On a decomposition of Banach spaces

Jakub Duda¹

¹ Department of Mathematics Weizmann Institute of Science Rehovot 76100, Israel and Faculty of Mathematics and Physics Charles University Sokolovská 83 186 75 Praha 8, Czech Republic

Colloquium Mathematicum, Tome 108 (2007) no. 1, pp. 147-157

Résumé

By using D. Preiss' approach to a construction from a paper by J. Matoušek and E. Matoušková, and some results of E. Matoušková, we prove that we can decompose a separable Banach space with modulus of convexity of power type $p$ as a union of a ball small set (in a rather strong symmetric sense) and a set which is Aronszajn null. This improves an earlier unpublished result of E. Matoušková. As a corollary, in each separable Banach space with modulus of convexity of power type $p$, there exists a closed nonempty set $A$ and a Borel non-Haar null set $Q$ such that no point from $Q$ has a nearest point in $A$. Another corollary is that $\ell_1$ and $L_1$ can be decomposed as unions of a ball small set and an Aronszajn null set.

Détail

DOI : 10.4064/cm108-1-13

Keywords: using nbsp preiss approach construction paper nbsp matou nbsp matou kov results nbsp matou kov prove decompose separable banach space modulus convexity power type union ball small set rather strong symmetric sense set which aronszajn null improves earlier unpublished result nbsp matou kov corollary each separable banach space modulus convexity power type nbsp there exists closed nonempty set nbsp borel non haar null set point has nearest point nbsp another corollary ell decomposed unions ball small set aronszajn null set