Dunford--Pettis type properties of locally convex spaces

Abstract

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 $p\in[1,\infty]$ 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 strict Dunford--Pettis property of order $p$ (the strict $DP_p$ 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 iff the space $E$ endowed with the Grothendieck topology $τ_{Σ'}$ has the weak Glicksberg property, and $E$ has the strict $DP_p$ property iff 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.