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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
DOI: https://doi.org/10.1090/proc/13861
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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  • [1] Trond A. Abrahamsen, Olav Nygaard, and Märt Põldvere, On weak integrability and boundedness in Banach spaces, J. Math. Anal. Appl. 314 (2006), no. 1, 67-74. MR 2183537, https://doi.org/10.1016/j.jmaa.2005.03.071
  • [2] Tomek Bartoszyński and Haim Judah, Set theory, On the structure of the real line, A K Peters, Ltd., Wellesley, MA, 1995. MR 1350295
  • [3] Murray G. Bell, On the combinatorial principle $ P({\mathfrak{c}})$, Fund. Math. 114 (1981), no. 2, 149-157. MR 643555
  • [4] Andreas Blass, Combinatorial cardinal characteristics of the continuum, Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 395-489. MR 2768685, https://doi.org/10.1007/978-1-4020-5764-9_7
  • [5] Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004
  • [6] Justin Tatch Moore, Michael Hrušák, and Mirna Džamonja, Parametrized $ \diamondsuit$ principles, Trans. Amer. Math. Soc. 356 (2004), no. 6, 2281-2306. MR 2048518, https://doi.org/10.1090/S0002-9947-03-03446-9
  • [7] Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos, and Václav Zizler, Banach space theory, The basis for linear and nonlinear analysis, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, 2011. MR 2766381
  • [8] M. Ĭ. Kadets and V. P. Fonf, Two theorems on the massiveness of the boundary in a reflexive Banach space, Funktsional. Anal. i Prilozhen. 17 (1983), no. 3, 77-78 (Russian). MR 714229
  • [9] D. H. Fremlin, Consequences of Martin's axiom, Cambridge Tracts in Mathematics, vol. 84, Cambridge University Press, Cambridge, 1984. MR 780933
  • [10] Kenneth Kunen and Franklin D. Tall, Between Martin's axiom and Souslin's hypothesis, Fund. Math. 102 (1979), no. 3, 173-181. MR 532951
  • [11] Kenneth Kunen, Set theory, An introduction to independence proofs; reprint of the 1980 original, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam, 1983. MR 756630
  • [12] Olav Nygaard, A strong uniform boundedness principle in Banach spaces, Proc. Amer. Math. Soc. 129 (2001), no. 3, 861-863. MR 1626466, https://doi.org/10.1090/S0002-9939-00-05607-0
  • [13] Olav Nygaard, Boundedness and surjectivity in normed spaces, Int. J. Math. Math. Sci. 32 (2002), no. 3, 149-165. MR 1938513, https://doi.org/10.1155/S0161171202011596
  • [14] Olav Nygaard, Thick sets in Banach spaces and their properties, Quaest. Math. 29 (2006), no. 1, 59-72. MR 2209791, https://doi.org/10.2989/16073600609486149
  • [15] Olav Nygaard and Märt Põldvere, Restricted uniform boundedness in Banach spaces, Quaest. Math. 32 (2009), no. 1, 5-14. MR 2521508, https://doi.org/10.2989/QM.2009.32.1.2.704
  • [16] R. S. Phillips, On linear transformations, Trans. Amer. Math. Soc. 48 (1940), 516-541. MR 0004094, https://doi.org/10.2307/1990096
  • [17] Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
  • [18] J. Schur, Über lineare Transformationen in der Theorie der unendlichen Reihen, J. Reine Angew. Math. 151 (1921), 79-111 (German). MR 1580985, https://doi.org/10.1515/crll.1921.151.79
  • [19] D. Sobota, Cardinal invariants of the continuum and convergence of measures on compact spaces, PhD thesis, Institute of Mathematics, Polish Academy of Sciences, 2016.

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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

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

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