Stability of isometries
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- by Peter M. Gruber
- Trans. Amer. Math. Soc. 245 (1978), 263-277
- DOI: https://doi.org/10.1090/S0002-9947-1978-0511409-2
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Abstract:
A map $T:E \to F$ (E, F Banach spaces) is called an $\varepsilon$-isometry if $\left | {\left \| {T(x)-T(y)} \right \|-\left \|{x -y}\right \|} \right | \leqslant \varepsilon$ 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 ${\left \| {T(x) - I(x)} \right \|}\leqslant \kappa \varepsilon$ 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.References
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Bibliographic Information
- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 245 (1978), 263-277
- MSC: Primary 41A65; Secondary 46B99
- DOI: https://doi.org/10.1090/S0002-9947-1978-0511409-2
- MathSciNet review: 511409