Weak and unbounded order convergence in Banach lattices

Abstract

AbstractA net (xy) in a vector lattice is unbounded order convergent (uo-convergent) to 0 if u ∧ |xv| order converges to 0 for all u ≧ 0. We consider, in a Banach lattice, the relationship between weak and uo-convergence. We characterise those Banach lattices in which weak convergence implies uo-convergence and those in which uo-convergence of a bounded net implies weak convergence. Finally we combine the results to characterise those Banach lattices in which weak and uo-convergence coincide for bounded nets.