Extreme points and strongly extreme points in Orlicz spaces equipped with the Orlicz norm
type: Item
instance of: scholarly article
title: Extreme points and strongly extreme points in Orlicz spaces equipped with the Orlicz norm
zbMATH Open document ID: 1060.46020
DOI: 10.4171/ZAA/1174
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2004-06-15T00:00:00Z
review text: The two main results of the present paper give a full description of the extreme points and the strongly extreme points of the unit ball of Orlicz spaces with the Orlicz norm, denoted by \( L_{ \Phi}^0 \) for convenient \( \Phi .\) The Amemiya formula for the Orlicz norm plays an important role not only in the proofs, which are very laborious, but in the statement of the two basic results. The authors use the result on extreme points for characterizing the spaces \( L_{ \Phi}^0 \) which are strictly convex and for getting some known results about extreme points in the unit balls of \( L_{ \infty},\) \( L_1 + L_{ \infty}\) and \( L_1 \cap L_{ \infty}.\) They characterize the strongly extreme points in the unit ball of \( L_1 \cap L_{ \infty}\) and prove that in \(L_{ \infty}\) and \( L_1 + L_{ \infty}\) every extreme point is strongly extreme. They also apply the main results to prove that \( L_p \cap L_{ \infty}\) \( ( 1<p< \infty )\) is strictly convex and to describe the strongly extreme points of its unit ball. The paper finishes with an application to the study of the space \( E_{ \Phi},\) the subspace of finite elements of \( L_{ \Phi}^0 ,\) where it is shown that every extreme point of the unit ball is an extreme point of the unit ball of \( L_{ \Phi}^0 \). From this it can be deduced that the set of strongly extreme points either is empty or coincides with the set of extreme points.
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 2075231
zbMATH Keywords: Orlicz space
zbMATH Keywords: Orlicz norm
zbMATH Keywords: extreme point
zbMATH Keywords: strongly extreme point
author: Yunan Cui
author: Henryk Hudzik
author: Ryszard Płuciennik
reviewed by: Juan Francisco Mena Jurado
MaRDI profile type: Publication
cites work: Q4338822
cites work: Isometric copies of 𝑙¹ and 𝑙^{∞} in Orlicz spaces equipped with the Orlicz norm
description: scientific article; zbMATH DE number 2075231