Weak compactness in the order dual of a vector lattice

Abstract:

A sequence $\{ {x_n}\}$ in a vector lattice E will be called an l’-sequence if there exists an x in E such that $\Sigma _{k = 1}^n|{x_k}| \leq x$ for all n. Denote the order dual of E by ${E^b}$. For a set $A \subset {E^b}$, let ${\left \| \cdot \right \|_{{A^ \circ }}}$ denote the Minkowski functional on E defined by its polar ${A^ \circ }$ in E. A set $A \subset {E^b}$ will be called equi-l’-continuous on E if $\lim {\left \| {{x_n}} \right \|_{{A^ \circ }}} = 0$ for each l’-sequence $\{ {x_n}\}$ in E. The main objective of this paper will be to characterize compactness in ${E^b}$ in terms of the order structure on E and ${E^b}$. In particular, the relationship of equi-l’-continuity to compactness is studied. §2 extends to ${E^{\sigma c}}$ the results in Kaplan [8] on vague compactness in ${E^C}$. Then this is used to study vague convergence of sequences in ${E^b}$.