Small-dimensional normed barrelled spaces

TL;DR

This paper explores the existence and limitations of barrelled subspaces within Banach spaces, linking their algebraic dimension to set-theoretic cardinal invariants and providing new insights into their structure.

Contribution

It proves the existence of large barrelled subspaces in Banach spaces with certain dimensions and establishes limitations based on dual space properties and set-theoretic assumptions.

Findings

Every separable Banach space has a barrelled subspace with dimension non(rac{M})

Banach spaces with density rac{}() contain barrelled subspaces with dimension []^  non(rac{M})

Certain classical Banach spaces do not contain barrelled subspaces with dimension less than ()

Abstract

We prove that every separable Banach space has a barrelled subspace with algebraic dimension $\mathrm{non}(\mathcal M)$, 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 $\kappa$ contains a barrelled subspace with algebraic dimension $\mathrm{cf}[\kappa]^\omega \cdot \mathrm{non}(\mathcal M)$, and in particular it is consistent with $\mathsf{ZFC}$ that every Banach space with density character $<\!\mathfrak{c}$ has a barrelled subspace with dimension $<\!\mathfrak{c}$. We also prove that if the dual of a Banach space contains either $c_0$ or $\ell^p$ for some $p \geq 1$, then that space does not have a barrelled subspace with dimension $<\!\mathrm{cov}(\mathcal N)$, which denotes the smallest cardinality of a collection of Lebesgue null sets covering $\mathbb R$. In particular, it is consistent with $\mathsf{ZFC}$ that no classical Banach spaces contain barrelled subspaces with dimension $\mathfrak{b}$. This partly answers a question of S\'anchez Ruiz and Saxon.