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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Strong algebrability of sets of sequences and functions


Authors: Artur Bartoszewicz and Szymon Głab
Journal: Proc. Amer. Math. Soc. 141 (2013), 827-835
MSC (2010): Primary 15A03; Secondary 28A20, 46J10
Published electronically: July 20, 2012
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Abstract: We introduce a notion of strong algebrability of subsets of linear algebras. Our main results are the following. The set of all sequences from $ c_0$ which are not summable with any power is densely strongly $ \mathfrak{c}$-algebrable. The set of all sequences in $ l^\infty $ whose sets of limit points are homeomorphic to the Cantor set is comeager and strongly $ \mathfrak{c}$-algebrable. The set of all non-measurable functions from $ \mathbb{R}^{\mathbb{R}}$ is $ 2^\mathfrak{c}$-algebrable. These results complete several by other authors, within the modern context of lineability.


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Additional Information

Artur Bartoszewicz
Affiliation: Institute of Mathematics, Technical University of Łódź, Wólczańska 215, 93-005 Łódź, Poland
Email: arturbar@p.lodz.pl

Szymon Głab
Affiliation: Institute of Mathematics, Technical University of Łódź, Wólczańska 215, 93-005 Łódź, Poland
Email: szymon.glab@p.lodz.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11377-2
PII: S 0002-9939(2012)11377-2
Keywords: Algebrability, non-summable sequences, non-measurable functions, Banach-Mazur game, accumulation points, Cantor sets
Received by editor(s): May 24, 2011
Received by editor(s) in revised form: July 19, 2011, and July 23, 2011
Published electronically: July 20, 2012
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.