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 strict Dunford--Pettis property of order p (the strict DPₚ 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 iff the space E endowed with the Grothendieck topology τΣ' has the weak Glicksberg property, and E has the strict DPₚ property iff the space (E,τΣ') 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.