Some geometric properties related to smoothness of Banach spaces Original Research Article

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