Local rotundity structure of Cesàro–Orlicz sequence spaces

Abstract

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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