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

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

1994-03-31

arxiv.org/abs/math/9403211v1

Mathematics

Functional Analysis

Scientific paper

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.

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