Hyers-Ulam stability of isometries and non-expansive maps between spaces of continuous functions
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Abstract:
We introduce a notion of an $\varepsilon$–non-expansive map and study the problem of uniform approximation of such a map by a non-expansive map. We apply then obtained results to show the following Hyers-Ulam stability of 𝜖–iso
metries: Let $X$ be a Hausdorff compact space and $Y$ be a metric compact space. Let ${F\colon C(X)\to C(Y)}$ be an 𝜖–iso
metry. Then there is an isometry ${H\colon C(X) \to C(Y)}$ such that \[ \| F(f) - H(f) \| \leq 5\varepsilon , \quad f\in C(X). \] If in addition for every proper closed subset $S\subset Y$ there is an $f\in C(X)$ with $|F(f)(z)|<\|F(f)\| - 3.5\varepsilon$ for every $z\in S$, then $H$ can be chosen linear.
This assertion does not hold for the $\ell _p$ norm with $1< p<\infty$.
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Additional Information
- Igor A. Vestfrid
- Affiliation: Nehemya Street, 21/6, 32294 Haifa, Israel
- MR Author ID: 721201
- Email: igor.vestfrid@gmail.com
- Received by editor(s): May 12, 2016
- Received by editor(s) in revised form: July 9, 2016
- Published electronically: February 10, 2017
- Communicated by: Thomas Schlumprecht
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 2481-2494
- MSC (2010): Primary 46B04, 46E15; Secondary 41A65
- DOI: https://doi.org/10.1090/proc/13383
- MathSciNet review: 3626505