Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 

 

Stability of isometries on Banach spaces


Author: Julian Gevirtz
Journal: Proc. Amer. Math. Soc. 89 (1983), 633-636
MSC: Primary 46B20
DOI: https://doi.org/10.1090/S0002-9939-1983-0718987-6
MathSciNet review: 718987
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X$ and $ Y$ be Banach spaces. A mapping $ f:X \to Y$ is called an $ \varepsilon $-isometry if $ \vert\left\Vert {f({x_0}) - f({x_1})} \right\Vert - \left\Vert {{x_0} - {x_1}} \right\Vert\vert \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\Vert {f(({x_0} + {x_1})/2) - (f({x_0}) + f({x_1}))/2} \right\Vert \leqslant A{(\varepsilon \left\Vert {{x_0} - {x_1}} \right\Vert)^{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\Vert {f(x) - I(x)} \right\Vert \leqslant 5\varepsilon $, thus answering a question of Hyers and Ulam about the stability of isometries on Banach spaces.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46B20

Retrieve articles in all journals with MSC: 46B20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0718987-6
Keywords: $ \varepsilon $-isometry, stability of isometries
Article copyright: © Copyright 1983 American Mathematical Society