Dunford–Pettis type properties of locally convex spaces

In 1953, Grothendieck introduced and studied the Dunford–Pettis property (the \({\textrm{DP}}\) property) and the strict Dunford–Pettis property (the strict \({\textrm{DP}}\) property). The \({\textrm{DP}}\) property of order \(p\in [1,\infty ]\) for Banach spaces was introduced by Castillo and Sanchez in 1993. Being motivated by these notions, for \(p,q\in [1,\infty ],\) we define the quasi-Dunford–Pettis property of order p (the quasi \({\textrm{DP}}_p\) property) and the sequential Dunford–Pettis property of order (p, q) (the sequential \({\textrm{DP}}_{(p,q)}\) property). We show that a locally convex space (lcs) E has the \({\textrm{DP}}\) property if the space E endowed with the Grothendieck topology \(\tau _{\Sigma '}\) has the weak Glicksberg property, and E has the quasi \({\textrm{DP}}_p\) property if the space \((E,\tau _{\Sigma '}) \) has the p-Schur property. We also characterize lcs with the sequential \({\textrm{DP}}_{(p,q)}\) property. Some permanent properties and relationships between Dunford–Pettis type properties are studied. Numerous (counter)examples are given. In particular, we give the first example of an lcs with the strict \({\textrm{DP}}\) property but without the \({\textrm{DP}}\) property and show that the completion of even normed spaces with the \({\textrm{DP}}\) property may not have the \({\textrm{DP}}\) property.