The universal separable metric space of Urysohn and isometric embeddings thereof in Вanach spaces

M. Holmes

Fundamenta Mathematicae (1992)

  • Volume: 140, Issue: 3, page 199-223
  • ISSN: 0016-2736

Abstract

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This paper is an investigation of the universal separable metric space up to isometry U discovered by Urysohn. A concrete construction of U as a metric subspace of the space C[0,1] of functions from [0,1] to the reals with the supremum metric is given. An answer is given to a question of Sierpiński on isometric embeddings of U in C[0,1]. It is shown that the closed linear span of an isometric copy of U in a Banach space which contains the zero of the Banach space is determined up to linear isometry. The question of what Banach spaces can be embedded in a linear isometric fashion in this uniquely determined closed linear span of U is investigated.

How to cite

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Holmes, M.. "The universal separable metric space of Urysohn and isometric embeddings thereof in Вanach spaces." Fundamenta Mathematicae 140.3 (1992): 199-223. <http://eudml.org/doc/211941>.

@article{Holmes1992,
abstract = {This paper is an investigation of the universal separable metric space up to isometry U discovered by Urysohn. A concrete construction of U as a metric subspace of the space C[0,1] of functions from [0,1] to the reals with the supremum metric is given. An answer is given to a question of Sierpiński on isometric embeddings of U in C[0,1]. It is shown that the closed linear span of an isometric copy of U in a Banach space which contains the zero of the Banach space is determined up to linear isometry. The question of what Banach spaces can be embedded in a linear isometric fashion in this uniquely determined closed linear span of U is investigated.},
author = {Holmes, M.},
journal = {Fundamenta Mathematicae},
keywords = {inflatable space; isometric embedding; triple midpoint property; universal separable metric space},
language = {eng},
number = {3},
pages = {199-223},
title = {The universal separable metric space of Urysohn and isometric embeddings thereof in Вanach spaces},
url = {http://eudml.org/doc/211941},
volume = {140},
year = {1992},
}

TY - JOUR
AU - Holmes, M.
TI - The universal separable metric space of Urysohn and isometric embeddings thereof in Вanach spaces
JO - Fundamenta Mathematicae
PY - 1992
VL - 140
IS - 3
SP - 199
EP - 223
AB - This paper is an investigation of the universal separable metric space up to isometry U discovered by Urysohn. A concrete construction of U as a metric subspace of the space C[0,1] of functions from [0,1] to the reals with the supremum metric is given. An answer is given to a question of Sierpiński on isometric embeddings of U in C[0,1]. It is shown that the closed linear span of an isometric copy of U in a Banach space which contains the zero of the Banach space is determined up to linear isometry. The question of what Banach spaces can be embedded in a linear isometric fashion in this uniquely determined closed linear span of U is investigated.
LA - eng
KW - inflatable space; isometric embedding; triple midpoint property; universal separable metric space
UR - http://eudml.org/doc/211941
ER -

References

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  1. [B] S. Banach, Théorie des Opérations Linéaires, Hafner, New York 1932, p. 185. Zbl0005.20901
  2. [B-P] C. Bessaga and A. Pełczyński, Selected Topics in Infinite-dimensional Topology, Monograf. Mat. 58, Polish Scientific Publishers, Warszawa 1975, pp. 48-51. Zbl0304.57001
  3. [G] B. Grünbaum, Convex Polytopes, Wiley, London 1967, pp. 72-73. 
  4. [H] G. E. Huhunaishvili, A property of the universal metric space of Urysohn, Dokl. Akad. Nauk SSSR 101 (1955), 607-610 (in Russian). 
  5. [J] C. Joiner, On Urysohn's universal separable metric space, Fund. Math. 73 (1971), 51-58. Zbl0223.54015
  6. [L] J. Lindenstrauss, On the extension of operators with a finite-dimensional range, Illinois J. Math. 8 (1964), 488-499. Zbl0132.09803
  7. [S] W. Sierpiński, Sur un espace métrique séparable universel, Fund. Math. 33 (1945), 115-122. Also see his General Topology, Univ. of Toronto Press, Toronto 1952, pp. 159-162. Zbl0061.40001
  8. [U] P. Urysohn, Sur un espace métrique universel, Bull. Sci. Math. 51 (1927), 43-64, 74-90. Zbl53.0556.01
  9. [Z] M. Ziegler, Ein problem von Urysohn, preprint, 1978. 

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