Stability of isometries

Gruber, Peter M.

doi:10.1090/S0002-9947-1978-0511409-2

Stability of isometries
HTML articles powered by AMS MathViewer

by Peter M. Gruber PDF

Trans. Amer. Math. Soc. 245 (1978), 263-277 Request permission

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

P. Alexandroff and H. Hopf, Topologie. I, Die Grundlehren der mathematischen Wissenschaften, Band 45, Springer-Verlag, Berlin-New York, 1974. Berichtigter Reprint. MR 0345087, DOI 10.1007/978-3-642-65614-9

Théorie des opérations linéaires

D. G. Bourgin, Approximate isometries, Bull. Amer. Math. Soc. 52 (1946), 704–714. MR 17465, DOI 10.1090/S0002-9904-1946-08638-3
D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J. 16 (1949), 385–397. MR 31194, DOI 10.1215/S0012-7094-49-01639-7
D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223–237. MR 42613, DOI 10.1090/S0002-9904-1951-09511-7
Richard D. Bourgin, Approximate isometries on finite dimensional Banach spaces, Trans. Amer. Math. Soc. 207 (1975), 309–328. MR 370137, DOI 10.1090/S0002-9947-1975-0370137-4
Ludwig Danzer, Detlef Laugwitz, and Hanfried Lenz, Über das Löwnersche Ellipsoid und sein Analogon unter den einem Eikörper einbeschriebenen Ellipsoiden, Arch. Math. (Basel) 8 (1957), 214–219 (German). MR 94771, DOI 10.1007/BF01899996

Helly’s theorem and its relatives

Mahlon M. Day, Normed linear spaces, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21, Springer-Verlag, New York-Heidelberg, 1973. MR 0344849
T. Figiel, On nonlinear isometric embeddings of normed linear spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 185–188 (English, with Russian summary). MR 231179
D. G. deFigueiredo and L. A. Karlovitz, The extension of contractions and the intersection of balls in Banach spaces, J. Functional Analysis 11 (1972), 168–178. MR 0372587, DOI 10.1016/0022-1236(72)90085-7
D. H. Hyers and S. M. Ulam, On approximate isometries, Bull. Amer. Math. Soc. 51 (1945), 288–292. MR 13219, DOI 10.1090/S0002-9904-1945-08337-2
D. H. Hyers and S. M. Ulam, Approximate isometries of the space of continuous functions, Ann. of Math. (2) 48 (1947), 285–289. MR 20717, DOI 10.2307/1969171
Eugene Lukacs, Über die Stabilität einer Charakterisierung der Normalverteilung, Österreich. Akad. Wiss. Math.-Naturwiss. Kl. S.-B. II 185 (1976), no. 4-7, 167–179. MR 0436412
S. M. Ulam, A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York-London, 1960. MR 0120127