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 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.

Mathematics Subject Classification (2020):

• 46A40
• 46B42

Key words:

• AM-space
• Carleman operator
• local Egoroff property
• discrete Banach lattice with order continuous norms
• relative uniform convergence