Dunford–Pettis type properties of locally convex spaces

Abstract

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.