\mathrm{Lip}\varphi )\right)\subset U.$$ |

*Remark.* The inclusion above implies that $A+\varphi :E\to E$ is invertible^{} if $r=\mathrm{\infty}$.

*Remark.* $\mathrm{Lip}\varphi $ denotes the smallest Lipschitz constant of $\varphi $.

Title | Lipschitz inverse mapping theorem |
---|---|

Canonical name | LipschitzInverseMappingTheorem |

Date of creation | 2013-03-22 14:25:13 |

Last modified on | 2013-03-22 14:25:13 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 7 |

Author | Koro (127) |

Entry type | Theorem |

Classification | msc 46B07 |

Classification | msc 47J07 |