On the geometric properities of new type modular space

Abstract

Abstract. In this paper, using a modular, we have defined the modular space M m * (p) and we have shown that the sequence space M m * (p) equipped with the Luxemburg norm is rotund and possesses H-property (or Kadec-Klee property) when p = (p k ) is bounded with p k > 1 for all k ∈ N. Introduction and PreliminariesRecently, Sanhan and Suantai [11] have generalized normed Cesàro sequence spaces to paranormed sequence spaces by making use of the Köthe sequence spaces. They have defined and studied modular structure and some geometrical properties of these generalized sequence spaces. Now, let's give some well-known descriptions to understand the subject better.Let us denote the set of all real numbers and the set of all natural numbers R and N, respectively. A sequence space is linear subspace of w , whereLet λ be a subset space of w . For a Banach space λ we denote by S(λ) and B (λ) the unit sphere and unit ball of λ, respectively. A point x 0 ∈ S(λ) is called: a) an extreme point if for every x, y ∈ S(λ) the equality 2x 0 = x + y implies x = y; b) an H-point if for any sequence (x n ) in λ such that ∥ x ∥→ 1 as n → ∞, the weak convergence of (x n ) to x 0 implies that ∥ x n − x ∥→ 0 as n → ∞.A Banach space λ is said to be rotund, if every point of S(λ) is an extreme point. A Banach space λ is said to possess H-property(or Kadec-Klee property) provided every point of S(λ) is an H-point. 2010 Mathematics Subject Classification. 46E20, 46E30, 46E40.