Hyers-Ulam stability of isometries and non-expansive maps between spaces of continuous functions

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 $1< p<\infty$.