Normed lattices majorizing in their norm completions
Eugene Bilokopytov, Viktor Bohdanskyi · Proceedings of the American Mathematical Society · 2026
This note is a follow-up to [Positivity 30 (2026), no. 3, Paper No. 51] by Vladimir Troitsky and the first author. We focus on conditions under which a normed lattice X X is majorizing in its norm completion. We show that Question 8.17 from the aforementioned paper – namely, whether this holds whenever every norm-null sequence in X X has an order-bounded subsequence – is equivalent to the question whether every P-ideal on N \mathbb {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.