商空间的k-端点和k-严格凸性

On the Extreme Point and Rotundity of Quotient Spaces

证明了商空间X/M单位球面上的点[x]为闭单位球的k-端点的充分条件是[x]与X的单位球面的交集中任一点均为闭单位球的k-端点,其中M是Banach空间X的可逼近子空间.进而推出了Banach空间X以它的可逼近子空间M为模的商空间X/M对X的k-严格凸性的继承性.同时,以由N-函数生成的Orlicz空间为例,说明了上述结论成立可逼近条件是必要的.

This paper presents that a point in the unit sphere of X/M is k-extreme point of closed unit ball if every point on [x]∩S（X） is a k-extreme point of closed unit ball . Here M is subpace of Banach space X and it is proximal in X. And by it,k-rotundity of X may be lifted to the quotient space X/M if X is proximal in X. Moreover, taking the example of Orlicz space generalized N-function, it points out that the condition making the above conclusion tenable is essential in general .