Let $A$ be a Banach space. The weak topology on $A$ is a topology which produced by the following family of seminorms:
$~~~~~~~~~~~~~~~~~~~~P_f(x)=|f(x)|,\qquad$ where $f\in A^*$ and $A^*$ is dual of $A$.
Please itroduce some conditions under them, weakly compact sets are necessarily compact under original topology( norm-topology of Banach space).