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

Piotr Niemiec¹

¹Institute of Mathematics Jagiellonian University /Lojasiewicza 6 30-348 Kraków, Poland

Studia Mathematica, Tome 192 (2009) no. 2, pp. 97-110

Résumé

It is proved (independently of the result of Holmes [Fund. Math. 140 (1992)]) that the dual space of the uniform closure ${\rm CFL}(\mathbb U_r)$ of the linear span of the maps $x \mapsto d(x,a) - d(x,b)$, where $d$ is the metric of the Urysohn space ${\mathbb U}_r$ of diameter $r$, is (isometrically if $r = +\infty$) isomorphic to the space ${\rm LIP}({\mathbb U}_r)$ of equivalence classes of all real-valued Lipschitz maps on ${\mathbb U}_r$. The space of all signed (real-valued) Borel measures on ${\mathbb U}_r$ is isometrically embedded in the dual space of ${\rm CFL}({\mathbb U}_r)$ and it is shown that the image of the embedding is a proper weak$^{*}$ dense subspace of ${\rm CFL}({\mathbb U}_r)^*$. Some special properties of the space ${\rm CFL}({\mathbb U}_r)$ are established.

Détail

DOI : 10.4064/sm192-2-1

Keywords: proved independently result holmes fund math dual space uniform closure cfl mathbb linear span maps mapsto where metric urysohn space mathbb diameter nbsp isometrically infty isomorphic space lip mathbb equivalence classes real valued lipschitz maps mathbb space signed real valued borel measures mathbb isometrically embedded dual space cfl mathbb shown image embedding proper weak * dense subspace cfl mathbb * special properties space cfl mathbb established