Schur property of l1
= Schur property of l1
{c}
{title2=$\ell^1$}
The <l-p sequence space> $\ell^1$ has the <Schur property>. If a weakly null sequence stayed bounded below in norm, coordinatewise convergence and summability would select disjoint blocks containing almost all of successive terms. A sequence in $\ell^\infty$ matching their signs on those blocks would pair uniformly positively with a subsequence, contradicting <weak convergence> through the <duality of l1 and l infinity>.