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

Piotr Niemiec

Studia Mathematica, Tome 192 (2009), p. 97-110 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

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Publié le : 2009-01-01

Zbl 1176.46031

EUDML-ID : urn:eudml:doc:286072