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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Almost linearity of $\epsilon $-bi-Lipschitz maps
between real Banach spaces

Authors: Kil-Woung Jun and Dal-Won Park
Journal: Proc. Amer. Math. Soc. 124 (1996), 217-225
MSC (1991): Primary 46B20
MathSciNet review: 1317040
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Abstract: Let $X$ and $Y$ be real Banach spaces. A map $f$ between $X$ and $Y$ is called an $\epsilon $-bi-Lipschitz map if $(1-\epsilon )\|x-y\| \le \|f(x) -f(y)\| \le (1+\epsilon )\|x- y\|$ for all $x, y\in X$. In this note we show that if $f$ is an $\epsilon $-bi-Lipschitz map with $f(0)=0$ from $X$ onto $Y$, then $f$ is almost linear. We also show that if $f:X\longrightarrow Y$ is a surjective $\epsilon $-bi-Lipschitz map with $f(0)=0$, then there exists a linear isomorphism $I:X\to Y$ such that

\begin{equation*}\|I(x)-f(x)\| \le E(\epsilon , \alpha )(\|x\|^\alpha +\|x\|^{2-\alpha })\end{equation*}

where $E(\epsilon ,\alpha )\to 0$ as $\epsilon \to 0 $ and $0<\alpha <1$.

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Kil-Woung Jun

Keywords: $\epsilon $-bi-Lipschitz map, almost linear map, real Banach spaces
Received by editor(s): August 8, 1994
Additional Notes: This work was partially supported by KOSEF, Grant No 91-08-00-01.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society