Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
linear extension of isometry; theorem of Banach and Mazur; Hilbert cube; Cantor set
linear extension of isometry; theorem of Banach and Mazur; Hilbert cube; Cantor set
Summary:
In this note, we prove that any “bounded” isometries of separable metric spaces can be represented as restrictions of linear isometries of function spaces $C(Q)$ and $C(\Delta)$, where $Q$ and $\Delta$ denote the Hilbert cube $[0,1]^{\infty}$ and a Cantor set, respectively.
References:
[1] Banach S.: Théories des Opérations Linéaires. Hafner, New York, 1932, p. 185.
[2] Ikegami Y., Kato H., Ueda A.: Dynamical systems of finite-dimensional metric spaces and zero-dimensional covers. Topology Appl. 160 (2013), 564–574. DOI 10.1016/j.topol.2013.01.010 | MR 3010364 | Zbl 1295.54036
[3] Mazur S., Ulam S.: Sur les transformation isométriques d'espace vectoriel normés. C.R. Acad. Sci. Paris 194 (1932), 946–948.
[4] van Mill J.: Infinite-dimensional Topology: Prerequisites and Introduction. North-Holland, Amsterdam, 1989. MR 0977744 | Zbl 0663.57001
[5] Sierpiński W.: Sur un espace métrique séparable universel. Fund. Math. 33 (1945), 115–122. MR 0015451 | Zbl 0061.40001
[6] Stone M.H.: Applications of the theory of Boolean rings to general topology. Trans. Amer. Math. Soc. 41 (1937), 375–381. DOI 10.1090/S0002-9947-1937-1501905-7 | MR 1501905 | Zbl 0017.13502
[7] Urysohn P.: Sur un espace métrique universel. Bull. Sci. Math. 51 (1927), 43–64.
[8] Uspenskij V.V.: On the group of isometries of the Urysohn universal metric space. Comment. Math. Univ. Carolin. 31 (1990), 181–182. MR 1056185 | Zbl 0699.54011
[9] Uspenskij V.V.: A universal topological group with a countable base. Funct. Anal. Appl. 20 (1986), 86–87. MR 0847156
Search
Partner of
