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 $\mathfrak{ b}$.
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