On uniform Kadec-Klee properties and rotundity in generalized Cesàro sequence spaces

We consider the generalized Cesàro sequence spaces defined by Suantai (2003) and consider it equipped with the Amemiya norm. The main purpose of this paper is to show that ces (p) equipped with the Amemiya norm is rotund and has uniform Kadec-Klee property. 1. Introduction. In the whole paper, N and R stand for the sets of natural numbers and real numbers, respectively. Let (X, · ·) be a real normed space and B(X)(S(X)) the closed unit ball (the unit sphere) of X. A point x ∈ S(X) is called an extreme point if for any y,z ∈ B(X) the equality 2x = y + z implies y = z.