Some geometric properties related to smoothness of Banach spaces

Abstract

In this paper, we introduce a new class of Banach spaces, locally k nearly uniformly smooth spaces, which is the dual notion to locally k nearly uniformly convex spaces introduced by Denka Kutzarova. The relationships between locally k nearly uniformly smooth spaces, locally k uniformly smooth Banach spaces, and locally fully k smooth Banach spaces are discussed. We also introduce the two new smooth properties, ( S k - I ) and ( S k - II ), which are the extensions of smooth properties (S-I) and (S-II) that were introduced by Suyalatu, and discuss the relations between ( S k - I ), ( S k - II ) and other smooth properties. At last, the locally fully k -smoothness, compact locally fully k -smoothness, smooth properties ( S k - I ) and ( S k - II ) in product space l p ( X i ) are discussed.