A small remark on small-dimensional normed barrelled spaces

TL;DR

This paper establishes lower bounds on the dimension of barrelled subspaces in infinite-dimensional Banach spaces using set-theoretic methods, linking functional analysis with cardinal invariants.

Contribution

It introduces a novel combination of classical theorems and set theory to determine minimal dimensions of barrelled subspaces in Banach spaces.

Findings

No infinite-dimensional Banach space contains a barrelled subspace of dimension less than cov(N).

Every infinite-dimensional normed barrelled space has dimension at least cov(N).

It is consistent with ZFC that no Banach space contains a barrelled subspace of dimension equal to the bounding number.

Abstract

Combining the methods of Brian and Stuart with the classical Dvoretzky theorem, we show that no infinite-dimensional Banach space contains a barrelled subspace of (algebraic) dimension $<\mbox{cov}(\mathcal{N})$, the covering number of the Lebesgue null ideal $\mathcal{N}$. Consequently, every infinite-dimensional normed barrelled space has dimension $\ge\mbox{cov}(\mathcal{N})$ and it is consistent with ZFC that no Banach space contains a barrelled subspace of dimension equal to the bounding number $\mathfrak{b}$.