SOME NEW SEQUENCE SPACES WHICH INCLUDE THE SPACES lP AND l∞

Abstract

Abstract. In the present paper, we introduce the sequence space a£ of non-absolute type and prove that the spaces a r v and lv are linearly isomorphic for 0 < ρ < oo. We also show that dp, which includes the space lp, is a p-normed space and a Β Κ space in the cases of 0 < ρ < 1 and 1 < ρ < oo, respectively. Furthermore, we give some inclusion relations and determine the α-, β-and 7-duals of the space a£ and construct its basis. We devote the last section of the paper to the characterization of the matrix mappings from the space a T v to some of the known sequence spaces and to some new sequence spaces. IntroductionBy w, we denote the space of all real or complex valued sequences. Any vector subspace of w is called as a sequence space. We write £oo, c and co for the spaces of all bounded, convergent and null sequences, respectively. Also by 6s, cs, £\ and £p, we denote the spaces of all bounded, convergent, absolutely and p-absolutely convergent series, respectively; whereA sequence space λ with a linear topology is called a K-space provided each of the maps p¿ : λ -> C defined by Pi(x) = Xi is continuous for all i 6 Ν; where C denotes the complex field and Ν = {0, 1, 2,...}. A Kspace Λ is called an FK-space provided λ is a complete linear metric space. An FK-space whose topology is normable is called a BK-space (see [9, pp. 272-273]).Let A = (anjt) be an infinite matrix of real or complex numbers αη*; where n, k € N. For the arbitrary sequence spaces λ and μ, the matrix A defines a mapping from λ into μ if for every sequence χ = (χ*) € λ the sequence Ax = {(ylx)n}) the ^4-transform of x, exists and is in μ; where {Ax)n = a nk^k-For simplicity in notation, here and in what follows, 1991 Mathematics Subject Classification: 46A45, 46B45, 46A35. Key words and phrases: sequence spaces of non-absolute type, Schauder basis, the α-, β-and 7-duals, matrix mappings.