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More on stability of almost surjective \(\varepsilon\)-isometries of Banach spaces. (English) Zbl 1378.46010

Let \(X\) and \(Y\) be two Banach spaces. A mapping \(f: X \to Y\) is called \(\varepsilon\)-isometry for some \(\varepsilon>0\) if \(|\,\|f(x)-f(y)\| - \|x-y\|\,|\leq \varepsilon\) for all \(x, y \in X\). Given two Banach spaces \(X\) and \(Y\) and a standard (i.e., \(f(0)=0\)) \(\varepsilon\)-isometry \(f: X \to Y\), the authors present a sufficient condition guaranteeing the following sharp stability: There is a surjective linear operator \(T : Y \to X\) of norm one such that \(\|Tf(x) - x\|\leq 2\varepsilon\) for all \(x\in X\). They also give some equivalent statements for standard \(\varepsilon\)-isometry.

MSC:

46B04 Isometric theory of Banach spaces
46B20 Geometry and structure of normed linear spaces
39B82 Stability, separation, extension, and related topics for functional equations
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