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.