New proofs of Rosenthal's ${\ell }^1$--theorem and the Josefson--Nissenzweig theorem

Abstract

We give elementary proofs of the theorems mentioned in the title. Our methods rely on a simple version of Ramsey theory and a martingale difference lemma. They also provide quantitative results: if a Banach space contains ${\ell }^1$ only with a bad constant then every bounded sequence admits a subsequence which is ``nearly'' a weak Cauchy sequence.

Cited by in corpus (2)

• Tsirelson-like spaces and complexity of classes of Banach spaces
• The Josefson--Nissenzweig theorem, Grothendieck property, and finitely supported measures on compact spaces