Small-dimensional normed barrelled spaces

2024/10/17 by Will Brian, Brian, Will, Christopher Stuart +1 · 2 citations
Mathematics · #Advanced Banach Space Theory #Fixed Point Theorems Analysis #Approximation Theory and Sequence Spaces

paper · pdf · doi:10.48550/arxiv.2410.13970

Abstract

We prove that every separable Banach space has a barrelled subspace with algebraic dimension non(\mathcal M), which denotes the smallest cardinality of a non-meager subset of \mathbb R. This strengthens a theorem of Sobota. More generally, we prove that every Banach space with density character κ contains a barrelled subspace with algebraic dimension cf[κ]^ω⋅ non(\mathcal M), and in particular it is consistent with ZFC that every Banach space with density character < \mathfrakc has a barrelled subspace with dimension < \mathfrakc. We also prove that if the dual of a Banach space contains either c₀ or ℓ^p for some p ≥ 1, then that space does not have a barrelled subspace with dimension < cov(\mathcal N), which denotes the smallest cardinality of a collection of Lebesgue null sets covering \mathbb R. In particular, it is consistent with ZFC that no classical Banach spaces contain barrelled subspaces with dimension \mathfrakb. This partly answers a question of Sánchez Ruiz and Saxon.

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