The Baire category of ideal convergent subseries and rearrangements. (English) Zbl 1378.54031
Let \(\mathcal{I}\) be an ideal, let \(S=\{ s\in \mathbb{N}^\mathbb{N}: \forall n\in\mathbb{N} \;s(n)<s(n+1)\}\) and \(P=\{ p\in\mathbb{N}^\mathbb{N}: \;p \text{ is a bijection}\}\). These are Polish spaces with the standard Polish topology in \(\mathbb{N}^\mathbb{N}\). An ideal \(\mathcal{I}\) on \(\mathbb{N}\) is called 1-shift invariant if for any \(A\subset\mathbb{N}, A\in\mathcal{I}\) implies \(\mathbb{N} \setminus (A+1) \notin \mathcal{I}\).
In the paper it is shown that if \(\mathcal{I}\) is a 1-shift invariant ideal on \(\mathbb{N}\) with the Baire property and a series \(\sum_n x_n\) with terms in a real Banach space \(X\) is not unconditionally convergent, then the sets \(\{ s\in S: \sum_n x_{s(n)} \text{ is\;} \mathcal{I}\text{-convergent}\}\) and \(\{ p\in P: \sum_n x_{p(n)} \text{ is } \mathcal{I}\text{-convergent}\}\) are meager in \(S\) and \(P\), respectively. Further, if \(\mathcal{I}\) is an ideal on \(\mathbb{N}\) with the Baire property and a series \(\sum_n x_n\) with terms in \(\mathbb{R}\) is not unconditionally convergent, then the sets \(\{ s\in S: (\sum_{i=1}^n x_{s(i)})_n \text{ is } \mathcal{I}\text{-bounded}\}\) and \(\{ p\in P: (\sum_{i=1}^n x_{p(i)})_n \text{ is } \mathcal{I}\text{-bounded}\}\) are meager in \(S\) and \(P\), respectively.
In the paper it is shown that if \(\mathcal{I}\) is a 1-shift invariant ideal on \(\mathbb{N}\) with the Baire property and a series \(\sum_n x_n\) with terms in a real Banach space \(X\) is not unconditionally convergent, then the sets \(\{ s\in S: \sum_n x_{s(n)} \text{ is\;} \mathcal{I}\text{-convergent}\}\) and \(\{ p\in P: \sum_n x_{p(n)} \text{ is } \mathcal{I}\text{-convergent}\}\) are meager in \(S\) and \(P\), respectively. Further, if \(\mathcal{I}\) is an ideal on \(\mathbb{N}\) with the Baire property and a series \(\sum_n x_n\) with terms in \(\mathbb{R}\) is not unconditionally convergent, then the sets \(\{ s\in S: (\sum_{i=1}^n x_{s(i)})_n \text{ is } \mathcal{I}\text{-bounded}\}\) and \(\{ p\in P: (\sum_{i=1}^n x_{p(i)})_n \text{ is } \mathcal{I}\text{-bounded}\}\) are meager in \(S\) and \(P\), respectively.
Reviewer: Jan Borsik (Kosice)
MSC:
54E52 | Baire category, Baire spaces |
40A30 | Convergence and divergence of series and sequences of functions |
40A05 | Convergence and divergence of series and sequences |
28A05 | Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets |
46B25 | Classical Banach spaces in the general theory |
Keywords:
ideal convergence; Baire category; unconditional convergence of series; subseries; rearrangementReferences:
[1] | Balcerzak, M.; Głąb, Sz.; Wachowicz, A., Qualitative properties of ideal convergent subsequences and rearrangements, Acta Math. Hung., 150, 312-323 (2016) · Zbl 1399.40010 |
[2] | Balcerzak, M.; Popławski, M.; Wachowicz, A., Ideal convergent subsequences and rearrangements for divergent sequences of functions, Math. Slovaca (2017), accepted · Zbl 1459.40002 |
[3] | Bernstein, A. R., A new kind of compactness for topological spaces, Fundam. Math., 66, 185-193 (1970) · Zbl 0198.55401 |
[4] | Bhaskara Rao, M.; Bhaskara Rao, K. P.S.; Rao, B. V., Remarks on subsequences, subseries and rearrangements, Proc. Am. Math. Soc., 67, 293-296 (1977) · Zbl 0412.40001 |
[5] | Bilyeu, R. G.; Kallman, R. R.; Lewis, P. W., Rearrangements and category, Pac. J. Math., 131, 41-46 (1986) · Zbl 0546.42005 |
[6] | Blass, A.; Brendle, J.; Brian, W.; Hamkins, J. D.; Hardy, M.; Larson, P. B., The rearrangement number · Zbl 1516.03015 |
[7] | Červeneňanský, J.; Šalát, T.; Toma, V., Remarks on statistical and \(I\)-convergence of series, Math. Bohem., 130, 177-184 (2005) · Zbl 1110.40001 |
[8] | Dems, K., On \(I\)-Cauchy sequences, Real Anal. Exch., 30, 123-128 (2004/2005) · Zbl 1070.26003 |
[9] | Dindoš, M.; Martišovitš, I.; Šalát, T., Remarks on infinite series in linear normed spaces, Tatra Mt. Math. Publ., 19, 31-46 (2000) · Zbl 0984.40001 |
[10] | Dindoš, M.; Šalát, T.; Toma, V., Statistical convergence of infinite series, Czechoslov. Math. J., 53, 989-1000 (2003) · Zbl 1080.40500 |
[11] | Farah, I., Analytic quotients. Theory of lifting for quotients over analytic ideals on integers, Mem. Am. Math. Soc., 148 (2000) · Zbl 0966.03045 |
[12] | Fast, H., Sur la convergence statistique, Colloq. Math., 2, 241-244 (1951) · Zbl 0044.33605 |
[13] | Fridy, J. A., On statistical convergence, Analysis, 5, 301-313 (1985) · Zbl 0588.40001 |
[14] | Furstenberg, H., Recurrence in Ergodic Theory and Combinatorial Number Theory (1981), Princeton University Press · Zbl 0459.28023 |
[15] | Głąb, Sz.; Olczyk, M., Convergence of series on large set of indices, Math. Slovaca, 65, 1095-1106 (2015) · Zbl 1363.40006 |
[16] | Jalali-Naini, S. A., The Monotone Subsets of the Cantor Space, Filters and Desriptive Set Theory (1976), University of Oxford, PhD thesis |
[17] | Kadets, M. I.; Kadets, V. M., Series in Banach Spaces (1997), Birkhäuser: Birkhäuser Basel · Zbl 0876.46009 |
[18] | Kallman, R. R., Subseries and category, J. Math. Anal. Appl., 132, 234-237 (1988) · Zbl 0676.46004 |
[19] | Kostyrko, P.; Šalát, T.; Wilczyński, W., \(I\)-convergence, Real Anal. Exch., 26, 669-685 (2000-2001) · Zbl 1021.40001 |
[20] | Kostyrko, P.; Mačaj, M.; Šalát, T.; Sleziak, M., \(I\)-convergence and extremal \(I\)-points, Math. Slovaca, 55, 443-464 (2005) · Zbl 1113.40001 |
[21] | Leonov, A., On the coincidence of the limit point range and the sum range along a filter of filter convergent series, Visn. Khark. Univ. Ser. Math. Prykl. Mat. Mekh., 826, 134-140 (2008) · Zbl 1164.40301 |
[22] | Leonov, A.; Orhan, C., On filter convergence of series, Real Anal. Exch., 40, 459-474 (2001) · Zbl 1457.40006 |
[23] | Nurrey, F.; Ruckle, W. H., Generalized statistical convergence and convergence free spaces, J. Math. Anal. Appl., 245, 513-527 (2000) · Zbl 0955.40001 |
[24] | Oxtoby, J. C., Measure and Category (1980), Springer: Springer New York · Zbl 0217.09201 |
[25] | Šalát, T., On subseries of divergent series, Mat. Čas. SAV, 18, 312-338 (1968) · Zbl 0169.07201 |
[26] | Šalát, T., On statistically convergent sequences of real numbers, Math. Slovaca, 30, 139-150 (1980) · Zbl 0437.40003 |
[27] | Srivastava, S. M., A Course of Borel Sets (1998), Springer: Springer New York · Zbl 0903.28001 |
[28] | Talagrand, M., Compacts de fonctions mesurables et filtres non mesurables, Stud. Math., 150, 13-43 (1980) · Zbl 0435.46023 |
[29] | Tripathy, B. C., Statistically convergent series, Punjab Univ. J. Math., 32, 1-8 (1999) · Zbl 0966.40003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.