中文摘要：

本文研究了ω-非常凸空间和ω-非常光滑空间的问题.利用局部自反原理和切片证明了ω-非常凸空间和ω-非常光滑空间的对偶关系,讨论了ω-非常凸空间和ω-非常光滑空间与其它凸性和光滑性的关系,给出了ω-非常凸空间与ω-非常光滑空间的若干特征刻画,所得结果完善了关于Banach空间凸性与光滑性理论的研究.

英文摘要：

In this article,we study the problems of about ω-very convex spaces and ω-very smooth spaces.Using the local reflexive principle and the slice of unit sphere,we show that theω-very convex spaces and ω-very smooth spaces are dual notions and study the relation betweenω-very convexity,ω-very smoothness with various convexity and smoothness,and give some characteristic descriptions of ω-very smoothness and ω-very convexity.These results perfect the research on convexity and smoothness about Banach spaces.