I am interested in examples of dual Banach spaces $X$ with the Schur property (weakly convergent sequences in $X$ are norm convergent) like $\ell\_1$.
The Lorentz spaces $d(w,1)$ [Lindenstrauss and Tzafriri. Classical Banach spaces I. Sequence spaces. Section 4.3] are candidates because they admit a predual and are hereditarily-$\ell\_1$ (Proposition 4.e.3 in the cited reference). 


Do the spaces $d(w,1)$ have the Schur property? 



On the Lorentz sequence space $d(w,1)$
No. The unit vector basis $(e\_n)$ is unconditional and symmetric but not equivalent to the unit vector basis for $\ell\_1$, hence $(e\_n)$ converges weakly to zero.


