Banach空间的广义平均一致凸性与光滑性

The smoothness and the generalized average uniform convexity of Banach spaces

给出了广义平均一致凸,广义平均局部一致凸,广义平均弱局部一致凸等概念。讨论了这些凸性与一致光滑、非常光滑、一致极光滑、很极光滑等光滑性之间的关系。证明了:若X是光滑的,则X* 是广义平均一致凸的;若X* 是广义平均弱局部一致凸的,则X是光滑的;若X* 是广义平均一致凸的,则X是非常光滑的和很极光滑的;若X是一致极光滑的,X* 是广义平均弱局部一致凸的,则X*是局部一致凸的。

The concepts of generalized average uniform convexity, generalized average local uniform convexity and generalized average weak local uniform convexity are given. The relations among these uniform convexity and uniform smoothness, very smoothness, uniform extreme smoothness and extreme smoothness are studed. If X is uniform smooth, then X~* is generalized average uniform convex; If X~* is generalized average weak local uniform convex, then X is smooth; If X~* is generalized average uniform convex, then X is very smooth and extremely smooth; If X is uniform extremely smooth and X~* is generalized average weak local uniform convex, then X~* is local uniform convex.