Normed lattices majorizing in their norm completions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908954412449792 |
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| author | Bilokopytov, Eugene Bohdanskyi, Viktor |
| author_facet | Bilokopytov, Eugene Bohdanskyi, Viktor |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_09939 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Normed lattices majorizing in their norm completions Bilokopytov, Eugene Bohdanskyi, Viktor Functional Analysis Logic 46A19, 46A40, 46B42, 54G99, 03E05, 03E35 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. |
| title | Normed lattices majorizing in their norm completions |
| topic | Functional Analysis Logic 46A19, 46A40, 46B42, 54G99, 03E05, 03E35 |
| url | https://arxiv.org/abs/2604.09939 |