ON THE H–PROPERTY OF SOME BANACH SEQUENCE SPACES

Abstra t. In this paper we define a generalized Cesàro sequence space ces (p) and consider it equipped with the Luxemburg norm under which it is a Banach space, and we show that the space ces (p) posses property (H) and property (G), and it is rotund, where p = (pk) is a bounded sequence of positive real numbers with pk> 1 for all k ∈ N. 1. Preliminaries For a Banach space X, we denote by S(X) and B(X) the unit sphere and unit ball of X, respectively. A point x0 ∈ S(X) is called a) an extreme point if for every x, y ∈ S(X) the equality 2x0 = x + y implies x = y; b) an H-point if for any sequence (xn) in X such that ‖xn ‖ → 1 as n → ∞, the weak convergence of (xn) to x0 (write xn w → x0) implies that ‖xn − x ‖ → 0 as n→∞; c) a denting point if for every ǫ> 0, x0 / ∈ conv{B(X)\(x0 + ǫB(X))}. A Banach space X is said to be rotund (R), if every point of S(X) is an extreme point. A Banach space X is said to posses property (H) (property (G)) provided every point of S(X) is H-point (denting point). For these geometric notions and their role in mathematics we refer to the mono-graphs [1], [2], [6] and [13]. Some of them were studied for Orlicz spaces in [3], [7], [8], [9] and [114]. Let us denote by l0 the space of all real sequences. For 1 ≤ p <∞, the Cesàro sequence space (ces p, for short) is defined by ces p = x ∈ l0: n=