Skip to Main Content

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 spaces
HTML articles powered by AMS MathViewer

by Julian Gevirtz PDF
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.
References
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46B20
  • Retrieve articles in all journals with MSC: 46B20
Additional Information
  • © Copyright 1983 American Mathematical Society
  • 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