On the notion of stability of order convergence in vector lattices

Abstract

In the theory of Banach lattices the following criterion for a norm to be order continuous is established: a norm is order continuous if and only if every order bounded sequence of positive pairwise disjoint elements in a lattice converges to zero in norm. In this paper we give a criterion for order convergence to be stable in a rather wide class of vector lattices which includes all Köthe spaces. The formulation of the criterion is analogous to that of the above-mentioned criterion for a norm to be order continuous. Namely, under certain conditions imposed on a vector lattice, stability of order convergence is equivalent to the condition that every order bounded sequence of positive pairwise disjoint elements converges relatively uniformly to zero. Furthermore, we study some types of order ideals in vector lattices. In terms of these ideals we give clarified versions of the above-stated criterions. As for notation and terminology, see for example [1,2].