关于一致凸Banach空间的注记

Remarks on Uniformly Convex Banach Spaces

给出了Banach空间一致凸的几个新的充要条件.定理 设1<p<+∞,λ,μ∈(0,1),λ+μ=1,X是Banach空间,则下列条件等价: X是一致凸的; 对任意ε>0,存在δ>0,使得当‖x‖≤1,‖y‖≤1,‖x-y‖≥ε时,有‖λx+μy‖≤1-δ 对任意满足‖xn‖≤1,‖yn‖≤1,limn∞‖λxn+μyn‖1的序列{xn},{yn}都有limn∞‖xn-yn‖=0 对任意ε>0,存在δ>0,使得当‖x‖≤1,‖x-y‖≥ε时。

Some new sufficient and necessary conditions are given for uniformly convex Banach spaces. The main result is the following theorem.Theorem Suppose that 1<p<+∞, λ,μ∈(0, 1), λ+μ=1 and X is a Banach space. Then the following conditions are equivalent:? X is uniformly convex;? For every ε>0 there exists δ>0 such that‖λx+μy‖≤1-δfor all ‖x‖≤1, ‖y‖≤1 satisfying ‖x-y‖≥ε;? For every sequence {xn}, {yn} satisfying ‖xn‖≤1, ‖yn‖≤1 and limn∞‖λxn+μyn‖1, one has limn∞‖xn-yn‖=0 ? For every ε>0 there exists δ>0 such that‖λx+μy‖p<λ‖x‖p+μ‖y‖p-δfor all ‖x‖≤1 and y∈X satisfying ‖x-y‖≥ε.