Saak Gabriyelyan - ATTI 2026 | Algebra, Topology and Their Interactions

Dunford—Pettis type properties of locally convex spaces.

Being motivated by the Dunford—Pettis property (the $DP$ property) and the strict Dunford—Pettis property (the strict $DP$ property) introduced by Grothendieck in the realm of locally convex spaces and by the $DP$ property of order $p\in[1,\infty]$ for Banach spaces introduced by Castillo and Sanchez, 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 iff the space $E$ endowed with the Grothendieck topology $\tau_{\Sigma'}$ has the weak Glicksberg property, and $E$ has the quasi $DP_p$ property iff the space $(E,\tau_{\Sigma'})$ has the $p$ -Schur property. We also characterize lcs with the sequential $DP_{(p,q)}$ property. 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.

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