Michal Wojciechowski
Math Institute of Polish Academy of Sciences
Title
Failure of the Dunford–Pettis Property for W^{1,1}, BV , and the Brudnyi–Brudnyi Preduals
Abstract
We prove that, for every d ≥ 2 and every nonempty open set Ω ⊂ R^d, the spaces W ^{1,1}(Ω) and BV (Ω) fail the Dunford–Pettis property. We also prove that the Brudnyi–Brudnyi preduals of BV(R^d)/R fail the Dunford–Pettis property for every d ≥ 2. The proof is based on a random two-dimensional construction; complemented embeddings yield the higher-dimensional results and, for W^{1,1} and BV , the extension to arbitrary open sets. In particular, since every L^1(μ)-space has the Dunford–Pettis property, W ^{1,1}(Ω) is not isomorphic to any L^1(μ) for d ≥ 2
