Extendability to summable ideals

Klinga, Paweł; Nowik, Andrzej

doi:10.1007/s10474-017-0704-8

Cited by 7 publications

(5 citation statements)

References 10 publications

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“…A classical theorem of Riemann says that any conditional convergent series of reals numbers can be rearranged to converge to any given real number or to diverge to +∞ or −∞. In other words, if (a n ) n is a conditional convergent series and r ∈ R ∪ {+∞, −∞}, there is a permutation π : N → N such that n a π(n) = r. In [18,34] is considered a property of ideals motivated by Riemann's theorem. Let us say that an ideal I has the property R, if for any conditionally convergent series n a n of real numbers and for any r ∈ R ∪ {+∞, −∞}, there is a permutation π : N → N such that n a π(n) = r and {n ∈ N : π(n) = n} ∈ I.…”

Section: Ramsey and Convergence Propertiesmentioning

confidence: 99%

“…Similarly, I has property W, if for any conditionally convergent series of reals n a n , there exists A ∈ I such that the restricted series n∈A a n is still conditionally convergent. In [34] it is studied similar properties but for series of vectors in R 2 . [34]) Suppose that (i) I has the R property; (ii) n a n is a conditionally convergent series of reals; (iii) n b n is divergent and all b n are positive reals.…”

Section: Ramsey and Convergence Propertiesmentioning

confidence: 99%

“…In [34] it is studied similar properties but for series of vectors in R 2 . [34]) Suppose that (i) I has the R property; (ii) n a n is a conditionally convergent series of reals; (iii) n b n is divergent and all b n are positive reals. Does there exist W ∈ I such that n∈W a n is conditionally convergent and n∈W b n = ∞?…”

Section: Ramsey and Convergence Propertiesmentioning

confidence: 99%

Section: Ramsey and Convergence Propertiesmentioning

confidence: 99%

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Ideals on countable sets: a survey with questions

Aylwin

2019

revint

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Section: Ramsey and Convergence Propertiesmentioning

confidence: 99%

Section: Ramsey and Convergence Propertiesmentioning

confidence: 99%

Section: Ramsey and Convergence Propertiesmentioning

confidence: 99%

Section: Ramsey and Convergence Propertiesmentioning

confidence: 99%

See 2 more Smart Citations

Ideals on countable sets: a survey with questions

Aylwin

2019

revint

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“…We will show that I does not have PLI. Note that this ideal was studied in [18] and denoted by Lac. Suppose that I has PLI.…”

Section: Subseries From the Measure Viewpointmentioning

confidence: 99%

Ideal convergent subseries in Banach spaces

Balcerzak¹,

Popławski²,

Wachowicz³

2018

Preprint

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Assume that I is an ideal on N, and n xn is a divergent series in a Banach space X. We study the Baire category, and the measure of the set A(I) := t ∈ {0, 1} N : n t(n)xn is I-convergent . In the category case, we assume that I has the Baire property and n xn is not unconditionally convergent, and we deduce that A(I) is meager. We also study the smallness of A(I) in the measure case when the Haar probability measure λ on {0, 1} N is considered. If I is analytic or coanalytic, and n xn is I-divergent, then λ(A(I)) = 0 which extends the theorem of Dindoš, Šalát and Toma.Generalizing one of their examples, we show that, for every ideal I on N, with the property of long intervals, there is a divergent series of reals such that λ(A(Fin)) = 0 and λ(A(I)) = 1.

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On some properties of Lévy vectors and their variations

Klinga

Nowik

2023

Lith Math J

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We investigate some properties of Lévy vectors of vector series Σn∈ω vn. We consider a generalization of an example from [P. Klinga, Rearranging series of vectors on a small set, J. Math. Anal. Appl., 424(2):966–974, 2015] proving that for a special class of vector series, there exists a subseries that has no convergent subseries with the sum range equal to the plane. To prove this result, we introduce a useful notion of a Lévy supervector and prove some of its properties.

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Extendability to summable ideals

Cited by 7 publications

References 10 publications

Ideals on countable sets: a survey with questions

Ideals on countable sets: a survey with questions

Ideal convergent subseries in Banach spaces

On some properties of Lévy vectors and their variations

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