Basic topological and geometric properties of Cesàro-Orlicz spaces

Cui, Yunan; Hudzik, Henryk; Petrot, Narin; Suantai, Suthep; Szymaszkiewicz, Alicja

doi:10.1007/bf02829808

Cited by 27 publications

(47 citation statements)

References 13 publications

(16 reference statements)

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“…The Cesàro-Orlicz sequence spaces ces M are the spaces of all real sequence x = {x k } such that The proofs of this fact and some other properties of Cesàro-Orlicz sequence spaces with the Luxemburg-Nakano norm can be found in the recent paper by Cui, Hudzik, Petrot, Suantai and Szymaszkiewicz [11]. We can also consider the Orlicz norm in the Amemiya form…”

Section: Introductionmentioning

confidence: 94%

On the James constant and B-convexity of Cesàro and Cesàro–Orlicz sequence spaces

Maligrańda

Petrot

Suantai

2007

Journal of Mathematical Analysis and Applications

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The classical James constant and the nth James constants, which are measure of B-convexity for the Cesàro sequence spaces ces p and the Cesàro-Orlicz sequence spaces ces M , are calculated. These investigations show that ces p , ces M are not uniformly non-square and even they are not B-convex. Therefore the classical Cesàro sequence spaces ces p are natural examples of reflexive spaces which are not B-convex.Moreover, the James constant for the two-dimensional Cesàro space ces(2) 2 is calculated.

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Section: Introductionmentioning

confidence: 94%

On the James constant and B-convexity of Cesàro and Cesàro–Orlicz sequence spaces

Maligrańda

Petrot

Suantai

2007

Journal of Mathematical Analysis and Applications

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“…The geometry of Cesàro sequence spaces have been extensively studied in [13][14][15][16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

The Ptolemy constant of absolute normalized norms on ℝ2

Zuo

2012

J Inequal Appl

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“…In recent years the theory of Cesàro-Orlicz sequence spaces has been studied intensively. Some basic topological properties (nontriviality, order continuity, separability and relationships between the modular and the norm defined itself) as well as some geometric properties (Fatou property, strict monotonicity and rotundity) were considered in [9]. Recently Maligranda, Petrot and Suantai in their remarkable paper [25] calculated n-dimensional James constans in Cesàro and Cesàro-Orlicz sequence spaces.…”

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confidence: 97%

“…Given any Orlicz function ϕ, we define on l 0 another convex modular ϕ : l 0 → [0, ∞], by ϕ (x) = I ϕ σ (x) and the Cesàro-Orlicz sequence space ces ϕ = x ∈ l 0 : σ x ∈ l ϕ (see [9,25]). We equip this space with the norm x ϕ = |||σ (x)||| ϕ .…”

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confidence: 99%

Local rotundity structure of Cesàro–Orlicz sequence spaces

Foralewski

Hudzik

Szymaszkiewicz

2008

Journal of Mathematical Analysis and Applications

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Some criteria for extreme points and strong U-points in Cesàro-Orlicz spaces are given. In consequence we find a Cesàro-Orlicz sequence space different from c 0 which has no extreme points. Some examples show that in these spaces the notion of the strong U-point is essentially stronger than the notion of the extreme point. Various examples presented in this paper show that there are some differences between criteria for extreme points and strong U-points in Orlicz spaces and in Cesàro-Orlicz spaces. We also show that the uniqueness of the local best approximation needs the notion of SU-point, that is, the notion of the extreme point is not strong enough here. PreliminariesLet (X, · ) be a real Banach space and let B( X) and S( X) be the closed unit ball and the unit sphere of X , respectively. A point x ∈ S( X) is called an extreme point of B( X) if for every y, z ∈ B( X) with x = y+z 2 , we have y = z. The notion of extreme point plays an important role in some branches of mathematics. For example, the Krein-Milman theorem, Choquet integral representation theorem, Rainwater theorem on convergence in the weak topology, Bessaga-Pełczyński theorem and Elton test for unconditional convergence are formulated in terms of extreme points (see [12, Chapter IX]).A point x ∈ S( X) is said to be a strong U-point (SU-point for short) of B( X) if for any y ∈ S( X) with x + y = 2, we have x = y (cf. [2], where SU-points are called rotund points). Recall that the nature of an SU-point is such that a point x ∈ S( X) is a point of local uniform rotundity if and only if x is a point of compact local uniform rotundity and an SU-point (see [8]). It is obvious that a Banach space X is rotund if and only if every point of S( X) is an extreme point of B( X), as well as if and only if any point of S( X) is an SU-point of B( X), but the notion of SU-point is essentially stronger than the notion of extreme point. Namely in l 2 ∞ (two-dimensional l ∞ space) the points x = (1, 1) and y = (1, −1) are extreme points of B(l 2 ∞ ). Since x = y and x + y = 2, neither x nor y is a strong U-point of B(l 2 ∞ ).It is well known that rotundity of a normed space X is important for the uniqueness of the best approximation element in any bounded closed convex and nonempty set A ⊂ X for any x ∈ X \ A. Namely, if X is a rotund normed space, A is a bounded closed convex and nonempty set in X and x ∈ X \ A, then if y ∈ A is such that x − y = d(x, A) := inf{ x − z : z ∈ A}, then for any z ∈ A, z = y, we have d(x, A) < x − z . If X is not rotund, then there is a bounded closed convex and nonempty set A in X and x ∈ X \ A such that there is a continuum of points in A that realize the distance d(x, A).

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Basic topological and geometric properties of Cesàro-Orlicz spaces

Cited by 27 publications

References 13 publications

On the James constant and B-convexity of Cesàro and Cesàro–Orlicz sequence spaces

On the James constant and B-convexity of Cesàro and Cesàro–Orlicz sequence spaces

The Ptolemy constant of absolute normalized norms on ℝ2

Local rotundity structure of Cesàro–Orlicz sequence spaces

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