On a decomposition of Banach spaces

Jakub Duda
2007 Colloquium Mathematicum  
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 This improves an earlier unpublished result of E. Matoušková.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
more » ... , 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 ℓ 1 and L 1 can be decomposed as unions of a ball small set and an Aronszajn null set. 2000 Mathematics Subject Classification: Primary 28A05; Secondary 46B04.
doi:10.4064/cm108-1-13 fatcat:mcpi37wjqje77bwo22cshwpuny