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

DOI:
https://doi.org/10.1090/S0002-9939-2012-11377-2

Published electronically:
July 20, 2012

MathSciNet review:
3003676

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Abstract | References | Similar Articles | Additional Information

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 which are not summable with any power is densely strongly -algebrable. The set of all sequences in whose sets of limit points are homeomorphic to the Cantor set is comeager and strongly -algebrable. The set of all non-measurable functions from is -algebrable. These results complete several by other authors, within the modern context of lineability.

**[1]**A. Aizpuru, C. Pérez-Eslava and J. B. Seoane-Sepúlveda,*Linear structure of sets of divergent sequences and series*, Linear Algebra Appl.**418**(2006), no. 2-3, 595-598. MR**2260214 (2008h:40001)****[2]**R. M. Aron, J. A. Conejero, A. Peris and J. B. Seoane-Sepúlveda,*Uncountably generated algebras of everywhere surjective functions*, Bull. Belg. Math. Soc. Simon Stevin**17**(2010), 571-575. MR**2731374 (2011g:46041)****[3]**R. Aron, V. I. Gurariy and J. B. Seoane-Sepúlveda,*Lineability and spaceability of sets of functions on*, Proc. Amer. Math. Soc.**133**(2005), no. 3, 795-803. MR**2113929 (2006i:26004)****[4]**R. M. Aron, D. Pérez-García and J. B. Seoane-Sepúlveda,*Algebrability of the set of non-convergent Fourier series*, Studia Math.**175**(2006), no. 1, 83-90. MR**2261701 (2007k:42007)****[5]**R. M. Aron and J. B. Seoane-Sepúlveda,*Algebrability of the set of everywhere surjective functions on*, Bull. Belg. Math. Soc. Simon Stevin**14**(2007), no. 1, 25-31. MR**2327324 (2008d:26016)****[6]**B. Balcar and F. Franěk,*Independent families in complete Boolean algebras*, Trans. Amer. Math. Soc.**274**(1982), no. 2, 607-618. MR**675069 (83m:06020)****[7]**A. Bartoszewicz, S. Głab and T. Poreda,*On algebrability of nonabsolutely convergent series*, Linear Algebra Appl.**435**(2011), no. 5, 1025-1028. MR**2807216****[8]**F. Bayart,*Topological and algebraic genericity of divergence and universality*, Studia. Math.**167**(2005), 161-181. MR**2134382 (2006b:46024)****[9]**L. Bernal-González,*Dense-lineability in spaces of continuous functions*, Proc. Amer. Math. Soc.**136**(2008), 3163-3169. MR**2407080 (2009c:46038)****[10]**G. Botelho, D. Diniz, V. V. Favaro and D. Pellegrino,*Spaceability in Banach and quasi-Banach sequence spaces*, Linear Algebra Appl.**434**(2011), 1255-1260. MR**2763584****[11]**F. J. García-Pacheco, M. Martín and J. B. Seoane-Sepúlveda,*Lineability, spaceability, and algebrability of certain subsets of function spaces*, Taiwanese J. Math.**13**(2009), no. 4, 1257-1269. MR**2543741 (2010h:46072)****[12]**F. J. García-Pacheco, N. Palmberg and J. B. Seoane-Sepúlveda,*Lineability and algebrability of pathological phenomena in analysis*, J. Math. Anal. Appl.**326**(2007), no. 2, 929-939. MR**2280953 (2007i:26003)****[13]**F. J. García-Pacheco and J. B. Seoane-Sepúlveda,*Vector spaces of non-measurable functions*, Acta Math. Sin. (Engl. Ser.)**22**(2006), no. 6, 1805-1808. MR**2262440 (2007i:28006)****[14]**V. I. Gurariy and L. Quarta,*On lineability of sets of continuous functions*, J. Math. Anal. Appl.**294**(2004), no. 1, 62-72. MR**2059788 (2005c:46026)****[15]**A. S. Kechris,*Classical descriptive set theory*, Graduate Texts in Mathematics,**156**. Springer-Verlag, New York, 1995. MR**1321597 (96e:03057)****[16]**J.C. Oxtoby,*Measure and Category*, Springer-Verlag, New York, 1980. MR**584443 (81j:28003)**

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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:
https://doi.org/10.1090/S0002-9939-2012-11377-2

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.