Banach-Saks Property and Property (β) in Cesàro Sequence Spaces

Cui, Yunan; Meng, Chenghui; Płuciennik, Ryszard

doi:10.1007/s100120070003

Cited by 19 publications

(21 citation statements)

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“…Since formulas (12) and (17) do not depend on the set A, (15) and (18) remain true if we replace set A by B or C. Hence we also get the estimates I f (t)g j (t)/(1 + ǫ) dt 1 − 3ǫ > 1 − η, j = 1, 2. It follows that g j /(1 + ǫ) ∈ s(F ; η), j = 1, 2.…”

Section: Description Of the Dual Spacementioning

confidence: 71%

“…For a long time Cesàro function spaces have not attracted a lot of attention contrary to their sequence counterparts. In fact there is quite rich literature concerning different topics studied in Cesàro sequence spaces as for instance in [11][12][13][14][15][16]. However recently in a series of papers [2][3][4], Astashkin and Maligranda started to study thoroughly the structure of Cesàro function spaces.…”

Section: Introductionmentioning

confidence: 99%

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On the dual of Cesàro function space

Kamińska

Kubiak

2012

Nonlinear Analysis: Theory, Methods & Applications

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Abstract. The goal of this paper is to present an isometric representation of the dual space to Cesàro function space Cp,w, 1 < p < ∞, induced by arbitrary positive weight function w on interval (0, l) where 0 < l ∞. For this purpose given a strictly decreasing nonnegative function Ψ on (0, l), the notion of essential Ψ-concave majorantf of a measurable function f is introduced and investigated. As applications it is shown that every slice of the unit ball of the Cesàro function space has diameter 2. Consequently Cesàro function spaces do not have the Radon-Nikodym property, are not locally uniformly convex and they are not dual spaces.

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Section: Description Of the Dual Spacementioning

confidence: 71%

Section: Introductionmentioning

confidence: 99%

On the dual of Cesàro function space

Kamińska

Kubiak

2012

Nonlinear Analysis: Theory, Methods & Applications

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“…In 1999-2000, it was proved by Cui-Hudzik [26], Cui-Hudzik-Li [29] and Cui-MengPłuciennik [31] that the Cesàro sequence spaces ces p for 1 < p < ∞ have the fixed point property (cf. also [24,Part 9]).…”

Section: Proof (A): Settingmentioning

confidence: 99%

“…[24], [26], [27], [28], [29], [30], [31], [55]). In particular, in 2007 Maligranda-Petrot-Suantai [69] proved that ces p for 1 < p < ∞ are not uniformly non-square, that is, there are sequences {x n } and {y n } from ces p such that x n c(p) = y n c(p) = 1 and lim n→∞ min( x n + y n c(p) , x n − y n c(p) ) = 2.…”

mentioning

confidence: 99%

Structure of Cesàro function spaces: a survey

Асташкин

Maligrańda

2014

Banach Center Publ.

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“…It is known that ce s p is locally uniform rotund and possess H-property [8]. Cui and Hudzik [4] proved that ce s p has the Banach-Saks of type p if p > 1, and it was shown in [5] that ce s p has β-property. Also, we know that Nakano sequence spaces are special cases of Musielak-Orlicz sequence spaces.…”

Section: On the Geometric Properties Of New Type Modular Space 161mentioning

confidence: 99%

On the geometric properities of new type modular space

Şengönül

2012

Tamkang J. Math.

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Abstract. In this paper, using a modular, we have defined the modular space M m * (p) and we have shown that the sequence space M m * (p) equipped with the Luxemburg norm is rotund and possesses H-property (or Kadec-Klee property) when p = (p k ) is bounded with p k > 1 for all k ∈ N. Introduction and PreliminariesRecently, Sanhan and Suantai [11] have generalized normed Cesàro sequence spaces to paranormed sequence spaces by making use of the Köthe sequence spaces. They have defined and studied modular structure and some geometrical properties of these generalized sequence spaces. Now, let's give some well-known descriptions to understand the subject better.Let us denote the set of all real numbers and the set of all natural numbers R and N, respectively. A sequence space is linear subspace of w , whereLet λ be a subset space of w . For a Banach space λ we denote by S(λ) and B (λ) the unit sphere and unit ball of λ, respectively. A point x 0 ∈ S(λ) is called: a) an extreme point if for every x, y ∈ S(λ) the equality 2x 0 = x + y implies x = y; b) an H-point if for any sequence (x n ) in λ such that ∥ x ∥→ 1 as n → ∞, the weak convergence of (x n ) to x 0 implies that ∥ x n − x ∥→ 0 as n → ∞.A Banach space λ is said to be rotund, if every point of S(λ) is an extreme point. A Banach space λ is said to possess H-property(or Kadec-Klee property) provided every point of S(λ) is an H-point. 2010 Mathematics Subject Classification. 46E20, 46E30, 46E40.

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Banach-Saks Property and Property (β) in Cesàro Sequence Spaces

Cited by 19 publications

References 0 publications

On the dual of Cesàro function space

On the dual of Cesàro function space

Structure of Cesàro function spaces: a survey

On the geometric properities of new type modular space

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