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 $\hat{I}º$ contains a barrelled subspace with algebraic dimension $\mathrm{cf}[\hat{I}º{]}^{\ddot{I}}‰\cdot \mathrm{non}(\mathcal{M})$, 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 $p\ge 1$, 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 Sánchez Ruiz and Saxon.