Norm and relative uniform convergence

Abstract

In this paper, we show that if H is a Dedekind complete normed Riesz space with the local Egoroff property, then H is lattice isomorphic to  an ideal of an AM-space if and only if xn o−→ 0 for any order bounded norm null sequence in H. Based on this, several  characterizations of discrete Banach lattices with order continuous norms are provided. As an application, we find the special regulator  for every relative uniform convergent sequence in discrete Banach lattices with order continuous norms.