Summary: Let \(X\) be a Banach space whose dual has the property \((V^*)\) and \(Y\) be a Banach space whose dual does not contain an isomorphic copy of \(l_\infty\). We show that every bounded linear operator from \(Y\) to \(X^*\) is weakly compact. Several results are given on dual Banach spaces concerning some geometric properties.