Rosenthal l1 theorem
= Rosenthal l1 theorem
{c}
{title2=$\ell^1$}
Every bounded sequence in a <Banach space> has either a <weakly Cauchy sequence> as a subsequence or a subsequence equivalent to the unit vector basis of $\ell^1$. The real-scalar proof represents vectors as continuous functions on a compact dual ball and uses infinite Ramsey theory to select independent lower and upper level sets when pointwise convergence cannot be obtained.