K(o)the-Bochner空间的凸性

本文利用K(o)the函数空间的性质以及K(o)the函数空间与K(o)the-Bochner空间的关系,讨论了K(o)the-Bochner空间E(X)的凸性,主要结果如下:(a)给出E(X)的端点的充分条件,得到了E(X)严格凸的判据,相应地推广了Lp(μ,X)以及LΦ(X)的结果;(b)讨论了E(X)的弱局部一致凸和局部完全k-凸;(c)刻画了E(X)的强凸,给出了E(X)强凸的充要条件.

Rotundity in K(o)the-Bochner Spaces

In this paper, using the properties of K(o)the function spaces and the relation between K(o)the function spaces and K(o)the-Bochner spaces, we discuss rotundity in K(o)theBochner spaces E(X), the main results are as follows:(a) We give the sufficient condition of extreme points of E(X), obtain a criterion for strict rotundity of these spaces, and accordingly generalize the results of Lp(μ, X) and LΦ(X);(b) We discuss weak local uniform rotundity and local full k-rotundity of E(X) and give their criteria;(c) We characterize strong rotundity of E(X) and obtain a sufficient and necessary condition of strong rotundity of E(X).