Arithmetic separation and Banach–Saks sets

Kryczka, Andrzej

doi:10.1016/j.jmaa.2012.04.084

Cited by 4 publications

(7 citation statements)

References 26 publications

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“…The subadditivity of the measure ϕ for convex sets is implied by the fact that the variations of the arithmetic separation of a sequence can be reduced to an arbitrary small interval by passing to a certain sequence of arithmetic means over successive equipollent blocks. A similar stabilising result holds for the quantity φ related to the ABS property (see [13]). Both results are based on Ramsey's theorem and the spreading models of Brunel and Sucheston [8].…”

Section: Quantitative Resultssupporting

confidence: 61%

“…We consider a quantity based on a modification of Beauzamy's condition, called the arithmetic separation of a sequence (see [13]), which measures distances between the arithmetic means over successive equipollent blocks of (x n ):…”

Section: Quantitative Resultsmentioning

confidence: 99%

“…Nishiura and Waterman [16] proved that every Banach space with the BS property is reflexive. This result also has a counterpart for sets: every BS subset of a Banach space is relatively weakly compact (see [13]). On the other hand, the reflexive version of the Schreier space given by Baernstein [2] shows that, in general, these implications cannot be reversed.…”

Section: Introductionmentioning

confidence: 85%

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A Quantitative Extension of Szlenk’s Theorem

Kryczka

2019

Bull. Aust. Math. Soc.

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We show that for a bounded subset $A$ of the $L_{1}(\unicode[STIX]{x1D707})$ space with finite measure $\unicode[STIX]{x1D707}$, the measure of weak noncompactness of $A$ based on the convex separation of sequences coincides with the measure of deviation from the Banach–Saks property expressed by the arithmetic separation of sequences. A similar result holds for a related quantity with the alternating signs Banach–Saks property. The results provide a geometric and quantitative extension of Szlenk’s theorem saying that every weakly convergent sequence in the Lebesgue space $L_{1}$ has a subsequence whose arithmetic means are norm convergent.

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Section: Quantitative Resultssupporting

confidence: 61%

Section: Quantitative Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 85%

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A Quantitative Extension of Szlenk’s Theorem

Kryczka

2019

Bull. Aust. Math. Soc.

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“…Hence, by (9) and Theorem 4.1 the quantity ϕ ′ inspired by [24] is equivalent to our quantities. Further, since clearly ϕ ′ ≤ ϕ, we get…”

Section: Quantitative Characterization Of Weak Banach-saks Setsmentioning

confidence: 90%

“…Some quantities related to the Banach-Saks property were defined and used in [24]. Let us recall them.…”

Section: Preliminariesmentioning

confidence: 99%

Quantification of the Banach–Saks property

Bendová

Kalenda

Spurný

2015

Journal of Functional Analysis

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We investigate possible quantifications of the Banach-Saks property and the weak Banach-Saks property. We prove quantitative versions of relationships of the Banach-Saks property of a set with norm compactness and weak compactness. We further establish a quantitative version of the characterization of the weak Banach-Saks property of a set using uniform weak convergence and ℓ 1 -spreading models. We also study the case of the unit ball and in this case we prove a dichotomy which is an analogue of the James distortion theorem for ℓ 1 -spreading models.2010 Mathematics Subject Classification. 46B20.

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Quantitative weakly compact sets and Banach-Saks sets in $$\ell _1$$

2021

Arch. Math.

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Arithmetic separation and Banach–Saks sets

Cited by 4 publications

References 26 publications

A Quantitative Extension of Szlenk’s Theorem

A Quantitative Extension of Szlenk’s Theorem

Quantification of the Banach–Saks property

Quantitative weakly compact sets and Banach-Saks sets in $$\ell _1$$

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