Lipschitz inverse mapping theorem

Let (E,) be a Banach spaceMathworldPlanetmath and let A:EE be a bounded linear isomorphism with bounded inverse (i.e. a topological linear automorphismPlanetmathPlanetmath); let B(r) be the ball with center 0 and radius r (we allow r=). Then for any Lipschitz map ϕ:B(r)E such that Lipϕ<A-1-1 and ϕ(0)=0, there are open sets UE and VB(r) and a map T:UV such that T(A+ϕ)=I|V and (A+ϕ)T=I|U. In other words, there is a local inverse of A+ϕ near zero. Furthermore, the inverse T is Lipschitz with LipT(A+Lipϕ)-1 and


Remark. The inclusion above implies that A+ϕ:EE is invertiblePlanetmathPlanetmath if r=.

Remark. Lipϕ denotes the smallest Lipschitz constant of ϕ.

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