Weak Unbounded Norm Topology and Dounford-Pettis Operators

In this paper, we study $un$-dual (in symbol, $\text{ud}E$) of Banach lattice $E$ and compare it with topological dual $E^{\ast }$. If $E^{\ast }$ has order continuous norm, then $E^{\ast }=\text{ud}E$. We introduce and study weakly unbounded norm topology ($wun$-topology) on Banach lattices and compare it with weak topology and $uaw$-topology. In the final, we introduce and study $wun$-Dunford-Pettis opertors from a Banach lattice $E$ into Banach space $X$ and we investigate some of its properties and its relationships with others known operators.