PlanetMath: Lipschitz inverse mapping theorem

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Lipschitz inverse mapping theorem

(Theorem)

Let $(E,\Vert\cdot\Vert)$ 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 be the ball with center 0 and radius (we allow $r=\infty$ ). Then for any Lipschitz map $\phi\colon B(r)\to E$ such that $\operatorname{Lip}\phi < \Vert A^{-1}\Vert^{-1}$ 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 $T(A+\phi) = I\vert _V$ and $(A+\phi)T = I\vert _U$ . In other words, there is a local inverse of $A+\phi$ near zero. Furthermore, the inverse is Lipschitz with $\operatorname{Lip}T \leq (\Vert A\Vert+\operatorname{Lip}\phi)^{-1}$ and

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

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 $\phi$ .

"Lipschitz inverse mapping theorem" is owned by Koro.

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Cross-references: Lipschitz constant, invertible, implies, inclusion, near, open sets, map, Lipschitz, radius, center, ball, automorphism, inverse, linear isomorphism, bounded, Banach space
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This is version 4 of Lipschitz inverse mapping theorem, born on 2004-06-17, modified 2008-07-19.
Object id is 5927, canonical name is LipschitzInverseMappingTheorem.
Accessed 2373 times total.

Classification:

AMS MSC:	46B07 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Local theory of Banach spaces)
	47J07 (Operator theory :: Equations and inequalities involving nonlinear operators :: Abstract inverse mapping and implicit function theorems)

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