| Title:
|
On isometrical extension properties of function spaces (English) |
| Author:
|
Kato, Hisao |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
56 |
| Issue:
|
1 |
| Year:
|
2015 |
| Pages:
|
105-115 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
linear extension of isometry |
| Keyword:
|
theorem of Banach and Mazur |
| Keyword:
|
Hilbert cube |
| Keyword:
|
Cantor set |
| MSC:
|
46B04 |
| MSC:
|
54C35 |
| MSC:
|
54H20 |
| idZBL:
|
Zbl 06433809 |
| idMR:
|
MR3311581 |
| DOI:
|
10.14712/1213-7243.015.109 |
| . |
| Date available:
|
2015-03-10T17:40:16Z |
| Last updated:
|
2017-04-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144192 |
| . |
| Reference:
|
[1] Banach S.: Théories des Opérations Linéaires.Hafner, New York, 1932, p. 185. |
| Reference:
|
[2] Ikegami Y., Kato H., Ueda A.: Dynamical systems of finite-dimensional metric spaces and zero-dimensional covers.Topology Appl. 160 (2013), 564–574. Zbl 1295.54036, MR 3010364, 10.1016/j.topol.2013.01.010 |
| Reference:
|
[3] Mazur S., Ulam S.: Sur les transformation isométriques d'espace vectoriel normés.C.R. Acad. Sci. Paris 194 (1932), 946–948. |
| Reference:
|
[4] van Mill J.: Infinite-dimensional Topology: Prerequisites and Introduction.North-Holland, Amsterdam, 1989. Zbl 0663.57001, MR 0977744 |
| Reference:
|
[5] Sierpiński W.: Sur un espace métrique séparable universel.Fund. Math. 33 (1945), 115–122. Zbl 0061.40001, MR 0015451 |
| Reference:
|
[6] Stone M.H.: Applications of the theory of Boolean rings to general topology.Trans. Amer. Math. Soc. 41 (1937), 375–381. Zbl 0017.13502, MR 1501905, 10.1090/S0002-9947-1937-1501905-7 |
| Reference:
|
[7] Urysohn P.: Sur un espace métrique universel.Bull. Sci. Math. 51 (1927), 43–64. |
| Reference:
|
[8] Uspenskij V.V.: On the group of isometries of the Urysohn universal metric space.Comment. Math. Univ. Carolin. 31 (1990), 181–182. Zbl 0699.54011, MR 1056185 |
| Reference:
|
[9] Uspenskij V.V.: A universal topological group with a countable base.Funct. Anal. Appl. 20 (1986), 86–87. MR 0847156 |
| . |
