In this paper we present some results about $wV$ (weak property $V$ of Peł czyński) or property $wV^*$ (weak property $V^*$ of Peł czyński) in Banach spaces. We show that $E$ has property $wV$ if for any reflexive subspace $F$ of $E^*$, $^{\perp} {F}$ has property $wV$. It is shown that $G$ has property $wV$ if under some condition $K_{w^*}(E^*, F^*)$ contains the dual of $G$. Moreover, it is proved that $E^*$ contains a copy of $c_0$ if and only if $E$ contains a copy of $\ell_1$ where $E$ has property $wV^*$. Finally, the identity between $L(C(\Omega, E), F)$ and $WP(C(\Omega, E), F)$ is investigated.