If $x_n'$ is weak-$\ast$-ly null and $x_n$ is weakly null in a Banach lattice, do we always have
$$\sup_m |x_m'|(|x_n|)\to 0\,?$$
Thank you!
Background:
Definition weak(-$\ast$) topology:
On the dual $E'$ of a Banach lattice $E$ we define two topologies:
- $(x_n')$ in $E'$ converges to $x'$ in $E'$ weak-$\ast$-ly if and only if $$\forall x\in E:x_n'(x)\to x'(x)\,,$$
- and it converges weakly if and only if $$\forall x''\in E'':x''(x_n')\to x''(x)\,.$$
Definition absolute value:
The absolute value $|x|$ of an element of a Banach lattice is defined by $|x|=x\vee -x$.