On families of subsets of natural numbers deciding the norm convergence in $\ell _1$
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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 $2^\omega$.References
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Additional Information
- Damian Sobota
- Affiliation: Kurt Gödel Research Center for Mathematical Logic, Universität Wien, Währinger Straße 25, 1090 Wien, Austria
- Email: damian.sobota@univie.ac.at
- Received by editor(s): February 11, 2017
- Received by editor(s) in revised form: June 6, 2017
- Published electronically: November 13, 2017
- Additional Notes: The author was supported by the FWF Grant I 2374-N35.
- Communicated by: Thomas Schlumprecht
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 1673-1680
- MSC (2010): Primary 46B45, 03E75, 03E17; Secondary 28E15, 46B10
- DOI: https://doi.org/10.1090/proc/13861
- MathSciNet review: 3754351