Strong algebrability of sets of sequences and functions
Authors:
Artur Bartoszewicz and Szymon Głab
Journal:
Proc. Amer. Math. Soc. 141 (2013), 827835
MSC (2010):
Primary 15A03; Secondary 28A20, 46J10
Published electronically:
July 20, 2012
MathSciNet review:
3003676
Fulltext PDF
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References 
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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 nonmeasurable functions from is algebrable. These results complete several by other authors, within the modern context of lineability.
 [1]
A.
Aizpuru, C.
PérezEslava, and J.
B. SeoaneSepúlveda, Linear structure of sets of divergent
sequences and series, Linear Algebra Appl. 418
(2006), no. 23, 595–598. MR 2260214
(2008h:40001), 10.1016/j.laa.2006.02.041
 [2]
Richard
M. Aron, José
A. Conejero, Alfredo
Peris, and Juan
B. SeoaneSepúlveda, Uncountably generated algebras of
everywhere surjective functions, Bull. Belg. Math. Soc. Simon Stevin
17 (2010), no. 3, 571–575. MR 2731374
(2011g:46041)
 [3]
Richard
Aron, V.
I. Gurariy, and J.
B. Seoane, Lineability and spaceability of sets
of functions on ℝ, Proc. Amer. Math.
Soc. 133 (2005), no. 3, 795–803 (electronic). MR 2113929
(2006i:26004), 10.1090/S0002993904075331
 [4]
Richard
M. Aron, David
PérezGarcía, and Juan
B. SeoaneSepúlveda, Algebrability of the set of
nonconvergent Fourier series, Studia Math. 175
(2006), no. 1, 83–90. MR 2261701
(2007k:42007), 10.4064/sm17515
 [5]
Richard
M. Aron and Juan
B. SeoaneSepúlveda, Algebrability of the set of everywhere
surjective functions on ℂ, 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), 10.1090/S00029947198206750693
 [7]
Artur
Bartoszewicz, Szymon
Głc ab, and Tadeusz
Poreda, On algebrability of nonabsolutely convergent series,
Linear Algebra Appl. 435 (2011), no. 5,
1025–1028. MR 2807216
(2012e:40001), 10.1016/j.laa.2011.02.008
 [8]
Frédéric
Bayart, Topological and algebraic genericity of divergence and
universality, Studia Math. 167 (2005), no. 2,
161–181. MR 2134382
(2006b:46024), 10.4064/sm16724
 [9]
L.
BernalGonzález, Denselineability in spaces of
continuous functions, Proc. Amer. Math.
Soc. 136 (2008), no. 9, 3163–3169. MR 2407080
(2009c:46038), 10.1090/S0002993908094951
 [10]
G.
Botelho, D.
Diniz, V.
V. Fávaro, and D.
Pellegrino, Spaceability in Banach and quasiBanach sequence
spaces, Linear Algebra Appl. 434 (2011), no. 5,
1255–1260. MR 2763584
(2011m:46006), 10.1016/j.laa.2010.11.012
 [11]
F.
J. GarcíaPacheco, M.
Martín, and J.
B. SeoaneSepúlveda, 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. GarcíaPacheco, N.
Palmberg, and J.
B. SeoaneSepúlveda, Lineability and algebrability of
pathological phenomena in analysis, J. Math. Anal. Appl.
326 (2007), no. 2, 929–939. MR 2280953
(2007i:26003), 10.1016/j.jmaa.2006.03.025
 [13]
Francisco
J. GarcíaPacheco and Juan
B. SeoaneSepúlveda, Vector spaces of nonmeasurable
functions, Acta Math. Sin. (Engl. Ser.) 22 (2006),
no. 6, 1805–1808. MR 2262440
(2007i:28006), 10.1007/s101140050726y
 [14]
Vladimir
I. Gurariy and Lucas
Quarta, On lineability of sets of continuous functions, J.
Math. Anal. Appl. 294 (2004), no. 1, 62–72. MR 2059788
(2005c:46026), 10.1016/j.jmaa.2004.01.036
 [15]
Alexander
S. Kechris, Classical descriptive set theory, Graduate Texts
in Mathematics, vol. 156, SpringerVerlag, New York, 1995. MR 1321597
(96e:03057)
 [16]
John
C. Oxtoby, Measure and category, 2nd ed., Graduate Texts in
Mathematics, vol. 2, SpringerVerlag, New YorkBerlin, 1980. A survey
of the analogies between topological and measure spaces. MR 584443
(81j:28003)
 [1]
 A. Aizpuru, C. PérezEslava and J. B. SeoaneSepúlveda, Linear structure of sets of divergent sequences and series, Linear Algebra Appl. 418 (2006), no. 23, 595598. MR 2260214 (2008h:40001)
 [2]
 R. M. Aron, J. A. Conejero, A. Peris and J. B. SeoaneSepúlveda, Uncountably generated algebras of everywhere surjective functions, Bull. Belg. Math. Soc. Simon Stevin 17 (2010), 571575. MR 2731374 (2011g:46041)
 [3]
 R. Aron, V. I. Gurariy and J. B. SeoaneSepúlveda, Lineability and spaceability of sets of functions on , Proc. Amer. Math. Soc. 133 (2005), no. 3, 795803. MR 2113929 (2006i:26004)
 [4]
 R. M. Aron, D. PérezGarcía and J. B. SeoaneSepúlveda, Algebrability of the set of nonconvergent Fourier series, Studia Math. 175 (2006), no. 1, 8390. MR 2261701 (2007k:42007)
 [5]
 R. M. Aron and J. B. SeoaneSepúlveda, Algebrability of the set of everywhere surjective functions on , Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 1, 2531. MR 2327324 (2008d:26016)
 [6]
 B. Balcar and F. Franěk, Independent families in complete Boolean algebras, Trans. Amer. Math. Soc. 274 (1982), no. 2, 607618. 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, 10251028. MR 2807216
 [8]
 F. Bayart, Topological and algebraic genericity of divergence and universality, Studia. Math. 167 (2005), 161181. MR 2134382 (2006b:46024)
 [9]
 L. BernalGonzález, Denselineability in spaces of continuous functions, Proc. Amer. Math. Soc. 136 (2008), 31633169. MR 2407080 (2009c:46038)
 [10]
 G. Botelho, D. Diniz, V. V. Favaro and D. Pellegrino, Spaceability in Banach and quasiBanach sequence spaces, Linear Algebra Appl. 434 (2011), 12551260. MR 2763584
 [11]
 F. J. GarcíaPacheco, M. Martín and J. B. SeoaneSepúlveda, Lineability, spaceability, and algebrability of certain subsets of function spaces, Taiwanese J. Math. 13 (2009), no. 4, 12571269. MR 2543741 (2010h:46072)
 [12]
 F. J. GarcíaPacheco, N. Palmberg and J. B. SeoaneSepúlveda, Lineability and algebrability of pathological phenomena in analysis, J. Math. Anal. Appl. 326 (2007), no. 2, 929939. MR 2280953 (2007i:26003)
 [13]
 F. J. GarcíaPacheco and J. B. SeoaneSepúlveda, Vector spaces of nonmeasurable functions, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 6, 18051808. 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, 6272. MR 2059788 (2005c:46026)
 [15]
 A. S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, 156. SpringerVerlag, New York, 1995. MR 1321597 (96e:03057)
 [16]
 J.C. Oxtoby, Measure and Category, SpringerVerlag, 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, 93005 Łódź, Poland
Email:
arturbar@p.lodz.pl
Szymon Głab
Affiliation:
Institute of Mathematics, Technical University of Łódź, Wólczańska 215, 93005 Łódź, Poland
Email:
szymon.glab@p.lodz.pl
DOI:
http://dx.doi.org/10.1090/S000299392012113772
Keywords:
Algebrability,
nonsummable sequences,
nonmeasurable functions,
BanachMazur 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.
