Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/proc/13861
Published electronically: November 13, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46B45, 03E75, 03E17, 28E15, 46B10

Retrieve articles in all journals with MSC (2010): 46B45, 03E75, 03E17, 28E15, 46B10


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

DOI: https://doi.org/10.1090/proc/13861
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

American Mathematical Society