Paranormed Sequence Spaces Research Articles

On the B-difference sequence space derived by generalized weighted mean and compact operators

In the present paper, by using generalized weighted mean and difference matrix B, we introduce the paranormed sequence space ℓ(u,v,p;B) which consist of all sequences whose generalized weighted B-difference means are in the linear space ℓ(p) introduced by I.J. Maddox. Also, we give the basis of this space and compute its α-, β- and γ-duals. Further, we characterize the classes of matrix mappings from ℓ(u,v,p;B) to ℓ∞, c and c0. Finally, we apply the Hausdorff measure of noncompactness to characterize some classes of compact operators given by matrices on the space ℓp(u,v;B) (1⩽p<∞).

Some new paranormed difference sequence spaces and weighted core

In this study, we define new paranormed sequence spaces by combining a generalized weighted mean and a difference operator. Furthermore, we compute the α-,β- andγ- duals and obtain bases for these sequence spaces. Besides this, we characterize the matrix transformations from the new paranormed sequence spaces to the spaces c0(q),c(q),ℓ(q) and ℓ∞(q). Finally, weighted core of a complex-valued sequence has been introduced, and we prove some inclusion theorems related to this new type of core.

Some Matrix Transformations of Convex and Paranormed Sequence Spaces into the Spaces of Invariant Means

We determine the necessary and sufficient conditions to characterize the matrices which transform convex sequences and Maddox sequences into <svg style="vertical-align:-3.2316pt;width:34.75px;" id="M1" height="14.7125" version="1.1" viewBox="0 0 34.75 14.7125" width="34.75" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.7125)"> <g transform="translate(72,-60.23)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑉</tspan> </text> <text transform="matrix(1,0,0,-1,-64.76,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝜎</tspan> </text> <text transform="matrix(1,0,0,-1,-59.13,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝜃</tspan> <tspan style="font-size: 12.50px; " x="10.715071" y="0">)</tspan> </text> </g> </g> </svg> and <svg style="vertical-align:-3.2316pt;width:41.974998px;" id="M2" height="15.1125" version="1.1" viewBox="0 0 41.974998 15.1125" width="41.974998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.1125)"> <g transform="translate(72,-59.91)"> <text transform="matrix(1,0,0,-1,-71.95,63.19)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑉</tspan> </text> <text transform="matrix(1,0,0,-1,-61.95,68.19)"> <tspan style="font-size: 8.75px; " x="0" y="0">∞</tspan> <tspan style="font-size: 8.75px; " x="-2.8099999" y="8.1300001">𝜎</tspan> </text> <text transform="matrix(1,0,0,-1,-53.35,63.19)"> <tspan style="font-size: 12.50px; " x="0" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝜃</tspan> <tspan style="font-size: 12.50px; " x="10.715071" y="0">)</tspan> </text> </g> </g> </svg>.

On Some Properties of New Paranormed Sequence Space of Nonabsolute Type

We introduce some new generalized sequence space related to the space ℓ(p). Furthermore we investigate some topological properties as the completeness, the isomorphism, and also we give some inclusion relations between this sequence space and some of the other sequence spaces. In addition, we compute α‐, β‐, and γ‐duals of this space and characterize certain matrix transformations on this sequence space.

On Paranormed Type Fuzzy Real Valued Class of Sequences2ℓF(p)

Abstract. In this article we introduce the fuzzy real valued double sequence spaces 2 l F ( p ) where p = ( p nk ) is a double sequence of bounded strictly positive numbers. Westudy their different properties like completeness , solidness , symmetricity, convergencefree etc. We prove some inclusion results also. 1. IntroductionThe concept of fuzzy sets was introduced by Zadeh [19] in 1965. The poten-tial of the notion of fuzzy set was realized by different scientific groups and manyresearchers. The concept of fuzziness has been applied in various fields such ascybernetics, artificial intelligence, expert system and fuzzy control, pattern recog-nition, operation research, decision making, image analysis, projectiles, probabilitytheory, agriculture, weather forecasting, etc.Using the notion of fuzzy real numbers, different types of fuzzy real-valued sequencespaces have been introduced and studied by Nanda [5], Nuray and Savas [6], Tripa-thy and Dutta ([10], [11], [12]), Tripathy and Borgohain [18], Tripathy and Baruah([8], [9]), Tripathy and Sarma [15], Choudhary and Tripathy [2] and many others.The initial works on double sequences of real or complex terms is found in Bromwich[1]. The notion of regular convergence of double sequences of real or complex termsis introduced by Hardy [2]. Tripathy and Sarma ([13], [14]) studied different typesof double sequence spaces, while Tripathy and Dutta ([10], [11], [12]) introducedand investigated different types of double sequence spaces.The notion of paranormed sequence spaces is pity old. Different classes of para-normed sequence spaces have been introduced and studied by Choudhary and Tri-pathy [2], Tripathy and Sen[16], Tripathy[7]. Tripathy and Sen [17] have character-ized some matrix classes involving some paranormed sequence spaces, those unify

On Some New Generalized Difference Double Sequence Spaces Defined By Orlicz Functions

In this article, the author defines the generalized difference double paranormed sequence spaces $c^{2}\left( \Delta^{m},M,p,q,s\right)$, $c_{o}^{2}\left( \Delta^{m},M,p,q,s\right)$ and $l_{\infty}^{2}\left( \Delta^{m},M,p,q,s\right)$ defined over a seminormed sequence space $(X, q).$ The author also studies their properties and inclusion relations between them. Keywords: P-convergent; difference sequence; modulus function. 2010 Mathematics Subject Classification Primary 42B15; Secondary 40C05.

On Some Generalized Difference Paranormed Sequence Spaces Associated with Multiplier Sequence Defined by Modulus Function

In this article we introduce the paranormed sequence spaces (f, Λ, Δm, p), c0(f, Λ, Δm, p) and ℓ∞(f, Λ, Δm, p), associated with the multiplier sequence Λ = (λk), defined by a modulus function f. We study their different properties like solidness, symmetricity, completeness etc. and prove some inclusion results.

Some new paranormed sequence spaces defined by Euler and difference operators

The spaces Δc0(p), Δc(p) and Δℓ∞(p) were defined by Ahmad and Mursaleen [1]. In [2], Altay and Polat also defined the sequence spaces e r0 , e rc (Δ) and e r∞ (Δ), and determined the absolute Köthe-Toeplitz duals of these spaces. The main purpose of this work is to introduce some new paranormed sequence spaces defined by Euler and difference operators and to compute their α-, β-, γ-duals. Also some of the matrix transformations has been characterized.

Generalized vector-valued paranormed sequence space using modulus function

In this paper we introduce a generalized vector-valued paranormed sequence space N p ( E k , Δ m , f , s ) using modulus function f , where p = ( p k ) is a bounded sequence of positive real numbers such that inf k p k > 0 , ( E k , q k ) is a sequence of seminormed spaces with E k + 1 ⊆ E k for each k ∈ N and s ⩾ 0 . We have also studied sequence space N p ( E k , Δ m , f r , s ) , where f r = f ∘ f ∘ f ∘ , … , f ( r -times composition of f with itself) and r ∈ N = { 1 , 2 , 3 , … } . Results regarding completeness, K -space, normality, inclusion relations etc. are derived. Further, a study of multiplier of the set N p ( E k , f , s ) is also made by choosing ( E k , ‖ · ‖ k ) as sequence of normed algebras.

Properties of some sets of sequences defined by a modulus function

In this article, the author introduces the generalized difference paranormed sequence spaces c (Δ m v, f, p, q, s), c 0 (Δ m v, f, p, q, s) , and ℓ(Δ m v, f, p, q, s) defined over a seminormed sequence space (X, q). The author also studies their properties like completeness, solidity, symmetricity, etc.

On Generalized Paranormed Statistically Convergent Sequence Spaces Defined by Orlicz Function

We define generalized paranormed sequence spaces , , , and defined over a seminormed sequence space . We establish some inclusion relations between these spaces under some conditions.

On some new paranormed sequence spaces of fuzzy numbers defined by Orlicz functions and statistical convergence

Abstract In this paper we introduce the concept of strongly λ(p) convergence of fuzzy numbers with respect to an Orlicz function and examine some properties of the resulting sequence spaces and λ(p) – statistical convergence. It is also shown that if a sequence of fuzzy numbers is strong λ(p) convergent with respect to an Orlicz function then it is λ(p) – statistically convergent.

Characterization of Some Matrix Classes Involving Paranormed Sequence Spaces

In this article we characterize some matrix classes with one member as $ {m}(p) $ or $ {m}_0(p) $ or $ c(p) $ or $ c_0(p)$. Some of these results generalize the existing results. Some are new proved in the general setting.

Some paranormed sequence spaces of non-absolute type derived by weighted mean

The sequence spaces ℓ ∞ ( p ) , c ( p ) and c 0 ( p ) were introduced and studied by Maddox [I.J. Maddox, Paranormed sequence spaces generated by infinite matrices, Proc. Cambridge Philos. Soc. 64 (1968) 335–340]. In the present paper, the sequence spaces λ ( u , v ; p ) of non-absolute type which are derived by the generalized weighted mean are defined and proved that the spaces λ ( u , v ; p ) and λ ( p ) are linearly isomorphic, where λ denotes the one of the sequence spaces ℓ ∞ , c or c 0 . Besides this, the β- and γ-duals of the spaces λ ( u , v ; p ) are computed and the basis of the spaces c 0 ( u , v ; p ) and c ( u , v ; p ) is constructed. Additionally, it is established that the sequence space c 0 ( u , v ) has AD property and given the f-dual of the space c 0 ( u , v ; p ) . Finally, the matrix mappings from the sequence spaces λ ( u , v ; p ) to the sequence space μ and from the sequence space μ to the sequence spaces λ ( u , v ; p ) are characterized.

Some new paranormed sequence spaces

Maddox defined the sequence spaces ℓ ∞( p), c( p) and c 0( p) in [Proc. Camb. Philos. Soc. 64 (1968) 335, Quart. J. Math. Oxford (2) 18 (1967) 345]. In the present paper, the sequence spaces a 0 r ( u, p) and a c r ( u, p) of non-absolute type have been introduced and proved that the spaces a 0 r ( u, p) and a c r ( u, p) are linearly isomorphic to the spaces c 0( p) and c( p), respectively. Besides this, the α-, β- and γ-duals of the spaces a 0 r ( u, p) and a c r ( u, p) have been computed and their basis have been constructed. Finally, a basic theorem has been given and later some matrix mappings from a 0 r ( u, p) to the some sequence spaces of Maddox and to some new sequence spaces have been characterized.

Topological duals of some paranormed sequence spaces

Let P = (pk) be a bounded positive sequence and let A = (ank) be an infinite matrix with all ank ≥ 0. For normed spaces E and Ek, the matrix A generates the paranormed sequence spaces [A, P] ∞((Ek)), [A, P] 0((Ek)), and [A, P]((E)), which generalise almost all the well‐known sequence spaces such as c0, c, lp, l∞, and wp. In this paper, topological duals of these paranormed sequence spaces are constructed and general representation formulae for their bounded linear functionals are obtained in some special cases of matrix A.

Vector‐valued sequence spaces generated by infinite matrices

Let A = (ank) be an infinite matrix with all ank ≥ 0 and P a bounded, positive real sequence. For normed spaces E and Ek the matrix A generates paranormed sequence spaces such as [A,P]∞((Ek)), [A,P]0((Ek)), and [A, P](E) which generalize almost all the existing sequence spaces, such as l∞, c0, c, lp, wp, and several others. In this paper, conditions under which these three paranormed spaces are separable, complete, and r‐convex, are established.

An Addendum on Some Properties of Paranormed Sequence Spaces

An Addendum on Some Properties of Paranormed Sequence Spaces

Some Properties of Paranormed Sequence Spaces

Some Properties of Paranormed Sequence Spaces

Paranormed sequence spaces generated by infinite matrices

A paranormed space X = (X, g) is a topological linear space in which the topology is given by paranorm g—a real subadditive function on X such that g(θ) = 0, g(x) = g(−x) and such that multiplication is continuous. In the above, θ is the zero in the complex linear space X and continuity of multiplication means that λn → λ, xn → x(i.e. g(xn − x) → 0) imply λnxn → λx, for scalars λ and vectors x. We shall use the term semimetric function to describe a real subadditive function g on X such that g(θ) = 0, g(x) = g(−x). Two familiar paranormed sequence spaces, which have been extensively studied (3), are l(p) and m(p). For a given sequence p = (gk) of strictly positive numbers, l;(p) is the set of all complex sequences x = (xk) such that and m(p) is the set of x such that sup Throughout, sums and suprema without limits are taken from 1 to ∞. Simons (3) considered only the case in which 0 &lt; pk ≤ 1 so that natural paranorms would seem to be in m(p). In fact Simons showed that g1 was a paranorm for l(p), but that g2 did not satisfy the continuity of multiplication axiom.