On families of subsets of natural numbers deciding the norm convergence in $\ell _1$

Abstract:

The classical Schur theorem asserts that the weak convergence and the norm convergence in the Banach space $\ell _1$ coincide. In this paper we study complexity and cardinality of subfamilies $\mathcal {F}$ of $\wp (\omega )$ such that a sequence $\big \langle {x_n}\colon \ n\in \omega \big \rangle \subseteq \ell _1$ is norm convergent whenever $\lim _{n\to \infty }\sum _{j\in A}x_n(j)=0$ for every $A\in \mathcal {F}$. We call such families Schur and prove that they cannot have cardinality less than the pseudo-intersection number $\mathfrak {p}$. On the other hand, we also show that every non-meager subset of the Cantor space $2^\omega$ is a Schur family when thought of as a subset of $\wp (\omega )$, implying that the minimal size of a Schur family is bounded from above by $\text {non}(\mathcal {M})$, the uniformity number of the ideal of meager subsets of $2^\omega$.