Dual maps and the Dunford-Pettis property

We characterize the points of $\left\|\cdot\right\|$-$w^*$ continuity of dual maps, turning out to be the smooth points. We prove that a Banach space has the Schur property if and only if it has the Dunford-Pettis property and there exists a dual map that is sequentially $w$-$w$ continuous at $0$. As consequence, we show the existence of smooth Banach spaces on which the dual map is not $w$-$w$ continuous at $0$.