# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

## Stability of isometries on Banach spacesHTML articles powered by AMS MathViewer

by Julian Gevirtz
Proc. Amer. Math. Soc. 89 (1983), 633-636 Request permission

## Abstract:

Let $X$ and $Y$ be Banach spaces. A mapping $f:X \to Y$ is called an $\varepsilon$-isometry if $|\left \| {f({x_0}) - f({x_1})} \right \| - \left \| {{x_0} - {x_1}} \right \|| \leqslant \varepsilon$ for all ${x_0},{x_1} \in X$. It is shown that there exist constants $A$ and $B$ such that if $f:X \to Y$ is a surjective $\varepsilon$-isometry, then $\left \| {f(({x_0} + {x_1})/2) - (f({x_0}) + f({x_1}))/2} \right \| \leqslant A{(\varepsilon \left \| {{x_0} - {x_1}} \right \|)^{1/2}} + B\varepsilon$ for all ${x_0},{x_1} \in X$. This, together with a result of Peter M. Gruber, is used to show that if $f:X \to Y$ is a surjective $\varepsilon$-isometry, then there exists a surjective isometry $I:X \to Y$ for which $\left \| {f(x) - I(x)} \right \| \leqslant 5\varepsilon$, thus answering a question of Hyers and Ulam about the stability of isometries on Banach spaces.
Similar Articles
• Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46B20
• Retrieve articles in all journals with MSC: 46B20