Dunford–Pettis type properties of locally convex spaces

In 1953, Grothendieck introduced and studied the Dunford–Pettis property (the \({\textrm{DP}}\) $\text{DP}$ property) and the strict Dunford–Pettis property (the strict \({\textrm{DP}}\) $\text{DP}$ property). The \({\textrm{DP}}\) $\text{DP}$ property of order \(p\in [1,\infty ]\) $p\in [1,\infty ]$ for Banach spaces was introduced by Castillo and Sanchez in 1993. Being motivated by these notions, for \(p,q\in [1,\infty ],\) $p,q\in [1,\infty ],$ we define the quasi-Dunford–Pettis property of order p (the quasi \({\textrm{DP}}_p\) ${\text{DP}}_p$ property) and the sequential Dunford–Pettis property of order (p, q) (the sequential \({\textrm{DP}}_{(p,q)}\) ${\text{DP}}_{(p,q)}$ property). We show that a locally convex space (lcs) E has the \({\textrm{DP}}\) $\text{DP}$ property if the space E endowed with the Grothendieck topology \(\tau _{\Sigma '}\) ${\tau }_{{\mathrm{\Sigma }}^{\prime }}$ has the weak Glicksberg property, and E has the quasi \({\textrm{DP}}_p\) ${\text{DP}}_p$ property if the space \((E,\tau _{\Sigma '}) \) $(E,{\tau }_{{\mathrm{\Sigma }}^{\prime }})$ has the p-Schur property. We also characterize lcs with the sequential \({\textrm{DP}}_{(p,q)}\) ${\text{DP}}_{(p,q)}$ 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 \({\textrm{DP}}\) $\text{DP}$ property but without the \({\textrm{DP}}\) $\text{DP}$ property and show that the completion of even normed spaces with the \({\textrm{DP}}\) $\text{DP}$ property may not have the \({\textrm{DP}}\) $\text{DP}$ property.