K (?) The-point-state Properties Of The Bochner Space

Let E be a Kothe space which is defined on the completeσ-finite measure space (Ω,Σ,μ) and X be a Banach space. We call all strong measurable vector-valued function x:Ωâ†'â…©and its equivalent class Kothe-Bochner space, denoted by E(X). It is a Banach space when endowed with norm‖x‖=‖‖x(·)‖‖E.This is a very broad and abstract categories of space. It includes Orlicz-Lorentz space, Vector-valued Musielak-Orlicz space and so on. When the researchers discovered the common nature of these spaces,they began to explore the nature of inherent properties of Kothe-Bochner space. This paper investigated the pointwise geometric properties of Kothe-Bochner space and focus on the extreme points, locally uniform rotundity points and strongly extreme points, The main results are:(1) In 2006,Ren liwei and the other proved:when E is strictly monotonic, if (a)x=‖x(·)‖x is an extreme point in B(E); (b) for almost every t∈supp x, is an extreme point in B(X),then x is an extreme point in B(E(X)).This paper proved that (a) and (b) are just the necessary conditions. As an application, we also established Orlicz-Lorentz-Bochner space. We obtained the necessary and sufficient conditions when it is strictly convex. (2) We obtained the sufficient conditions and necessary conditions for locally uniform rotundity points in S(E(X))which improved R. Plucienniks' work. (3) For the strongly extreme points, H. Hudzik and other authors have given many different forms of sufficient conditions and necessary conditions. In this paper, we gave a perfect form of necessary condition. That is:if x is a strongly extreme points in S(E(X)),then (a)‖x(t)‖x is a strongly extreme point in S(E);(b) for almost every t∈supp x, is a strongly extreme point in S(X).