Banach空间的凸性和光滑性及其应用  

On the convexity and smoothness of Banach space and its application

作　　者：方习年[1] 

机构地区：[1]安徽机电学院应用数理系,安徽芜湖241000

出　　处：《安徽机电学院学报》2001年第3期1-8,共8页Journal of Anhui Institute of Mechanical and Electrical Engineering

摘　　要：简要概述国际和国内 Banach空间理论的研究状况，综述对 Banach空间的凸性和光滑性所做的工作：用线性泛函的函数值作元素构成的行列式、单位球的切片、一致凸定义的形式以及特征函数等形式 ,分别刻划及定义空间的 k—严格凸、 (局部 )k—致凸、 k—强凸、 (局部 )近—致凸、近强凸等凸性以及 (局部 )k—致光滑， k—强光滑 (局部 )近－致光滑等光滑性，并讨论以上空间的关系及性质；给出弱 Banach— Saks性质的新特征，引入 B— NUC和 g— NUCε空间概念.证明：具有 Banach— Saks性质的近－致凸空间等价于 B— NUC空间、具有 Banach— Saks性质的 NUCε空间等价于 g— NUCε空间.举例说明非 k— NUC空间的 B— NUC空间的存在性；研究近强凸、近非常凸空间的几何性质、拓朴性质，并得到：近强凸空间中的度量投影具有上半连续性，近非常凸空间中的度量投影具有弱上半连续性.This paper is to briefly introduce the study on Banach space theory both at home and in the world as well as the writers' research on convexity and smoothness of Banach space.By using the unit slice, definition form of uniform convexness, characteristic function of space、 and determinant which is composed of linear functional values, We describe and define the space convexity, such as k－ uniform convex, (local) k－ uniform convex, k－ strong convex, (local) nearly uniform convex and nearly strong convex, and the smoothnees of space, such as k－ uniform smoothness, k－ strong smoothness, (local) nearly uniform smoothness. Based on this, we discuss their property and their relationship. To introduce the conception of B－ NUC and g－ NUC space by presenting that the new characteristics of weak Banach－ Saks property equals B－ NUC space. The geometrical and topological property of nearly strong convex and nearly very convex space have been studied. We get the result that: Let A be a proximinal convex subset in a nearly strong convex (nearly very convex) Banach space, then the metric projection PA is norm－ norm upper semicontinous (normweak upper semicontinous).

关 键 词：BANACH空间 凸性 光滑性 BANACH-SAKS性质 度量投影 连续性 

分 类 号：O177.2[理学—数学]