Banach空间中的局部各向一致凸和局部弱紧一致凸(英文)

Banach Spaces That Are Locally UniformlyRotund in Every Directions and in WeaklyCompact Sets of Directions

1977年,Mark.A.Simith在其博土论文中(见[6])引入Banach空间中的弱紧、一致凸(URWC)凸性新概念,并给出了一个Banach空间是URED但不是URWC的例子,以及一个Banach空间是URWC但不是WUR的例子。本文继续Mark.A的工作,引入了Banach空间中的局部各向一致凸(LUREO)和局部弱紧一致凸(LURWC)两个凸性新概念。作为本文的重要结果,本文给出了一个Banach空间是LURED但既不是URED也不是LURWC的例子;一个Banach空间是LURWC但不是LURWC的例子。特别,本文还证明了熟知的严格凸(SR),实际上闵是本文所新引入的一种局部一致凸性,即:局部各向一致凸。从而使严格凸在凸性分类中找到了一个恰当的位置。通过本文的工作,使Banach空间中的凸性类型得到了完善。

Continuing the works of Mark. A. Sminth, this paper introduces the other two new notions of locally uniformly rotundity, that are locally uniformly rotund in every directions (LURED) and locally uniformly rotund in weakly compact sets of idrections (LURWC). As main resuly, this paper gives three examples, namely, a Banach space which is LURED but neither URED nor LURWC, a Banach space which is LURWC but not URWC, and a Banach space which is LURWC but not LURW C in a conjugate space. These examples show that the new notions LURED and LURWC are distinct from the other uniformly rotundity and locally uniformly rotundity that we have known before. Specifically, this paper shows that the well known strictly fotundity (SR) actually is one of the locally uniformly rotundities, namely, LURED given in this paper. It makes the SR find a property position in rotundity classfication in Banach spaces.