Canonical Banach function spaces generated by Urysohn universal spaces. Measures as Lipschitz maps
Volume 192 / 2009
Abstract
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.
