Almost linearity of $\epsilon$-bi-Lipschitz maps between real Banach spaces
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- by Kil-Woung Jun and Dal-Won Park PDF
- Proc. Amer. Math. Soc. 124 (1996), 217-225 Request permission
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 \[ \|I(x)-f(x)\| \le E(\epsilon , \alpha )(\|x\|^\alpha +\|x\|^{2-\alpha })\] where $E(\epsilon ,\alpha )\to 0$ as $\epsilon \to 0$ and $0<\alpha <1$.References
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Additional Information
- Kil-Woung Jun
- Email: kwjun@math.chungnam.ac.kr
- 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
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 217-225
- MSC (1991): Primary 46B20
- DOI: https://doi.org/10.1090/S0002-9939-96-03267-4
- MathSciNet review: 1317040