Various k-Smoothnesses of Norms on Banach Spaces

Abstract:

      We define a concept of very k-smoothness(kVS) for every natural number k, andgives the necessary and sufficient conditions for very k-smoothness to be k-smoothnessand weak Hahn-Banach smoothness (w (H-B)S).The results extend a conclusion of [3].Moreover, we give several equivalent conditions for k-smoothness, strong k-smoothness,uniform k-convexity, weak uniform k-convexity and weak uniform k-convexity.