Locally Convex Spaces with Sequential Dunford–Pettis Type Properties

Let $p,q,q^{\prime }\in [1,\infty ]$, $q^{\prime }\le q$. Several new characterizations of locally convex spaces with the sequential Dunford–Pettis property of order $(p,q)$ are given. We introduce and thoroughly study the sequential Dunford–Pettis${}^{*}$ property of order $(p,q)$ of locally convex spaces (in the realm of Banach spaces, the sequential $D{P}_{(p,\infty )}^{*}$ property coincides with the well-known $D{P}_p^{*}$ property). Being motivated by the coarse p-$D{P}^{*}$ property and the p-Dunford–Pettis relatively compact property for Banach spaces, we define and study the coarse sequential $D{P}_{(p,q)}^{*}$ property, the coarse $D{P}_p^{*}$ property and the p-Dunford–Pettis sequentially compact property of order $(q^{\prime },q)$ in the class of all locally convex spaces.