On the Aleksandrov-Rassias problems on linear \(n\)-normed spaces. (English) Zbl 1283.46010
Summary: This paper generalizes results of Th. M. Rassias and P. Šemrl [Proc. Am. Math. Soc. 118, No. 3, 919–925 (1993; Zbl 0780.51010)] to \(n\)-normed spaces. If \(X\) and \(Y\) are two real \(n\)-normed spaces and \(Y\) is \(n\)-strictly convex, a surjective mapping \(f : X \to Y\) preserving the unit distance in both directions and preserving any integer distance is an \(n\)-isometry.
MSC:
| 46B04 | Isometric theory of Banach spaces |
Citations:
Zbl 0780.51010References:
| [1] | A. D. Aleksandrov, “Mappings of families of sets,” Soviet Mathematics Doklady, vol. 11, pp. 116-120, 1970. · Zbl 0213.48903 |
| [2] | S. Mazur and S. Ulam, “Sur les transformationes isométriques d’espaces vectoriels normés,” Comptes Rendus de L’académie des Sciences, vol. 194, pp. 946-948, 1932. · Zbl 0004.02103 |
| [3] | T. M. Rassias and P. \vSemrl, “On the Mazur-Ulam theorem and the Aleksandrov problem for unit distance preserving mappings,” Proceedings of the American Mathematical Society, vol. 118, no. 3, pp. 919-925, 1993. · Zbl 0780.51010 · doi:10.2307/2160142 |
| [4] | H.-Y. Chu, C.-G. Park, and W.-G. Park, “The Aleksandrov problem in linear 2-normed spaces,” Journal of Mathematical Analysis and Applications, vol. 289, no. 2, pp. 666-672, 2004. · Zbl 1045.46002 · doi:10.1016/j.jmaa.2003.09.009 |
| [5] | H.-Y. Chu, S. K. Choi, and D. S. Kang, “Mappings of conservative distances in linear n-normed spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 3, pp. 1168-1174, 2009. · Zbl 1170.46022 · doi:10.1016/j.na.2008.02.002 |
| [6] | J. Gao, “On the Alexandrov problem of distance preserving mapping,” Journal of Mathematical Analysis and Applications, vol. 352, no. 2, pp. 583-590, 2009. · Zbl 1160.46004 · doi:10.1016/j.jmaa.2008.10.022 |
| [7] | H.-Y. Chu, K. Lee, and C.-G. Park, “On the Aleksandrov problem in linear n-normed spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 59, no. 7, pp. 1001-1011, 2004. · Zbl 1066.46005 · doi:10.1016/j.na.2004.07.046 |
| [8] | C.-G. Park and T. M. Rassias, “Isometries on linear n-normed spaces,” Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 5, pp. 1-17, 2006. · Zbl 1137.46005 |
| [9] | Y. Ma, “The Aleksandrov problem on linear n-normed spaces,” Acta Mathematica Scientia. In press. · Zbl 1283.46010 |
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