Dunford–Pettis type properties of locally convex spaces

In 1953, Grothendieck introduced and studied the Dunford–Pettis property (the DP property) and the strict Dunford–Pettis property (the strict DP property). The DP property of order p ∈ [ 1 , ∞ ] for Banach spaces was introduced by Castillo and Sanchez in 1993. Being motivated by these notions, for p , q ∈ [ 1 , ∞ ] , we define the quasi-Dunford–Pettis property of order p (the quasi DP p property) and the sequential Dunford–Pettis property of order ( p , q ) (the sequential DP ( p , q ) property). We show that a locally convex space (lcs) E has the DP property if the space E endowed with the Grothendieck topology τ Σ ′ has the weak Glicksberg property, and E has the quasi DP p property if the space ( E , τ Σ ′ ) has the p -Schur property. We also characterize lcs with the sequential 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 DP property but without the DP property and show that the completion of even normed spaces with the DP property may not have the DP property.