Lectures on Rosenthal's l^1-theorem

Abstract

Let $x\left(n\right)$ be a bounded sequence in a Banach space X. Rosenthal's $l^{1}$-theorem states that there is essentially only one exceptional situation where it is $\mathit{not}$ possible to extract a subsequence which is a weak Cauchy sequence: This happens if (x$_{n}$) is the sequence of unit vectors in $l^{1}$.The aim of these lectures is twofold: On the one hand results from the last few years centering around this theorem are presented, and on the other hand the opportunity is taken to introduce the audience to a number of techniques which are of importance in modern Banach space theory (Ramsey theory, Martin's axiom, ... ).