On the Paranormed Nörlund Sequence Space of Nonabsolute Type

Abstract

Maddox defined the spaceℓ(p)of the sequencesx=(xk)such that∑k=0∞‍|xk|pk<∞, in Maddox, 1967. In the present paper, the Nörlund sequence spaceNt(p)of nonabsolute type is introduced and proved that the spacesNt(p)andℓ(p)are linearly isomorphic. Besides this, the alpha-, beta-, and gamma-duals of the spaceNt(p)are computed and the basis of the spaceNt(p)is constructed. The classes(Nt(p):μ)and(μ:Nt(p))of infinite matrices are characterized. Finally, some geometric properties of the spaceNt(p)are investigated.

A New Paranormed Series Space and Matrix Transformations

“…Proof. We prove this theorem in a way similar to that in Yeşilkayagil and Başar [27]. Assume that µ is any given sequence space and take into account that the spaces |C −1 | (p) and (p) are linearly isomorphic.…”

“…For example, Başar and Altay have examined the space bs (p), Altay and Başar have studied the sequence spaces r t (p) , r t ∞ (p) , r t c (p) and r t 0 (p) in [4,1,2]. Also, some new paranormed sequence spaces have been employed by Malkowsky [20], Aydın and Başar [3], Kara and Demiriz [14], Başar et al [5], Yeşilkayagil and Başar [27], Maji and Srivastava [19] and Gökçe and Sarıgöl [8]. Additionally, Malkowsky and Rakočević [21] have obtained some general results on matrix domains of arbitrary triangles which is fundamental for our study.…”

Matrix transformations of Norlund–Orlicz difference sequence spaces of nonabsolute type and their Toeplitz duals

“…In this section we use the concept of rotundity and give some conditions to prove the rotundity of the space N t (F , m n , μ, q). For details about rotundity, Opial property, modularity, see [3,4,13,26]. Definition 5.1 Let S(X) be the unit sphere of a Banach space X.…”

On Domain of Nörlund Matrix

“…Also, f 0 (N t ) and f (N t ) were defined by Tug and Başar [35], where f 0 and f were almost null and almost convergent sequence spaces, respectively. Yeşilkayagil and Başar [36] investigated the paranormed Nörlund sequence space of the non-absolute type. Yeşilkayagil and Başar [37] worked on the domain of the Nörlund matrix in some Maddox's spaces.…”