Extreme Points and Rotundity in Musielak-Orlicz-Bochner Function Spaces Endowed with Orlicz Norm

The criteria for extreme point and rotundity of Musielak-Orlicz-Bochner function spaces equipped with Orlicz norm are given. Although criteria for extreme point of Musielak-Orlicz function spaces equipped with the Orlicz norm were known, we can easily deduce them from our main results. 1. Introduction Let be a real Banach space. and denote the unit sphere and unit ball, respectively. By denote the dual space of . Let , and denote the set natural number, reals, and nonnegative reals, respectively. A point is said to be extreme point of if and imply . The set of all extreme points of is denoted by . If , then is said to be rotund. A point is said to be strongly extreme point if for any with , and , there holds . If the set of all strongly extreme points of is equal to , then is said to be midpoint local uniform rotund. 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 weak topology, Bessaga-Pelczynski theorem, and Elton test unconditional convergence are strongly connected with this notion. In [1], using the principle of locally reflexivity, a remarkable theorem describing connections between extreme points of and strongly extreme points of is proved. Namely, a Banach space is midpoint local uniformly rotund if and only if every point of is an extreme point in . Another proof of this theorem based on Goldstein's theorem is given in [2]. Analyzing the proof of this fact one can easily see its local version, namely, if is a strongly extreme point in , then is an extreme point in , where is the mapping of canonical embedding of into . The criteria for extreme point and rotundity in the classical Musielak-Orlicz function spaces have been given in [3] already. However, because of the complication of Musielak-Orlicz-Bochner function spaces equipped with Orlicz norm, at present, the criteria for extreme point and rotundity have not been discussed yet. The aim of this paper is to give criteria for extreme point and rotundity of Musielak-Orlicz-Bochner function spaces equipped with Orlicz norm. By the result of this paper, it is easy to see that the result of [3] is true. Let be nonatomic measurable space. Suppose that a function satisfies the following conditions: (1)for , , , and for some ; (2)for , , is convex on with respect to ; (3)for each is a -measurable function of on . Let denote the right derivative of at (where if , let ) and let be the generalized inverse function of defined on by Then for any and -a.e.