Weak and Unbounded Order Convergence in Banach Lattices

Abstract

A net (xy) in a vector lattice is unbounded order convergent (uo-convergent) to 0 if u A i xv I order converges to 0 for all u g 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. © 1977, Australian Mathematical Society. All rights reserved.