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Properties of isometric mappings. (English) Zbl 0936.46009

This is a survey paper related to theorems of Mazur-Ulam type and approximate isometries on Banach and metric vector spaces.
Reviewer: D.Werner (Berlin)

MSC:

46B04 Isometric theory of Banach spaces
Full Text: DOI

References:

[1] Aczél, J.; Dhombres, J., Functional Equations in Several Variables, Encyclopedia of Mathematics and Its Applications (1989), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0685.39006
[2] Alexandrov, A. D., Mappings of families of sets, Soviet Math. Dokl., 11, 116-120 (1970) · Zbl 0213.48903
[3] Aronszajn, N., Characterization métrique de \(L\)-espace de Hiblert, des espaces vectoriels et de certains groupes métriques, C. R. Acad. Sci. Paris, 201, 811-813 (1935) · JFM 61.0633.01
[4] Baker, J. A., Isometries in normed spaces, Amer. Math. Monthly, 78, 655-658 (1971) · Zbl 0214.12704
[5] Banach, S., Théorie des Opérations Linéaires. Théorie des Opérations Linéaires, Monografie Matematyczne, 1 (1932) · JFM 58.0420.01
[6] Beckman, F. S.; Quarles, D. A., On isometries of Euclidean spaces, Proc. Amer. Math. Soc., 4, 810-815 (1953) · Zbl 0052.18204
[7] Bessaga, C.; Pelczynski, A.; Rolewicz, S., Some properties of the norm in \(F\)-spaces, Studia Math., 16, 183-192 (1957) · Zbl 0080.31301
[8] W. A. Beyer, Approximately Lorentz transformations and \(p\); W. A. Beyer, Approximately Lorentz transformations and \(p\)
[9] Bourgin, D. G., Approximate isometries, Bull. Amer. Math. Soc., 52, 704-714 (1946) · Zbl 0060.26405
[10] Bourgin, D. G., Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J., 16, 385-397 (1949) · Zbl 0033.37702
[11] Bourgin, D. G., Two dimensional ε-isometries, Trans. Amer. Math. Soc., 244, 85-102 (1978) · Zbl 0412.46011
[12] Bourgin, R. D., Approximate isometries on finite dimensional Banach spaces, Trans. Amer. Math. Soc., 207, 309-328 (1975) · Zbl 0327.46023
[13] Charzynski, Z., Sur les transformations isométriques des espaces du type \((F)\), Studia Math., 13, 94-121 (1953) · Zbl 0051.08504
[14] Ciesielski, K.; Rassias, Th. M., On some properties of isometric mappings, Facta Univ. Ser. Math. Inform., 7, 107-115 (1992) · Zbl 0827.51010
[15] Clarkson, J. A., Uniformly convex spaces, Trans. Amer. Math. Soc., 40, 396-414 (1936) · Zbl 0015.35604
[16] Day, M. M., Normed Linear Spaces (1958), Springer-Verlag: Springer-Verlag Berlin · Zbl 0082.10603
[17] Fickett, J. W., Approximate isometries on bounded sets with an application to measure theory, Studia Math., 72, 37-46 (1981) · Zbl 0411.28011
[18] Figiel, T., On non-linear isometric embedding of normed linear spaces, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys., 16, 185-188 (1968) · Zbl 0155.18301
[19] Fleming, R. J.; Jamison, J. E., Isometries on Banach Spaces—A survey, (Srivastava, H. M.; Rassias, Th. M., Analysis, Geometry and Groups: A Riemann Legacy Volume (1993), Hadronic Press: Hadronic Press Palm Harbor), 52-123 · Zbl 0913.46009
[20] Gevirtz, J., Stability of isometries on Banach spaces, Proc. Amer. Math. Soc., 89, 633-636 (1983) · Zbl 0561.46012
[21] Gruber, P. M., Stability of isometries, Trans. Amer. Math. Soc., 245, 263-277 (1978) · Zbl 0393.41020
[22] Hyers, D. H., A note on linear topological spaces, Bull. Amer. Math. Soc., 44, 76-80 (1938) · Zbl 0018.27702
[23] Hyers, D. H., Locally bounded linear topological spaces, Rev. Cienc. (Lima), 41, 555-574 (1939) · Zbl 0060.26502
[24] Hyers, D. H., The stability of homomorphisms and related topics, (Rassias, Th. M., Global Analysis—Analysis on Manifolds. Global Analysis—Analysis on Manifolds, Teubner-Texte zur Math., 57 (1983), Teubner: Teubner Leipzig), 140-153 · Zbl 0517.22001
[25] Hyers, D. H.; Ulam, S. M., On approximate isometries, Bull. Amer. Math. Soc., 51, 288-292 (1945) · Zbl 0060.26404
[26] Hyers, D. H.; Ulam, S. M., Approximate isometries of the space of continuous functions, Ann. of Math., 48, 285-289 (1947) · Zbl 0029.36701
[27] S.-M. Jung, Hyers-Ulam-Rassias stability of isometries on restricted domains, submitted.; S.-M. Jung, Hyers-Ulam-Rassias stability of isometries on restricted domains, submitted. · Zbl 1080.46508
[28] Kuz’minyh, A. V., On a characteristic property of isometric mappings, Soviet Math. Dokl., 17, 43-45 (1976) · Zbl 0336.54029
[29] Lindenstrauss, J.; Szankowski, A., Nonlinear perturbations of isometries, Colloq. in Honor of Laurent Schwartz (Palaiseau, 1983) (1985), Soc. Math. France: Soc. Math. France Marseille, p. 357-371 · Zbl 0585.47007
[30] Mazur, S.; Orlicz, W., Sur les espaces métriques linéaires (I), Studia Math., 10, 184-208 (1948) · Zbl 0036.07801
[31] Mazur, S.; Ulam, S. M., Sur les transformations isométriques des espaces vectoriels normés, C. R. Acad. Sci. Paris, 194, 946-948 (1932) · Zbl 0004.02103
[32] Mielnik, B., Phenomenon of mobility in non-linear theories, Comm. Math. Phys., 101, 323-339 (1985)
[33] Mielnik, B.; Rassias, Th. M., On the Aleksandrov problem of conservative distances, Proc. Amer. Math. Soc., 116, 1115-1118 (1992) · Zbl 0769.51005
[34] Modenov, P. S.; Parkhomenko, A. S., Geometric Transformations (1965), Academic Press: Academic Press New York · Zbl 0192.26701
[35] Von Neumann, J.; Schoenberg, I. J., Fourier integrals and metric geometry, Trans. Amer. Math. Soc., 50, 226-251 (1941) · Zbl 0028.41002
[36] M. Omladić, and, P. Šemrl, On non-linear perturbations of isometries, preprint.; M. Omladić, and, P. Šemrl, On non-linear perturbations of isometries, preprint. · Zbl 0836.46014
[37] Rassias, Th. M., Is a distance one preserving mapping between metric spaces always an isometry?, Amer. Math. Monthly, 90, 200 (1983) · Zbl 0512.54017
[38] Rassias, Th. M., Some remarks on isometric mappings, Facta Univ. Ser. Math. Inform., 2, 49-52 (1987) · Zbl 0661.54030
[39] Rassias, Th. M., On the stability of mappings, Rend. Sem. Mat. Fis. Milano, 58, 91-99 (1988) · Zbl 0711.47002
[40] Rassias, Th. M., Problems and remarks, Report of Meeting—The 27th International Symposium on Functional Equations. Report of Meeting—The 27th International Symposium on Functional Equations, Aequationes Math., 39 (1990)
[41] Rassias, Th. M.; Šemrl, P., On the Mazur-Ulam theorem and the Aleksandrov problem for unit distance preserving mappings, Proc. Amer. Math. Soc., 118, 919-925 (1993) · Zbl 0780.51010
[42] Rassias, Th. M.; Sharma, C. S., Properties of isometries, J. Natur. Geom., 3, 1-38 (1993) · Zbl 0776.51015
[43] Rolewicz, S., On a certain class of linear metric spaces, Bull. Acad. Polon. Sci., 5, 479-484 (1957) · Zbl 0078.29101
[44] Rolewicz, S., A generalization of the Mazur-Ulam theorem, Studia. Math., 31, 501-505 (1968) · Zbl 0185.20604
[45] Rolewicz, S., Metric Linear Spaces. Metric Linear Spaces, Monogr. Mat., 56 (1972), Polish Scientific Publishers: Polish Scientific Publishers Warszawa · Zbl 0226.46001
[46] Schoenberg, I. J., Metric spaces and completely monotone functions, Ann. of Math., 39, 811-841 (1938) · Zbl 0019.41503
[47] Swain, R., Approximate isometries in bounded spaces, Proc. Amer. Math., 2, 727-729 (1951) · Zbl 0044.19603
[48] Vogt, A., Maps which preserve equality of distance, Studia Math., 45, 43-48 (1973) · Zbl 0222.46015
[49] Wigner, E. P., On unitary representations of the inhomogeneous Lorentz group, Ann. of Math., 40, 149-204 (1939) · JFM 65.1129.01
[50] Wobst, R., Isometrien in metrischen Vektorräumen, Studia Math., 54, 41-54 (1975) · Zbl 0318.46019
[51] Yosida, K., Functional Analysis (1968), Springer-Verlag: Springer-Verlag New York · Zbl 0217.16001
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