Extreme Points and Strongly Extreme Points in Orlicz Spaces Equipped with the Orlicz Norm

Abstract

Criteria for extreme points and strongly extreme points of the unit ball in Orlicz spaces with the Orlicz norm are given. These results are applied to characterization of extreme points of B(L 1 + L ∞) which corresponds with the result obtained by R. Grzaślewicz and H. Schaefer [10] and H. Schaefer [25]. Moreover, we show that every extreme point of B(L 1 + L ∞) is strongly extreme. We also get criteria for extreme points of B(L p ∩ L ∞) (1 ≤ p < ∞) using Theorem 1 and for strongly extreme points of B(L p ∩ L ∞) (1 ≤ p < ∞) applying Theorem 2. Although, criteria for extreme points of B(L 1 ∩ L ∞) were known (see [13]), we can easily deduce them from our main results and we can extend those results to establish which among extreme points are strongly extreme. The descriptions of the extreme and strongly extreme points of B(L p ∩ L ∞) (1 < p < ∞) are original. Moreover, criteria for extreme points and strongly extreme points of the unit ball in the subspace of finite elements of an Orlicz space are deduced on the basis of our main results.

Orlicz spaces without extreme points

“…Defining Sufficiency. The sufficiency of the inequality I Ψ (p • u 0 χ T ) < 1 follows immediately from Corollary 3 in [11]. Suppose that such that Φ(u) = au − b for u u 0 .…”

“…Although methods developed in papers [10,11,13] are useful in our considerations, in many cases to solve the problems from this paper, developing of some new techniques was necessary.…”

“…Some sufficient conditions under which the set ext B(L 0 Φ ) is empty are given in [11]. It is natural to characterize all Orlicz function spaces L 0 Φ for which the set ext B(L 0 Φ ) is empty.…”

“…(See[11, Theorem 1].) Let (T , Σ, μ) be a non-atomic and complete measure space and Φ be an arbitrary Orliczfunction.…”

Complex rotundity of Orlicz sequence spaces equipped with the p-Amemiya norm

“…For more details on Orlicz spaces we refer to [1,5,9,[11][12][13][14]. Further details about the geometry of complex Banach spaces can be found in [10,[16][17][18][19][20].…”