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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Stability of isometries

Author: Peter M. Gruber
Journal: Trans. Amer. Math. Soc. 245 (1978), 263-277
MSC: Primary 41A65; Secondary 46B99
MathSciNet review: 511409
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Abstract: A map $ T:E \to F$ (E, F Banach spaces) is called an $ \varepsilon $-isometry if $ \left\vert\,{\left\Vert {T(x)-T(y)} \right\Vert-\left\Vert{x -y}\right\Vert}\,\right\vert\, \leqslant \varepsilon $ whenever $ x,\,y \in E$. Hyers and Ulam raised the problem whether there exists a constant $ \kappa $, depending only on E, F, such that for every surjective $ \varepsilon $-isometry $ T:E \to F$ there exists an isometry $ I:E \to F$ with $ {\left\Vert {T(x) - I(x)} \right\Vert}\leqslant \kappa \varepsilon $ for every $ x \in E$. It is shown that, whenever this problem has a solution for E, F, one can assume $ \kappa \leqslant 5$. In particular this holds true in the finite dimensional case.

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Keywords: Approximate isometry, isometry, stability of isometries, Banach space, normed space, finite dimensional normed space, stability of functional equations, stability, Löwner's ellipsoid
Article copyright: © Copyright 1978 American Mathematical Society

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