On the Mazur-Ulam theorem and the Aleksandrov problem for unit distance preserving mappings
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- by Themistocles M. Rassias and Peter Šemrl PDF
- Proc. Amer. Math. Soc. 118 (1993), 919-925 Request permission
Abstract:
Let $X$ and $Y$ be two real normed vector spaces. A mapping $f:X \to Y$ preserves unit distance in both directions iff for all $x,y \in X$ with $||x - y|| = 1$ it follows that $||f(x) - f(y)|| = 1$ and conversely. In this paper we shall study, instead of isometries, mappings satisfying the weaker assumption that they preserve unit distance in both directions. We shall prove that such mappings are not very far from being isometries. This problem was asked by A. D. Aleksandrov. The first classical result that characterizes isometries between normed real vector spaces goes back to S. Mazur and S. Ulam in 1932. We also obtain an extension of the Mazur-Ulam theorem.References
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S. Mazur et S. Ulam, Sur les transformations isométriques d’espaces vectoriels normés, C. R. Acad. Sci. Paris 194 (1932), 946-948.
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Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 118 (1993), 919-925
- MSC: Primary 46B20; Secondary 51K99, 54E40
- DOI: https://doi.org/10.1090/S0002-9939-1993-1111437-6
- MathSciNet review: 1111437