Canonical Banach function spaces generated by Urysohn universal spaces. Measures as Lipschitz maps

Piotr Niemiec

Studia Mathematica (2009)

  • Volume: 192, Issue: 2, page 97-110
  • ISSN: 0039-3223

Abstract

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It is proved (independently of the result of Holmes [Fund. Math. 140 (1992)]) that the dual space of the uniform closure C F L ( r ) of the linear span of the maps x ↦ d(x,a) - d(x,b), where d is the metric of the Urysohn space r of diameter r, is (isometrically if r = +∞) isomorphic to the space L I P ( r ) of equivalence classes of all real-valued Lipschitz maps on r . The space of all signed (real-valued) Borel measures on r is isometrically embedded in the dual space of C F L ( r ) and it is shown that the image of the embedding is a proper weak* dense subspace of C F L ( r ) * . Some special properties of the space C F L ( r ) are established.

How to cite

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Piotr Niemiec. "Canonical Banach function spaces generated by Urysohn universal spaces. Measures as Lipschitz maps." Studia Mathematica 192.2 (2009): 97-110. <http://eudml.org/doc/286072>.

@article{PiotrNiemiec2009,
abstract = {It is proved (independently of the result of Holmes [Fund. Math. 140 (1992)]) that the dual space of the uniform closure $CFL(_\{r\})$ of the linear span of the maps x ↦ d(x,a) - d(x,b), where d is the metric of the Urysohn space $_\{r\}$ of diameter r, is (isometrically if r = +∞) isomorphic to the space $ LIP(_\{r\})$ of equivalence classes of all real-valued Lipschitz maps on $_\{r\}$. The space of all signed (real-valued) Borel measures on $_\{r\}$ is isometrically embedded in the dual space of $CFL(_\{r\})$ and it is shown that the image of the embedding is a proper weak* dense subspace of $CFL(_\{r\})*$. Some special properties of the space $CFL(_\{r\})$ are established.},
author = {Piotr Niemiec},
journal = {Studia Mathematica},
keywords = {Urysohn's universal space; spaces of measures; spaces of Lipschitz maps},
language = {eng},
number = {2},
pages = {97-110},
title = {Canonical Banach function spaces generated by Urysohn universal spaces. Measures as Lipschitz maps},
url = {http://eudml.org/doc/286072},
volume = {192},
year = {2009},
}

TY - JOUR
AU - Piotr Niemiec
TI - Canonical Banach function spaces generated by Urysohn universal spaces. Measures as Lipschitz maps
JO - Studia Mathematica
PY - 2009
VL - 192
IS - 2
SP - 97
EP - 110
AB - It is proved (independently of the result of Holmes [Fund. Math. 140 (1992)]) that the dual space of the uniform closure $CFL(_{r})$ of the linear span of the maps x ↦ d(x,a) - d(x,b), where d is the metric of the Urysohn space $_{r}$ of diameter r, is (isometrically if r = +∞) isomorphic to the space $ LIP(_{r})$ of equivalence classes of all real-valued Lipschitz maps on $_{r}$. The space of all signed (real-valued) Borel measures on $_{r}$ is isometrically embedded in the dual space of $CFL(_{r})$ and it is shown that the image of the embedding is a proper weak* dense subspace of $CFL(_{r})*$. Some special properties of the space $CFL(_{r})$ are established.
LA - eng
KW - Urysohn's universal space; spaces of measures; spaces of Lipschitz maps
UR - http://eudml.org/doc/286072
ER -

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