Normed lattices majorizing in their norm completions

TL;DR

This paper investigates conditions for a normed lattice to be majorizing in its completion, linking it to a set-theoretic open problem about P-ideals and providing equivalent conditions and a classic characterization of completeness.

Contribution

It establishes the equivalence between a lattice majorizing condition and a set-theoretic problem, and offers new equivalent conditions and a simplified proof of a known characterization.

Findings

The majorizing condition is equivalent to a set-theoretic problem about P-ideals.

Several new equivalent conditions to the majorizing properties are presented.

A simple proof of the Riesz-Fischer characterization of completeness is provided.

Abstract

This note is a follow-up to \cite{bt}. We focus on conditions under which a normed lattice $X$ is majorizing in its norm completion. We show that \cite[Question 8.17]{bt} -- namely, whether this holds whenever every norm-null sequence in $X$ has an order-bounded subsequence -- is equivalent to the question whether every P-ideal on $\N$ is meager. This is a longstanding open problem in Set Theory, and it has a negative answer under various set-theoretical assumptions, in particular under the Continuum Hypothesis. We also present several equivalent conditions to both of the two aforementioned properties, and give a simple proof of a well-known Riesz-Fischer-style characterization of completeness of a normed lattice.