Small-dimensional normed barrelled spaces

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.