Weak compactness in the order dual of a vector lattice

Abstract

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