Some remarks on Cesàro-Orlicz sequence spaces

“…If x(m) = 0 for every m > n 2 , then, by condition (ii), we have a ϕ = 0. Proceeding analogously as in the proof of the theorem about rotundity of (ces 2 ϕ , · ϕ ) in [14], we get a contradiction with (1). Now, let us denote by m 2 the smallest number in supp x such that m 2 > n 2 .…”

Section: Remarkmentioning

confidence: 69%

“…If x(m + 1) = 0, then we get inequality (6) for m. If there exists p > m such that x(m + 1) = · · · = x(p) = 0 and x(p + 1) = 0 then, by p / ∈ B x , we get (6) on a pth coordinate. If we have x(n) = 0 for every n > m, then the proof can proceed analogously as in the theorem about rotundity of (ces 2 ϕ , · ϕ ) in [14]. Hence ϕ ( x+ y 2 ) < 1.…”

Section: Theorem 2 An Element X ∈ S(ces ϕ ) Is An Su-point Of B(ces mentioning

confidence: 88%

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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“…Note that in [5] there is another explicit example of an Orlicz function ϕ for which the space ces ϕ contains an order isometric copy of ∞ , however in that case it can be easily checked that α(ϕ) > 1 and it is not immediately clear whether the condition (ii) of Proposition 2 is satisfied or not by this function ϕ.…”

Section: Example 1 Letmentioning

confidence: 94%

“…Cesàro-Orlicz sequence spaces ces ϕ appeared for the first time in 1988 [16] and since then they have been studied by a number of authors [3,5,6,13,20]. We will consider in this paper the problem of existence of order linearly isometric copy of ∞ in ces ϕ under the Luxemburg norm.…”

Section: Introductionmentioning

confidence: 99%

On isometric copies of ℓ∞ and James constants in Cesàro–Orlicz sequence spaces

Kamińska

Kubiak

2010

Journal of Mathematical Analysis and Applications

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For a wide class of Orlicz functions not satisfying the growth condition δ 2 we show that the Cesàro-Orlicz sequence spaces ces ϕ equipped with the Luxemburg norm contain an order linearly isometric copy of ∞ . We also compute the n-th James constant in these spaces for any Orlicz function ϕ, under either the Luxemburg or Orlicz norm, showing that they are equal to n for any natural n 2. In particular, we prove that the non-trivial spaces ces ϕ are not B-convex for any Orlicz function ϕ.

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Some remarks on Cesàro-Orlicz sequence spaces

Cited by 6 publications

References 1 publication

The Ptolemy constant of absolute normalized norms on ℝ2

The Ptolemy constant of absolute normalized norms on ℝ2

Local rotundity structure of Cesàro–Orlicz sequence spaces

On isometric copies of ℓ∞ and James constants in Cesàro–Orlicz sequence spaces

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