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On families of subsets of natural numbers deciding the norm convergence in $ \ell_1$

Author: Damian Sobota
Journal: Proc. Amer. Math. Soc. 146 (2018), 1673-1680
MSC (2010): Primary 46B45, 03E75, 03E17; Secondary 28E15, 46B10
Published electronically: November 13, 2017
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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 $.

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Additional Information

Damian Sobota
Affiliation: Kurt Gödel Research Center for Mathematical Logic, Universität Wien, Währinger Straße 25, 1090 Wien, Austria

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
Article copyright: © Copyright 2017 American Mathematical Society

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