# On the Classical Paranormed Sequence Spaces and Related Duals over the Non-Newtonian Complex Field

Authors: Uğur Kadak; Murat Kirişci; Ahmet Faruk Çakmak
Journal: Journal of Function Spaces
 (2015)

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## Abstract

The studies on sequence spaces were extended by using the notion of associated multiplier sequences. A multiplier sequence can be used to accelerate the convergence of the sequences in some spaces. In some sense, it can be viewed as a catalyst, which is used to accelerate the process of chemical reaction. Sometimes the associated multiplier sequence delays the rate of convergence of a sequence. In the present paper, the classical paranormed sequence spaces have been introduced and proved that the spaces are⋆-complete. By using the notion of multiplier sequence, the α-, β-, and γ-duals of certain paranormed spaces have been computed and their basis has been constructed.


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## Body

## 1. Introduction

The theory of sequence spaces is the fundamental of summability. Summability is a wide field of mathematics, mainly in analysis and functional analysis, and has many applications, for instance in numerical analysis to speed up the rate of convergence, operator theory, the theory of orthogonal series, and approximation theory. The classical summability theory deals with the generalization of the convergence of sequences or series of real or complex numbers. Besides this, the studies on paranormed sequence spaces were initiated by Nakano [1] and Simons [2] at the initial stage. Later on it was further studied by Maddox [3], Lascarides [4], and Lascarides and Maddox [5]. In recent years, Mursaleen et al. [6–8] have investigated some matrix transformations of paranormed sequence spaces. Also Kirişçi and Başar [9, 10] motivated the notion of generalized difference matrix and Demiriz and Çakan [11] determined some new paranormed sequence spaces.In the period from 1967 till 1972, Grossman and Katz [12] introduced the non-Newtonian calculus consisting of the branches of geometric, bigeometric, quadratic, biquadratic calculus, and so forth. Also Grossman extended this notion to the other fields in [13, 14]. All these calculi can be described simultaneously within the framework of a general theory. We prefer to use the namenon-Newtonian to indicate any of the calculi other than the classical calculus. Every property in classical calculus has an analogue in non-Newtonian calculus which is a methodology that allows one to have a different look at problems which can be investigated via calculus. In some cases, for example for wage-rate (in dollars, euro, etc.) related problems, the use of bigeometric calculus which is a kind of non-Newtonian calculus is advocated instead of a traditional Newtonian one.Bashirov et al. [15, 16] have recently concentrated on non-Newtonian calculus and gave the results with applications corresponding to the well-known properties of derivatives and integrals in classical calculus. Further Misirli and Gurefe have introduced multiplicative Adams Bashforth-Moulton method for numerical solution of differential equations in [17]. Also some authors have also worked on classical sequence spaces and related topics by using non-Newtonian calculus [18, 19]. Further Kadak [20] and Kadak et al. [21–23] have determined Kothe-Toeplitz duals and matrix transformations between certain sequence spaces over the non-Newtonian complex field and have generalized Runge-Kutta method with respect to the non-Newtonian calculus.


## 2. Preliminaries, Background, and Notations

Agenerator is a one-to-one function whose domain is R and whose range is a subset Rα of R where Rα≔α{x}:x∈R. Each generator generates exactly one arithmetic, and conversely each arithmetic is generated by exactly one generator. For example, the identity function I generates classical arithmetic, and exponential function generates geometric (multiplicative) arithmetic. As a generator, we choose the function α such that those basic algebraic operations are defined as follows:(1)α-additionx+˙y=αα-1x+α-1yα-subtractionx-˙y=αα-1x-α-1yα-multiplicationx×˙y=αα-1x×α-1yα-divisionxy.=αα-1x÷α-1yα-orderx<˙y⟺α-1x<α-1yfor all x,y∈Rα⊆R. As an example if we choose the function α=exp: (2)α:R⟶Rexp⊆Rx⟼y=αx=ex,α-arithmetic can be interpreted as geometric arithmetic: (3)α-additionx+˙y=eln⁡x+ln⁡y=x·yα-subtractionx-˙y=eln⁡x-ln⁡y=x÷yα-multiplicationx×˙y=eln⁡xlny=xln⁡y=yln⁡xα-divisionxy.=eln⁡x/lny=x1/ln⁡y.By an arithmetic, we mean a complete ordered field whose realm is a subset of R. There are infinitely many arithmetics, all of which are isomorphic, that is, structurally equivalent. The α-positive real numbers, denoted by Rα+, are the numbers x in Rα such that 0˙<˙x; the α-negative real numbers, denoted by Rα-, are those for which x<˙0˙. The α-zero, 0˙, and the alpha-one, 1˙, turn out to be α(0) and α(1). Also α(-p)=α{-α-1(p˙)}=-˙p˙ holds for all p∈Z+. Thus the set of all α-integers can be given by (4)Zα=…,α-2,α-1,α0,α1,α2,…=…,-˙2˙,-˙1˙,0˙,1˙,2˙,….One can immediately conclude that the set of exp-integer can be written as (5)Zexp=…,1e2,1e,1,e,e2,….Besides, the α-summation is defined by(6)α∑k=0∞xk=α∑k=0∞α-1xk=αα-1x0+⋯+α-1xk+⋯for all xk∈Rα⊆R.Definition 1 (see [18]).
LetX be a nonempty set and dα:X×X→Rα⊆R be a function such that, for all x,y,z∈X, the following axioms hold:(NM1)dα(x,y)=0˙ if and only if x=y,(NM2)dα(x,y)=dα(y,x),(NM3)dα(x,y)≤˙dα(x,z)+˙dα(z,y).Then, the pair (X,dα) and dα are called an α-metric space and an α-metric on X, respectively.Throughout this paper, we define thepth α-exponent xpα and qth α-root x1/qα of x∈Rα⊆R as(7)x2α=x×˙x=αα-1x×α-1x=αα-1x2x3α=x2α×˙x=αα-1αα-1x×α-1x×α-1x=αα-1x3⋮xpα=xp-1α×˙x=αα-1xp.Hence xα=x1/2α=y provided there exists an y∈Rα⊆R such that y2α=x. For each α-nonnegative number x, the symbol xα will be used to denote α{[α-1(x)]1/2} which is the unique α-nonnegative number y whose α-square is equal to x. For each number x∈Rα, x2αα=|x|α=α(|α-1(x)|) where the absolute value |x|α of x∈Rα is defined by(8)xα=x,x>˙0˙0˙,x=0˙-˙x,x<˙0˙.Definition 2 (see [18]).
LetX=(X,dα) be an α-metric space. Then the basic notions can be defined as follows. (a)
A sequencex=(xk) is a function from the set N into the set Rα. The α-real number xk denotes the value of the function at k∈N and is called the kth term of the sequence.(b)
A sequence(xn) in X=(X,dα) is said to be α-convergent if, for every given ɛ>˙0˙ (ɛ∈Rα), there exist an n0=n0(ɛ)∈N and x∈X such that dα(xn,x)=|xn-˙x|α<˙ɛ for all n>n0 which is denoted by limαn→∞xn=x or xn→αx, as n→∞.(c)
A sequence(xn) in X=(X,dα) is said to be α-Cauchy if for every ɛ>˙0˙ there is an n0=n0(ɛ)∈N such that dα(xn,xm)<˙ɛ for all m,n>n0.Following [12], we give a new type of calculus by using the notion of non-Newtonian complex numbers, denoted by ⋆-calculus (“star-”), which is a branch of non-Newtonian calculus. From now on we will use the notation ⋆-calculus corresponding calculus which is based on two arbitrarily selected generator functions.


### 2.1.⋆-Arithmetic (“Star”-Arithmetic)

Suppose thatα and β are two arbitrarily selected generators and (“star-”) also is the ordered pair of arithmetics, that is, β-arithmetic and α-arithmetic. The sets (Rβ,+¨,-¨,×¨,/¨) and (Rα,+˙,-˙,×˙,/˙) are complete ordered fields (see [19]) andbeta- (alpha-) generator generatesbeta- (alpha-) arithmetics, respectively. Definitions given for β-arithmetic are also valid for α-arithmetic. The important point to note here is that α-arithmetic is used for arguments and β-arithmetic is used for values; in particular, changes in arguments and values are measured by α-differences and β-differences, respectively.Definition 3 (see [13]).
The⋆-limit of a function f at an element a in Rα is, if it exists, the unique number b in Rβ such that(9)limx→a⋆⁡fx=b⟺∀ɛ>¨0¨,∃δ>˙0˙∋fx-¨bβ<¨ɛ∀x∈Rα,x-˙aα<˙δand is denoted by limx→a⋆f(x)=b. Also we can give the definition for every sequence (xn) of arguments of f distinct from a; if (xn) is α-convergent to a, then {f(xn)}  β-converges to b.A functionf is ⋆-continuous at a point a in Rα if and only if a is an argument of f and limx→a⋆f(x)=f(a). When α and β are the identity function I, the concepts of ⋆-limit and ⋆-continuity are identical with those of classical limit and classical continuity.The isomorphism fromα-arithmetic to β-arithmetic is the unique function ι (iota) that possesses the following three properties.(i)
ι is one to one.(ii)
ι is from Rα to Rβ.(iii)
For any numbersu and v in Rα, (10)ιu+˙v=ιu+¨ιv,ιu-˙v=ιu-¨ιv,ιu×˙v=ιu×¨ιv,ιu/˙v=ιu/¨ιv;v≠0˙,u≤v⟺ιu≤¨ιv.It turns out that ι(x)=β{α-1(x)} for every x in Rα and that ι(n˙)=n¨ for every integer n. Since, for example, u+˙v=ι-1{ι(u)+¨ι(v)}, it should be clear that any statement in α-arithmetic can readily be transformed into a statement in β-arithmetic.


### 2.2. Non-Newtonian Complex Field

Leta˙∈(Rα,+˙,-˙,×˙,/˙) and b¨∈(Rβ,+¨,-¨,×¨,/¨) be arbitrarily chosen elements from corresponding arithmetics. Then the ordered pair (a˙,b¨) is called a ⋆-point and the set of all ⋆-points is called the set of ⋆-complex numbers, which is denoted by C⋆; that is,(11)C⋆≔z⋆=a˙,b¨∣a˙∈Rα⊆R,b¨∈Rβ⊆R.Define the binary operations addition (⊕) and multiplication (⊙) of ⋆-complex numbers z1⋆=(a˙1,b¨1) and z2⋆=(a˙2,b¨2) as(12)⊕:C⋆×C⋆⟶C⋆z1⋆,z2⋆⟼z1⋆⊕z2⋆=αa1+a2,βb1+b2=a˙1+˙a˙2,b¨1+¨b¨2⊙:C⋆×C⋆⟶C⋆z1⋆,z2⋆⟼z1⋆⊙z2⋆=αa1a2-b1b2,βa1b2+b1a2,where a˙1,a˙2∈Rα and b¨1,b¨2∈Rβ.Theorem 4 (see [19]).
(
C
⋆
,
⊕
,
⊙
) is a field.Following Grossman and Katz [12] we can give the definition of ⋆-distance regarding ⋆-calculus.Definition 5 (see [19]).
The⋆-distance d⋆ between two arbitrarily elements z1⋆=(a˙1,b¨1) and z2⋆=(a˙2,b¨2) of the set C⋆ is defined by(13)d⋆:C⋆×C⋆⟶0¨,∞=B′⊂Rβz1⋆,z2⋆⟼d⋆z1⋆,z2⋆=ιa˙1-˙a˙22α+¨b¨1-¨b¨22β1/2β=βa1-a22+b1-b22.Definition 6 (see [20]).
Given a sequence(zk⋆)=(x˙k,y¨k) of ⋆-complex numbers, the formal notation (14)⋆∑k=0∞zk⋆=z0⋆⊕z1⋆⊕z2⋆⊕⋯⊕zk⋆⊕⋯=α∑k=0∞x˙k,β∑k=0∞y¨k=α∑k=0∞α-1xk,β∑k=0∞β-1xk∈C⋆,for all k∈N, is called an infinite series with ⋆-complex terms or simply complex ⋆-series. Also, for integers n∈N, the finite ⋆-sums sn⋆=⋆∑k=0n‍zk⋆ are called the partial sums of complex ⋆-series. If the sequence ⋆-converges to a complex number s⋆, then we say that the series ⋆-converges and write s⋆=⋆∑n=0∞‍zn⋆. The number s⋆ is then called the ⋆-sum of this series. If (sn)  ⋆-diverges, we say that the series ⋆-diverges or that it is ⋆-divergent.Definition 7 (see [22]).
LetX be a real or complex linear space and let ∥¨·∥¨ be a function from X to the set Rβ+ of nonnegative β-real numbers. Then the pair (X,∥¨·∥¨) is called a ⋆-normed space and ‖¨·‖¨ is a ⋆-norm for X, if the following axioms are satisfied for all elements x,y∈X and for all scalars λ: (NN1)‖¨x‖¨=0¨⇔x=θ⋆ (θ⋆=(0˙,0¨)),(NN2)‖¨λ⊙x‖¨=|¨λ|¨×¨‖¨x‖¨,(NN3)‖¨x⊕y‖¨≤¨‖¨x‖¨+¨‖¨y‖¨.It is trivial that a ⋆-norm ‖¨·‖¨ on X defines a ⋆-metric d⋆ on X which is given by d⋆x,y=‖¨x⊖y‖¨, (x,y∈X),and is called the⋆-metric induced by the⋆-norm.Letz⋆∈C⋆ be an arbitrary element. The distance function d⋆(z⋆,θ⋆) is called ⋆-norm of z⋆. In other words, (15)∥¨z⋆∥¨=d⋆z⋆,θ⋆=ιa˙-˙0˙2α+¨b¨-¨0¨2β1/2β=βa2+b2,where z⋆=(a˙,b¨) and θ⋆=(0˙,0¨).In particular, in multiplicative calculus by takingα=I, the identity function and β=exp, the exponential function, and the axioms of ⋆-normed space turn into(N(MC)1)‖¨x‖¨=1⇔x=θ⋆ (θ⋆=(0,1)),(N(MC)2)‖¨λ⊙x‖¨=‖¨x‖¨|λ|,(N(MC)3)‖¨x⊕y‖¨≤‖¨x‖¨‖¨y‖¨.Then we say that(X,∥¨·∥¨) is multiplicative normed space.Definition 8 (see [21]).
Letz⋆=(a˙,b¨)∈C⋆. We define the ⋆-complex conjugate z¯⋆ of z⋆ by z¯⋆=(α{a},β{-β-1(b¨)})=(a˙,-¨b¨). Conjugation changes the sign of the imaginary part of z⋆ but leaves the real part the same. Thus(16)Rez¯⋆=Rez⋆=z⋆⊕z¯⋆/˙2˙=a˙,Imz¯⋆=-¨Imz⋆=z⋆⊖z¯⋆/¨2¨=b¨.Remark 9 (see [21]).
The following conditions hold.(i)
Letz1⋆=(a˙1,b¨1), z2⋆=(a˙2,b¨2)∈C⋆. We can give the ⋆-division of two ⋆-complex numbers z1⋆ and z2⋆ as (17)z1⋆⊘z2⋆=αa1a2+b1b2a22+b22,βb1a2-a1b2a22+b22.(ii)
Letα and β be the same generators and let z⋆=(a˙,b¨)∈C⋆. Then, the relation z⋆⊙z⋆¯=‖¨z⋆‖¨2β holds. Really, (18)z⋆⊙z⋆¯=a˙,b¨⊙a˙,-¨b¨=αa2+b2,β0=βa2+b2=ββ-1βa2+b22=∥¨z⋆∥¨2β.Theorem 10 (see [19]).
(
C
⋆
,
d
⋆
) is a complete metric space, where d⋆ is defined by (13).Corollary 11 (see [19]).
C
⋆ is a Banach space with the ⋆-norm ‖¨·‖¨ defined by ‖¨z⋆‖¨=(ι(a˙)2α+¨b¨2β)(1/2)β; z⋆=(a˙,b¨)∈C⋆.Following Tekin and Başar [19], we can give some examples of ⋆-normed sequence spaces. First, consider the following relations which are derived from the corresponding metrics given in (13) by putting, as usual, ‖¨z⋆‖¨=d⋆(z⋆,θ⋆).Theorem 12 (see [19]).
The following statements hold.(a)
The spacesl∞⋆, c⋆, and c0⋆ are Banach spaces with the norm ∥·∥∞⋆ defined by (19)z∞⋆≔supk∈N⁡∥¨zk⋆∥¨;z=zk⋆∈λ⋆,λ∈l∞,c,c0.(b)
The spacelp⋆ is Banach spaces with the norm ∥·∥p⋆ defined by (20)zp⋆≔⋆∑k∥¨zk⋆∥¨pβ1/pβ;p≥1,z=zk⋆∈lp⋆.Theorem 13 (see [20]).
(a) The spacesbs⋆, cs⋆, and cs0⋆ are Banach spaces with the norm ∥·∥bs⋆ defined by (21)xbs⋆≔xcs⋆≔supn∈N⁡∥¨⋆∑k=0nxk∥¨;x=xk∈μ⋆,μ∈bs,cs,cs0.
(b) The spacesbv⋆, bvp⋆(p≥1), and bv∞⋆ are Banach spaces with the corresponding norms defined by (22)xbv⋆≔⋆∑k∥¨Δ′xk∥¨,xbvp⋆≔⋆∑k∥¨Δxk∥¨pβ1/pβ,xbv∞⋆≔supk∈N⁡∥¨Δxk∥¨,where (Δ′x)k=(xk⊖xk+1) and (Δx)k=(xk⊖xk-1), x-1=θ⋆ for all k∈N.Analogous to classical analysis, a sequence spaceμ⋆ with a linear ⋆-metric topology (cf. [19]) is called a K⋆-space provided that each of the maps pi:μ⋆→C⋆ defined by pi(x)=xi is ⋆-continuous by (9) for all i∈N. Additionally, a K⋆-space μ⋆ is called an FK⋆-space provided that μ⋆ is a complete linear non-Newtonian metric space, denoted by ⋆-linear (see [20]). An FK⋆-space whose non-Newtonian topology is normable and is called a BK⋆-space.


## 2.1.⋆-Arithmetic (“Star”-Arithmetic)

Suppose thatα and β are two arbitrarily selected generators and (“star-”) also is the ordered pair of arithmetics, that is, β-arithmetic and α-arithmetic. The sets (Rβ,+¨,-¨,×¨,/¨) and (Rα,+˙,-˙,×˙,/˙) are complete ordered fields (see [19]) andbeta- (alpha-) generator generatesbeta- (alpha-) arithmetics, respectively. Definitions given for β-arithmetic are also valid for α-arithmetic. The important point to note here is that α-arithmetic is used for arguments and β-arithmetic is used for values; in particular, changes in arguments and values are measured by α-differences and β-differences, respectively.Definition 3 (see [13]).
The⋆-limit of a function f at an element a in Rα is, if it exists, the unique number b in Rβ such that(9)limx→a⋆⁡fx=b⟺∀ɛ>¨0¨,∃δ>˙0˙∋fx-¨bβ<¨ɛ∀x∈Rα,x-˙aα<˙δand is denoted by limx→a⋆f(x)=b. Also we can give the definition for every sequence (xn) of arguments of f distinct from a; if (xn) is α-convergent to a, then {f(xn)}  β-converges to b.A functionf is ⋆-continuous at a point a in Rα if and only if a is an argument of f and limx→a⋆f(x)=f(a). When α and β are the identity function I, the concepts of ⋆-limit and ⋆-continuity are identical with those of classical limit and classical continuity.The isomorphism fromα-arithmetic to β-arithmetic is the unique function ι (iota) that possesses the following three properties.(i)
ι is one to one.(ii)
ι is from Rα to Rβ.(iii)
For any numbersu and v in Rα, (10)ιu+˙v=ιu+¨ιv,ιu-˙v=ιu-¨ιv,ιu×˙v=ιu×¨ιv,ιu/˙v=ιu/¨ιv;v≠0˙,u≤v⟺ιu≤¨ιv.It turns out that ι(x)=β{α-1(x)} for every x in Rα and that ι(n˙)=n¨ for every integer n. Since, for example, u+˙v=ι-1{ι(u)+¨ι(v)}, it should be clear that any statement in α-arithmetic can readily be transformed into a statement in β-arithmetic.


## 2.2. Non-Newtonian Complex Field

Leta˙∈(Rα,+˙,-˙,×˙,/˙) and b¨∈(Rβ,+¨,-¨,×¨,/¨) be arbitrarily chosen elements from corresponding arithmetics. Then the ordered pair (a˙,b¨) is called a ⋆-point and the set of all ⋆-points is called the set of ⋆-complex numbers, which is denoted by C⋆; that is,(11)C⋆≔z⋆=a˙,b¨∣a˙∈Rα⊆R,b¨∈Rβ⊆R.Define the binary operations addition (⊕) and multiplication (⊙) of ⋆-complex numbers z1⋆=(a˙1,b¨1) and z2⋆=(a˙2,b¨2) as(12)⊕:C⋆×C⋆⟶C⋆z1⋆,z2⋆⟼z1⋆⊕z2⋆=αa1+a2,βb1+b2=a˙1+˙a˙2,b¨1+¨b¨2⊙:C⋆×C⋆⟶C⋆z1⋆,z2⋆⟼z1⋆⊙z2⋆=αa1a2-b1b2,βa1b2+b1a2,where a˙1,a˙2∈Rα and b¨1,b¨2∈Rβ.Theorem 4 (see [19]).
(
C
⋆
,
⊕
,
⊙
) is a field.Following Grossman and Katz [12] we can give the definition of ⋆-distance regarding ⋆-calculus.Definition 5 (see [19]).
The⋆-distance d⋆ between two arbitrarily elements z1⋆=(a˙1,b¨1) and z2⋆=(a˙2,b¨2) of the set C⋆ is defined by(13)d⋆:C⋆×C⋆⟶0¨,∞=B′⊂Rβz1⋆,z2⋆⟼d⋆z1⋆,z2⋆=ιa˙1-˙a˙22α+¨b¨1-¨b¨22β1/2β=βa1-a22+b1-b22.Definition 6 (see [20]).
Given a sequence(zk⋆)=(x˙k,y¨k) of ⋆-complex numbers, the formal notation (14)⋆∑k=0∞zk⋆=z0⋆⊕z1⋆⊕z2⋆⊕⋯⊕zk⋆⊕⋯=α∑k=0∞x˙k,β∑k=0∞y¨k=α∑k=0∞α-1xk,β∑k=0∞β-1xk∈C⋆,for all k∈N, is called an infinite series with ⋆-complex terms or simply complex ⋆-series. Also, for integers n∈N, the finite ⋆-sums sn⋆=⋆∑k=0n‍zk⋆ are called the partial sums of complex ⋆-series. If the sequence ⋆-converges to a complex number s⋆, then we say that the series ⋆-converges and write s⋆=⋆∑n=0∞‍zn⋆. The number s⋆ is then called the ⋆-sum of this series. If (sn)  ⋆-diverges, we say that the series ⋆-diverges or that it is ⋆-divergent.Definition 7 (see [22]).
LetX be a real or complex linear space and let ∥¨·∥¨ be a function from X to the set Rβ+ of nonnegative β-real numbers. Then the pair (X,∥¨·∥¨) is called a ⋆-normed space and ‖¨·‖¨ is a ⋆-norm for X, if the following axioms are satisfied for all elements x,y∈X and for all scalars λ: (NN1)‖¨x‖¨=0¨⇔x=θ⋆ (θ⋆=(0˙,0¨)),(NN2)‖¨λ⊙x‖¨=|¨λ|¨×¨‖¨x‖¨,(NN3)‖¨x⊕y‖¨≤¨‖¨x‖¨+¨‖¨y‖¨.It is trivial that a ⋆-norm ‖¨·‖¨ on X defines a ⋆-metric d⋆ on X which is given by d⋆x,y=‖¨x⊖y‖¨, (x,y∈X),and is called the⋆-metric induced by the⋆-norm.Letz⋆∈C⋆ be an arbitrary element. The distance function d⋆(z⋆,θ⋆) is called ⋆-norm of z⋆. In other words, (15)∥¨z⋆∥¨=d⋆z⋆,θ⋆=ιa˙-˙0˙2α+¨b¨-¨0¨2β1/2β=βa2+b2,where z⋆=(a˙,b¨) and θ⋆=(0˙,0¨).In particular, in multiplicative calculus by takingα=I, the identity function and β=exp, the exponential function, and the axioms of ⋆-normed space turn into(N(MC)1)‖¨x‖¨=1⇔x=θ⋆ (θ⋆=(0,1)),(N(MC)2)‖¨λ⊙x‖¨=‖¨x‖¨|λ|,(N(MC)3)‖¨x⊕y‖¨≤‖¨x‖¨‖¨y‖¨.Then we say that(X,∥¨·∥¨) is multiplicative normed space.Definition 8 (see [21]).
Letz⋆=(a˙,b¨)∈C⋆. We define the ⋆-complex conjugate z¯⋆ of z⋆ by z¯⋆=(α{a},β{-β-1(b¨)})=(a˙,-¨b¨). Conjugation changes the sign of the imaginary part of z⋆ but leaves the real part the same. Thus(16)Rez¯⋆=Rez⋆=z⋆⊕z¯⋆/˙2˙=a˙,Imz¯⋆=-¨Imz⋆=z⋆⊖z¯⋆/¨2¨=b¨.Remark 9 (see [21]).
The following conditions hold.(i)
Letz1⋆=(a˙1,b¨1), z2⋆=(a˙2,b¨2)∈C⋆. We can give the ⋆-division of two ⋆-complex numbers z1⋆ and z2⋆ as (17)z1⋆⊘z2⋆=αa1a2+b1b2a22+b22,βb1a2-a1b2a22+b22.(ii)
Letα and β be the same generators and let z⋆=(a˙,b¨)∈C⋆. Then, the relation z⋆⊙z⋆¯=‖¨z⋆‖¨2β holds. Really, (18)z⋆⊙z⋆¯=a˙,b¨⊙a˙,-¨b¨=αa2+b2,β0=βa2+b2=ββ-1βa2+b22=∥¨z⋆∥¨2β.Theorem 10 (see [19]).
(
C
⋆
,
d
⋆
) is a complete metric space, where d⋆ is defined by (13).Corollary 11 (see [19]).
C
⋆ is a Banach space with the ⋆-norm ‖¨·‖¨ defined by ‖¨z⋆‖¨=(ι(a˙)2α+¨b¨2β)(1/2)β; z⋆=(a˙,b¨)∈C⋆.Following Tekin and Başar [19], we can give some examples of ⋆-normed sequence spaces. First, consider the following relations which are derived from the corresponding metrics given in (13) by putting, as usual, ‖¨z⋆‖¨=d⋆(z⋆,θ⋆).Theorem 12 (see [19]).
The following statements hold.(a)
The spacesl∞⋆, c⋆, and c0⋆ are Banach spaces with the norm ∥·∥∞⋆ defined by (19)z∞⋆≔supk∈N⁡∥¨zk⋆∥¨;z=zk⋆∈λ⋆,λ∈l∞,c,c0.(b)
The spacelp⋆ is Banach spaces with the norm ∥·∥p⋆ defined by (20)zp⋆≔⋆∑k∥¨zk⋆∥¨pβ1/pβ;p≥1,z=zk⋆∈lp⋆.Theorem 13 (see [20]).
(a) The spacesbs⋆, cs⋆, and cs0⋆ are Banach spaces with the norm ∥·∥bs⋆ defined by (21)xbs⋆≔xcs⋆≔supn∈N⁡∥¨⋆∑k=0nxk∥¨;x=xk∈μ⋆,μ∈bs,cs,cs0.
(b) The spacesbv⋆, bvp⋆(p≥1), and bv∞⋆ are Banach spaces with the corresponding norms defined by (22)xbv⋆≔⋆∑k∥¨Δ′xk∥¨,xbvp⋆≔⋆∑k∥¨Δxk∥¨pβ1/pβ,xbv∞⋆≔supk∈N⁡∥¨Δxk∥¨,where (Δ′x)k=(xk⊖xk+1) and (Δx)k=(xk⊖xk-1), x-1=θ⋆ for all k∈N.Analogous to classical analysis, a sequence spaceμ⋆ with a linear ⋆-metric topology (cf. [19]) is called a K⋆-space provided that each of the maps pi:μ⋆→C⋆ defined by pi(x)=xi is ⋆-continuous by (9) for all i∈N. Additionally, a K⋆-space μ⋆ is called an FK⋆-space provided that μ⋆ is a complete linear non-Newtonian metric space, denoted by ⋆-linear (see [20]). An FK⋆-space whose non-Newtonian topology is normable and is called a BK⋆-space.


## 3. Some Inequalities and Inclusion Relations

Definition 14 (Schauder basis).
If a⋆-normed sequence space λ⋆ contains a sequence (bn) with the property that for every x∈λ⋆ there is a unique sequence of scalars (ξn) such that (23)limn→∞⋆x⊖ξ0⊙b0⊕ξ1⊙b1⊕⋯⊕ξn⊙bn⋆=θ⋆with corresponding norm, then (bn) is called a Schauder basis (in non-Newtonian sense) briefly ⋆-basis, for λ⋆. The series ⋆∑k‍ξk⊙bk which has the sum x is then called the expansion of x with respect to (bn) and is written as x=⋆∑k‍ξk⊙bk. The concepts of Schauder and algebraic ⋆-bases coincide for finite dimensional spaces. Nevertheless, there are ⋆-linear spaces without a Schauder ⋆-basis.Lete=(ek) and e(n)=(ek(n)), (n∈N) be the sequences with ek=1⋆ for all k∈N, and ek(n)=δnk⋆, where δnk⋆ denotes the non-Newtonian Kronecker delta defined by (24)δnk⋆=1⋆,n=k,θ⋆,n≠k.Example 15.
The sequence{e,e(0),e(1),…}k∈N is a Schauder ⋆-basis for the space c⋆ and any x=(xk) in c⋆ has a unique representation of the form (25)x=ξ⊙e⊕⋆∑kxk⊖ξ⊙ek,wherelimk→∞⋆⁡xk=ξ.Theorem 16.
The spacebs⋆ is norm isomorphic to the space l∞⋆; that is, bs⋆≅l∞⋆.Proof.
To prove this, we should show the existence of a⋆-norm preserving linear bijection between the spaces bs⋆ and l∞⋆.
Consider the transformationT defined from bs⋆ to l∞⋆ by Tx=⋆∑j=0k‍xj. By using the corresponding operations ⊕ and ⊙, the ⋆-linearity of T is obvious. Further, it is trivial that x=θ⋆ whenever Tx=θ⋆ and hence T is injective. Let y=(yk)∈l∞⋆ and define the sequence x=(xk) by xk=yk⊖yk-1 for all k∈N with y-1=θ⋆. Then, we obtain that (26)supk∈N⁡∥¨⋆∑j=0kxj∥¨=supk∈N⁡∥¨⋆∑j=0kyj⊖yj-1∥¨=supk∈N⁡∥¨yk∥¨=y∞⋆<∞.Thus, we observe that xbs⋆<∞ and hence x∈bs⋆. Consequently T is surjective and is norm preserving. Hence, T is a linear bijection which therefore says that the spaces bs⋆ and l∞⋆ are norm isomorphic, as desired.Theorem 17.
Then the following relations are satisfied.(i)
μ⊆μ⋆ holds for each μ∈{l∞,c,c0,lp,bs,cs,cs0,bv,bv∞}.(ii)
l1⋆⊆cs⋆⊆c0⋆⊆c⋆⊆l∞⋆⊆ω⋆ and l1⋆⊆bv0⋆⊆bv⋆⊆c⋆, where bv0⋆≔bv⋆∩c0⋆.(iii)
If the inverse functionβ-1 is bounded in classical mean, then c0⊆c0⋆⊆c⊆c⋆⊆l∞⊆l∞⋆⊆ω holds.Proof.
Since the proof is trivial for the conditions (i) and (ii), we prove only (iii).
(iii) Using (i) and (ii) we need only to showl∞⋆⊆ω, c⋆⊆l∞, and c0⋆⊆c. Now, consider z=(zk)∈c⋆ is given. Then for every ɛ>¨0¨ there exist an n0=n0(ɛ)∈N and l∈C⋆ such that d⋆(zk,l)<¨ɛ for all n>n0. Since β-1 is a bounded function there exists an element M>0 such that |β-1(x)|<M for all x∈R. On the other hand, by applying the well-known inequality(27)∥¨zk∥¨≤¨∥¨zk⊖l∥¨+¨∥¨l∥¨≤¨ϵ+¨∥¨l∥¨which implies that |zk|=β-1‖¨zk‖¨≤β-1(ϵ+¨‖¨l‖¨). Therefore, by taking into account the boundedness of β-1 there exists M0>0 such that |β-1(ϵ+¨‖¨l‖¨)|<M0; we obtain that (zk) is bounded in classical mean. Thus z∈l∞. Hence c⋆⊆l∞. The remaining part can be obtained similarly.Corollary 18.
The spacesl∞⋆, c⋆, c0⋆, bs⋆, cs⋆, bvp⋆, and lp⋆ are ⋆-norm isomorphic to the spaces l∞, c, c0, bs, cs, bvp, and lp, respectively.Now, we give some well-known inequalities in the non-Newtonian sense which are essential in the study.Lemma 19 (Young’s inequality).
Letp and q be conjugate real numbers. Then,(28)u×˙v≤˙upαp˙.+˙vqαq˙.holds for all u,v∈Rα+ and p>1.Proof.
For any generator functionα, we must show that the following inequality holds:(29)α-1uα-1v≤α-1upp+α-1vqq.It is trivial that (29) holds for u=0˙ or v=0˙. Let u,v be nonzero α-real numbers. Consider the function f:[0,∞]⊆Rα→Rβ⊆R defined by(30)ft=ιtλβ-¨λ¨×¨ιt=βα-1tλ-λα-1t,where ι=β∘α-1 and 0<λ<1. Then, the ⋆-derivative of f (see [23]) can be written as (31)f⋆t=ββ-1∘f′tα-1′t=βα-1λ-λα-1′tα-1′t=βλα-1tλ-1-λ.From the first derivative test in non-Newtonian sense, the condition f⋆(t)=0¨ holds and t=1˙ is a critial point of f. Besides this, (32)f⋆⋆t=ββ-1f⋆t′α-1′t=βλλ-1α-1tλand, by using the second derivative test in non-Newtonian sense we have f⋆⋆(1˙)=β{λ(λ-1)}<¨0¨ which implies that f has a maximum at 1˙; that is, f(1˙)=β{1-λ}. In other words, we say that(33)α-1tλ-λα-1t≤1-λ∀t∈0,∞⊆Rα.Now taking t=(u)pα/˙(v)qα=α{(α-1(u))p/(α-1(v))q} and λ=1/p in (33), we get (34)α-1upα-1vq1/p-1pα-1upα-1vq≤1-1p,α-1uα-1vq/pα-1vq≤1-1p+1pα-1upα-1vqα-1vq.Hence the inclusion (29) holds. This step completes the proof.Theorem 20 (Hölder’s inequality).
Letp and q be conjugate positive real numbers and uk⋆,vk⋆∈C⋆ for k∈{0,1,2,…,n}. Then, the following inequality holds:(35)β∑k=0n∥¨uk⋆⊙vk⋆∥¨≤¨β∑k=0n∥¨uk⋆∥¨pβ1/pβ×¨β∑k=0n∥¨vk⋆∥¨qβ1/qβ.Proof.
The inequality clearly holds whenu=θ⋆=(0˙,0¨) or v=(0˙,0¨). We may assume u,v≠θ⋆ in the following proof. Let(36)ɛ=β∑k=0n∥¨uk⋆∥¨pβ1/pβ,δ=β∑k=0n∥¨vk⋆∥¨qβ1/qβand, sk⋆=uk⋆⊘ɛ,tk⋆=vk⋆⊘δ where ɛ=(0˙,ε¨)∈Rβ⊆C⋆ and δ=(0˙,δ¨)∈Rβ⊆C⋆. By taking into account Lemma 19 for each k∈{0,1,2,…,n}, we obtain (37)∥¨sk⋆⊙tk⋆∥¨=∥¨sk⋆∥¨×¨∥¨tk⋆∥¨≤¨∥¨sk⋆∥¨pβp¨:+¨∥¨tk⋆∥¨qβq¨:which implies that (38)β∑k=0n∥¨sk⋆⊙tk⋆∥¨≤¨β∑k=0n∥¨sk⋆∥¨pβp¨:+¨β∑k=0n∥¨tk⋆∥¨qβq¨:.Then, as is easy to see,(39)β∑k=0n∥¨uk⋆⊘ɛ⊙vk⋆⊘δ∥¨≤¨β∑k=0n∥¨uk⋆∥¨pβɛ×¨p¨:+¨β∑k=0n∥¨vk⋆∥¨qβδ×¨q¨:=1¨p¨+¨1¨q¨=1¨.Therefore, we deduce by combining this with the inclusion (39) that (35) holds for every k∈{0,1,2,…,n}.In particular, forp=2 the inequality (35) turns out to be(40)β∑k=0n∥¨uk⋆⊙vk⋆∥¨≤¨β∑k=0n∥¨uk⋆∥¨2β×¨β∑k=0n∥¨vk⋆∥¨2β1/2βdenoted by Cauchy-Schwartz inequality in non-Newtonian sense.Theorem 21 (Minkowski’s inequality).
Letp≥1 and uk⋆,vk⋆∈C⋆ for all k∈{0,1,2,…,n}. Then,(41)β∑k=0n∥¨uk⋆⊕vk⋆∥¨pβ1/pβ≤¨β∑k=0n∥¨uk⋆∥¨pβ1/pβ+¨β∑k=0n∥¨vk⋆∥¨pβ1/pβ.Proof.
The casep=1 is trivial. Let p>1 and uk⋆,vk⋆∈C⋆. One can immediately conclude that(42)β∑k=0n∥¨uk⋆⊕vk⋆∥¨pβ≤¨β∑k=0n∥¨uk⋆∥¨×¨∥¨uk⋆⊕vk⋆∥¨p-1β+¨β∑k=0n∥¨vk⋆∥¨×¨∥¨uk⋆⊕vk⋆∥¨p-1β.This leads us with Theorem 20 to the consequence that (43)β∑k=0n∥¨uk⋆⊕vk⋆∥¨pβ≤¨β∑k=0n∥¨uk⋆∥¨pβ1/pβ+¨β∑k=0n∥¨vk⋆∥¨pβ1/pβ×¨β∑k=0n∥¨uk⋆⊕vk⋆∥¨qp-qβ1/qβ. This concludes the proof.


## 4. Non-Newtonian Paranormed Sequence Spaces

Firstly, we give the definition of non-Newtonian paranorm, briefly⋆-paranorm.Definition 22.
LetX be a real or complex ⋆-linear space and let g⋆ be a subadditive function from X to the subset Rβ+⊆R+. Then the pair (X,g⋆) is called a ⋆-paranormed space and g⋆ is a ⋆-paranorm, for X, if the following axioms are satisfied for all elements x,y∈X and for all scalars λ.(N(PN)1)g⋆(x)=0¨ if x=θ⋆, (θ⋆=(0⋆,0⋆,…)).(N(PN)2)g⋆(⊖x)=g⋆(x), (⊖x is opposite ⋆-vector of x).(N(PN)3)g⋆(x⊕y)≤¨g⋆(x)+¨g⋆(y).(N(PN)4) If(λn⋆) is a sequence of complex scalars, that is, λ⋆=(λ˙,λ¨) with λn⋆→d⋆λ⋆ as n→∞ and xn, x∈X for all n∈N with xn→g⋆x, then λn⋆⊙xn→g⋆λ⋆⊙x as n→∞.In particular, in bigeometric calculus case, that is,α=β=exp, the conditions (N(PN)1), (N(PN)2), and (N(PN)4) also hold with zero ⋆-vector θ⋆=1,1,1,1,…, and (N(PN)3) turns into(BG(PN)3)g⋆x⊕y≤g⋆xg⋆y.Assume hereafter thatp=(pk) is a bounded sequence of strictly positive real numbers, so that 0<pk≤suppk=H<∞ and M=max{1,H}. We will assume throughout that pk×pk′=pk+pk′ provided that 1<infpk≤H<∞ for all k∈N.Quite recently Tekin and Başar [19] have introduced the sets l∞⋆,c⋆,c0⋆, and lp⋆ of sequences over the complex field C⋆ which correspond to the sets l∞,c,c0, and lp over the complex field C, respectively. It is natural to expect that the Banach spaces l∞⋆,c⋆,c0⋆, and lp⋆ can be extended to the complete ⋆-paranormed sequence spaces so as the Maddox's spaces are derived on the real or complex field from the classical sequence spaces. Now, we may give the spaces l∞⋆(p),c⋆(p),c0⋆(p),l⋆(p),bs⋆(p),cs⋆(p),cs0⋆(p),bv⋆(p), and bv∞⋆(p) in non-Newtonian sense which correspond to the well-known examples of the paranormed sequence spaces in (CC):(44)l∞⋆p≔x=xk∈ω⋆:supk∈N⁡∥¨xk∥¨pkβ<¨∞,≔x=xk=ε˙k,δ¨k∈ω⋆:supk∈N⁡βεk2+δk2pk<∞,c⋆p≔x=xk∈ω⋆:∃l∈C⋆∋limk→∞⋆⁡∥¨xk⊖l∥¨pkβ=θ⋆,≔x=xk=ε˙k,δ¨k∈ω⋆:∃l=l˙,l¨∈C⋆∋limk→∞β⁡βl2εk2+δk2pk=0¨,c0⋆p=x=xk∈ω⋆:limk→∞⋆⁡∥¨xk∥¨pkβ=θ⋆,≔x=xk=ε˙k,δ¨k∈ω⋆:limk→∞β⁡βεk2+δk2pk<∞,l⋆p=x=xk∈ω⋆:β∑k∥¨xk∥¨pkβ<¨∞,0<pk<∞,≔x=xk=ε˙k,δ¨k∈ω⋆:β∑kεk2+δk2pk<∞.Following Kadak [20], we define the several sets bs⋆(p),cs⋆(p),cs0⋆(p),bv⋆(p), and bv∞⋆(p) of sequences in the sense of non-Newtonian calculus as follows: (45)bs⋆p≔x=xk∈ω⋆:⋆∑j=0kxj∈l∞⋆p,≔x=xk=ε˙k,δ¨k∈ω⋆:supk∈N⁡β∑j=0kεj2+∑j=0kδj2pk<∞,cs⋆p≔x=xk∈ω⋆:⋆∑j=0kxj∈c⋆p,≔x=xk=ε˙k,δ¨k∈ω⋆:limk→∞β⁡β∑j=0kεj-l2+∑j=0kδj-l2pk<∞,l=l˙,l¨∈C⋆,cs0⋆p≔x=xk∈ω⋆:⋆∑k=0nxk∈c0⋆p,bv⋆p=x=xk∈ω⋆:β∑k=0∞∥¨Δxk∥¨pkβ<∞,Δxk=xk⊖xk-1;x-1=θ⋆,≔x=xk=ε˙k,δ¨k∈ω⋆:β∑k=0∞εk-εk-12+δk-δk-12pk<∞,bv∞⋆p≔x=xk∈ω⋆:supk∈N∥¨Δxk∥¨pkβ<∞.It is a routine verification that each of the sets l∞⋆(p),c⋆(p),c0⋆(p),l⋆(p),bs⋆(p),cs⋆(p),cs0⋆(p),bv⋆(p), and bv∞⋆(p) is a ⋆-linear space.Theorem 23.
The following statements hold.(i)
Define the functionsg⋆ and g~⋆ by (46)g⋆x=supk∈N∥¨xk∥¨pk/Mβ,g~⋆x=β∑k∥¨xk∥¨pkβ1/Mβ.Then c⋆(p) and c0⋆(p) are complete ⋆-paranormed spaces by g⋆ if pk∈l∞⋆. Also the spaces l∞⋆(p) and l⋆(p) are complete ⋆-paranormed spaces paranormed by g⋆ and g~⋆, respectively if and only if infpk>0.(ii)
The setsbs⋆(p), cs⋆(p), and cs0⋆(p) of sequences are the complete ⋆-paranormed spaces paranormed by g1⋆ by (47)g1⋆x=supk∈N⁡∥¨⋆∑j=0kxj∥¨pk/Mβiffinf⁡pk>0.(iii)
The setsbv⋆(p) and bv∞⋆(p) are the complete ⋆-paranormed spaces by g2⋆ and g3⋆ defined by (48)g2⋆x=β∑k∥¨Δxk∥¨pkβ1/Mβ,g3⋆x=supk∈N⁡∥¨Δxk∥¨1/Mβiffinf⁡pk>0,respectively, where Δxk=xk⊖xk-1; x-1=θ⋆ for all k∈N.Proof.
To avoid repetition of similar statements, we give the proof only for the spacebv⋆(p) in case (iii). The remaining parts can be obtained similarly.
The⋆-linearity of bv⋆(p) with respect to coordinatewise addition and scalar multiplication follows from the following inequalities which are satisfied for u,x∈bv⋆(p) (see Theorem 21):(49)⋆∑k∥¨Δuk⊕xk∥¨pkβ1/Mβ≤¨⋆∑k∥¨Δuk∥¨pkβ1/Mβ+¨⋆∑k∥¨Δxk∥¨pkβ1/Mβand the condition(50)∥¨λ⋆∥¨pkβ≤¨max⁡1¨,∥¨λ⋆∥¨Mβholds for any scalar λ⋆=(λ˙,λ¨)∈C⋆ (cf. [3]). It is clear that g⋆(θ⋆)=0¨ and g⋆(⊖x)=g⋆(x) for all x∈bv⋆(p). Hence, by combining the inclusions (49) and (50) with subadditivity of g⋆ we get the inequality g⋆λ⋆⊙x≤¨max1¨,‖¨λ⋆‖¨×¨g⋆(x).
Let(xn) be any sequence of the points of the space bv⋆(p) such that g⋆(xn⊖x)→θ⋆ and let (λn) be any sequence of ⋆-complex scalars such that λn⋆→λ⋆ with corresponding ⋆-metric. Then, since the ⋆-triangle inequality g⋆(xn)≤¨g⋆(x)+¨g⋆(xn⊖x) holds, the sequence {g⋆(xn)} is β-bounded and we thus have (51)g⋆λn⋆⊙xn⊖λ⋆⊙x=β∑k∥¨Δλn⋆⊙xkn⊖λ⋆⊙xk∥¨pkβ1/Mβ≤¨∥¨λn⋆⊖λ⋆∥¨×¨g⋆xn+¨∥¨λ⋆∥¨×¨g⋆xn⊖x,which tends to 0¨ as n→∞. That is to say that the scalar multiplication is ⋆-continuous. Hence, g⋆ is a ⋆-paranorm on the space bv⋆(p).
It remains to prove the⋆-completeness of the space bv⋆(p). Let {xi} be any Cauchy sequence in the space bv⋆(p), where xi={x0(i),x1(i),x2(i),…}. Then, for a given ϵ>¨0¨ there exists a positive integer n0(ϵ) such that(52)g⋆xi⊖xj<¨ϵ¨∀i,j≥n0ϵ.By taking into account the definition of g⋆ for each fixed k∈N, we have that (53)∥¨Δxik⊖xjk∥¨≤¨β∑k∥¨Δxik⊖xjk∥¨pkβ1/Mβ<¨ϵ∀i,j≥n0ϵ,which leads us to the fact that {(Δx0)k,(Δx1)k,(Δx2)k,…} is a Cauchy sequence for every fixed k∈N. Since C⋆ is complete (see [19]), it ⋆-converges; that is, (Δxi)k→xk as i→∞. Using these infinitely many limits x0,x1,x2,…, we define the sequence {x0,x1,x2,…}. From the inclusion (52) for each m∈N and i,j≥n0(ϵ) we have(54)β∑k=0m∥¨Δxik⊖xjk∥¨pkβ≤¨g⋆xi⊖xjMβ<¨ϵMβ.Take any i≥n0(ϵ). First let j→∞ in (54) and then m→∞, to obtain g⋆(xi⊖x)≤¨ϵ. We have by Minkowski’s inequality for each m∈N that (55)β∑k∥¨Δxk∥¨pkβ1/Mβ≤¨g⋆xi⊖x+¨g⋆xi≤¨ϵ+¨g⋆xi,which implies that x∈bv⋆(p). Since g⋆(xi⊖x)≤¨ϵ for all i≥n0(ϵ) it follows that xi→x as i→∞. Therefore, we have shown that bv⋆(p) is ⋆-complete.It is trivial to show that the⋆-paranormed spaces l∞⋆(p),c⋆(p),c0⋆(p),l⋆(p),bs⋆(p),cs⋆(p),cs0⋆(p),bv⋆(p), and bv∞⋆(p) may be reduced to some new sequence spaces in the special cases of the sequences (pk) and generator functions. For instance, the sequence space bv⋆(p) corresponds in the case pk=p for all k∈N to the sequence space bvp⋆ of p-bounded variation sequences in [20].Now, as a consequence of Corollary18, the following corollary presents the relations between ⋆-paranormed and classic paranormed spaces.Corollary 24.
The following statements hold.(i)
The spaceμ⋆(p) is norm isomorphic to the usual paranormed space μ(p) where μ∈{l∞,c,c0,lp,bs,cs,cs0,bv,bv∞}.(ii)
μ(p)⊆μ⋆(p) and l1⋆(p)⊆cs⋆(p)⊆c0⋆(p)⊆c⋆(p)⊆l∞⋆(p)⊆ω⋆.


### 4.1. Duality Properties

Following [24], we give the alpha-, beta-, and gamma-duals of a ⋆-paranormed sequence space λ⋆(p)⊂ω⋆ which are, respectively, denoted by {λ⋆(p)}α, {λ⋆(p)}β, and {λ⋆(p)}γ as follows: (56)λ⋆pα≔w=wk∈ω⋆:w⊙z=wk⊙zk∈l1⋆p∀z=zk∈λ⋆p,λ⋆pβ≔w=wk∈ω⋆:w⊙z=wk⊙zk∈cs⋆p∀z=zk∈λ⋆p,λ⋆pγ≔w=wk∈ω⋆:w⊙z=wk⊙zk∈bs⋆p∀z=zk∈λ⋆p,where (wk⊙zk) is the coordinatewise product of ⋆-complex numbers w and z for all k∈N. Throughout the text, we also use the notation “<” for a ⋆-linear subspace which was created in [18].Theorem 25.
Let∅≠λ⋆(p)⊂ω⋆. Then the following statements are valid. (a)
{λ⋆(p)}β is a sequence space if {λ⋆(p)}β⊂ω⋆.(b)
Ifλ⋆(p)⊂μ⋆(p)⊂ω⋆, then {μ⋆(p)}β⊂{λ⋆(p)}β.(c)
λ⋆⊂{λ⋆}ββ≔({λ⋆}β)β.(d)
{c0⋆(p)}β={c⋆(p)}β={l∞⋆(p)}β=l1⋆(p).(e)
{l1⋆(p)}β=l∞⋆(p).Proof.
Since the proofs are trivial for the conditions (b) and (c), we prove only (a), (d), and (e). Letw=(wk), m=(mk), and n=(nk)∈{λ⋆(p)}β. (a)
It is trivial that{λ⋆(p)}β⊂ω⋆ holds from the hypothesis. We show that m⊕n∈{λ⋆(p)}β for m,n∈{λ⋆(p)}β. Suppose that l∈λ⋆(p). Then (mk⊙lk)∈cs⋆(p) and (nk⊙lk)∈cs⋆(p) for all l∈λ⋆(p). We can deduce that (57)mk⊕nk⊙lk=mk⊙lk⊕nk⊙lk∈cs⋆p∀l∈λ⋆p.Hence,m⊕n∈{λ⋆(p)}β. Now, we show that t⊙w∈{λ⋆(p)}β for any t∈C⋆ and w=(wk)∈{λ⋆(p)}β. Since (wk⊙lk)∈cs⋆(p) for all l∈λ⋆(p) and combining this with ((tk⊙wk)⊙lk)=tk⊙(wk⊙lk)∈cs⋆(p) for all l∈λ⋆(p), we get t⊙w∈{λ⋆(p)}β. Therefore, we have proved that {λ⋆(p)}β is a subspace of the space ω⋆.(d)
Obviously{l∞⋆(p)}β⊂{c⋆(p)}β⊂{c0⋆(p)}β by Theorem 25(b). Then we must show that l1⋆(p)⊂{l∞⋆(p)}β and {c0⋆(p)}β⊂l1⋆. Now, consider w=(wk)∈l1⋆(p) and z=(zk)∈l∞⋆(p) are given. By taking into account the cases ((a)-(b)) of Theorem 12, we have (58)⋆∑kwk⊙zkpkβ≤¨supk∈N⁡∥¨zk∥¨1/Mβ×¨β∑k∥¨wk∥¨pkβ1/Mβ≤¨g⋆z×¨g~⋆w<∞,wheresuppk≤1 which implies that w⊙z∈cs⋆(p). So the condition l1⋆(p)⊂{l∞⋆(p)}β holds.Conversely, for a giveny=(yk)∈ω⋆∖l1⋆(p) we prove the existence of an x∈c0⋆(p) with y⊙x∉cs⋆(p). According to y∉l1⋆(p) we can choose an index sequence (nr) which is strictly increasing with n0=0 and ⋆∑k=nr-1nr-1‍‖¨yk‖¨(pk)β>¨r; (r∈N). By taking into account Remark 9(i) we define x=(xk)∈c0⋆(p) by xk≔(sgn⋆yk⊘r) where the ⋆-complex signum function is defined by (59)sgn⋆y≔y¯⊘∥¨y∥¨,y≠θ⋆,θ⋆,y=θ⋆,for ally=(yk)∈C⋆. Finally, by using Remark 9(ii) and taking the generators α=β we get (60)⋆∑k=nr-1nr-1yk⊙xkpkβ=⋆∑k=nr-1nr-1yk⊙sgn⋆yk⊘rpkβ=1rpkβ⊙⋆∑k=nr-1nr-1∥¨yk∥¨pkβ≥¨1¨,sup⁡pk≤1.Therefore,y⊙x∉cs⋆(p) and thus y∉{c0⋆(p)}β. Hence {c0⋆(p)}β⊂l1⋆(p). The other part of this case can be obtained similarly.(e)
From the condition (c) we havel∞⋆(p)⊂({l∞⋆(p)}β)β={l1⋆(p)}β since {l∞⋆(p)}β=l1⋆(p). Now we assume the existence of a w=(wn)∈{l1⋆(p)}β∖l∞⋆(p). Since w is an ⋆-unbounded there exists a subsequence (wnk) of (wn) and we can find a number (k+1)2β such that ‖¨wnk‖¨(pk)β≥¨(k+1)2β for all k∈N1. The sequence (xn) is defined by xn≔(sgn⋆(wnk)⊘(k+1)2β) if n=nk, and θ⋆ otherwise. Then x∈l1⋆(p). However, (61)⋆∑nwn⊙xnpkβ=⋆∑k∥¨wnk∥¨pkβk+12pkβ≔∞.Hencew∉{l1⋆(p)}β, which contradicts our assumption and {l1⋆(p)}β⊂l∞⋆(p). This step completes the proof.In addition to Theorem25, we give the following corollary which is immediate consequences of the ζ-duals (ζ∈{α,β,γ}).Corollary 26.
For eachζ∈{α,β,γ}, the following statements hold. (a)
{λ⋆(p)}α⊂{λ⋆(p)}β⊂{λ⋆(p)}γ⊂ω⋆; in particular, {λ⋆(p)}ζ is a sequence space over C⋆.(b)
{l1⋆(p)}ζ=l∞⋆(p) and {l∞⋆(p)}ζ=l1⋆(p).(c)
Ifλ⋆(p)⊂μ⋆(p)⊂ω⋆, then {μ⋆(p)}ζ⊂{λ⋆(p)}ζ.


## 4.1. Duality Properties

Following [24], we give the alpha-, beta-, and gamma-duals of a ⋆-paranormed sequence space λ⋆(p)⊂ω⋆ which are, respectively, denoted by {λ⋆(p)}α, {λ⋆(p)}β, and {λ⋆(p)}γ as follows: (56)λ⋆pα≔w=wk∈ω⋆:w⊙z=wk⊙zk∈l1⋆p∀z=zk∈λ⋆p,λ⋆pβ≔w=wk∈ω⋆:w⊙z=wk⊙zk∈cs⋆p∀z=zk∈λ⋆p,λ⋆pγ≔w=wk∈ω⋆:w⊙z=wk⊙zk∈bs⋆p∀z=zk∈λ⋆p,where (wk⊙zk) is the coordinatewise product of ⋆-complex numbers w and z for all k∈N. Throughout the text, we also use the notation “<” for a ⋆-linear subspace which was created in [18].Theorem 25.
Let∅≠λ⋆(p)⊂ω⋆. Then the following statements are valid. (a)
{λ⋆(p)}β is a sequence space if {λ⋆(p)}β⊂ω⋆.(b)
Ifλ⋆(p)⊂μ⋆(p)⊂ω⋆, then {μ⋆(p)}β⊂{λ⋆(p)}β.(c)
λ⋆⊂{λ⋆}ββ≔({λ⋆}β)β.(d)
{c0⋆(p)}β={c⋆(p)}β={l∞⋆(p)}β=l1⋆(p).(e)
{l1⋆(p)}β=l∞⋆(p).Proof.
Since the proofs are trivial for the conditions (b) and (c), we prove only (a), (d), and (e). Letw=(wk), m=(mk), and n=(nk)∈{λ⋆(p)}β. (a)
It is trivial that{λ⋆(p)}β⊂ω⋆ holds from the hypothesis. We show that m⊕n∈{λ⋆(p)}β for m,n∈{λ⋆(p)}β. Suppose that l∈λ⋆(p). Then (mk⊙lk)∈cs⋆(p) and (nk⊙lk)∈cs⋆(p) for all l∈λ⋆(p). We can deduce that (57)mk⊕nk⊙lk=mk⊙lk⊕nk⊙lk∈cs⋆p∀l∈λ⋆p.Hence,m⊕n∈{λ⋆(p)}β. Now, we show that t⊙w∈{λ⋆(p)}β for any t∈C⋆ and w=(wk)∈{λ⋆(p)}β. Since (wk⊙lk)∈cs⋆(p) for all l∈λ⋆(p) and combining this with ((tk⊙wk)⊙lk)=tk⊙(wk⊙lk)∈cs⋆(p) for all l∈λ⋆(p), we get t⊙w∈{λ⋆(p)}β. Therefore, we have proved that {λ⋆(p)}β is a subspace of the space ω⋆.(d)
Obviously{l∞⋆(p)}β⊂{c⋆(p)}β⊂{c0⋆(p)}β by Theorem 25(b). Then we must show that l1⋆(p)⊂{l∞⋆(p)}β and {c0⋆(p)}β⊂l1⋆. Now, consider w=(wk)∈l1⋆(p) and z=(zk)∈l∞⋆(p) are given. By taking into account the cases ((a)-(b)) of Theorem 12, we have (58)⋆∑kwk⊙zkpkβ≤¨supk∈N⁡∥¨zk∥¨1/Mβ×¨β∑k∥¨wk∥¨pkβ1/Mβ≤¨g⋆z×¨g~⋆w<∞,wheresuppk≤1 which implies that w⊙z∈cs⋆(p). So the condition l1⋆(p)⊂{l∞⋆(p)}β holds.Conversely, for a giveny=(yk)∈ω⋆∖l1⋆(p) we prove the existence of an x∈c0⋆(p) with y⊙x∉cs⋆(p). According to y∉l1⋆(p) we can choose an index sequence (nr) which is strictly increasing with n0=0 and ⋆∑k=nr-1nr-1‍‖¨yk‖¨(pk)β>¨r; (r∈N). By taking into account Remark 9(i) we define x=(xk)∈c0⋆(p) by xk≔(sgn⋆yk⊘r) where the ⋆-complex signum function is defined by (59)sgn⋆y≔y¯⊘∥¨y∥¨,y≠θ⋆,θ⋆,y=θ⋆,for ally=(yk)∈C⋆. Finally, by using Remark 9(ii) and taking the generators α=β we get (60)⋆∑k=nr-1nr-1yk⊙xkpkβ=⋆∑k=nr-1nr-1yk⊙sgn⋆yk⊘rpkβ=1rpkβ⊙⋆∑k=nr-1nr-1∥¨yk∥¨pkβ≥¨1¨,sup⁡pk≤1.Therefore,y⊙x∉cs⋆(p) and thus y∉{c0⋆(p)}β. Hence {c0⋆(p)}β⊂l1⋆(p). The other part of this case can be obtained similarly.(e)
From the condition (c) we havel∞⋆(p)⊂({l∞⋆(p)}β)β={l1⋆(p)}β since {l∞⋆(p)}β=l1⋆(p). Now we assume the existence of a w=(wn)∈{l1⋆(p)}β∖l∞⋆(p). Since w is an ⋆-unbounded there exists a subsequence (wnk) of (wn) and we can find a number (k+1)2β such that ‖¨wnk‖¨(pk)β≥¨(k+1)2β for all k∈N1. The sequence (xn) is defined by xn≔(sgn⋆(wnk)⊘(k+1)2β) if n=nk, and θ⋆ otherwise. Then x∈l1⋆(p). However, (61)⋆∑nwn⊙xnpkβ=⋆∑k∥¨wnk∥¨pkβk+12pkβ≔∞.Hencew∉{l1⋆(p)}β, which contradicts our assumption and {l1⋆(p)}β⊂l∞⋆(p). This step completes the proof.In addition to Theorem25, we give the following corollary which is immediate consequences of the ζ-duals (ζ∈{α,β,γ}).Corollary 26.
For eachζ∈{α,β,γ}, the following statements hold. (a)
{λ⋆(p)}α⊂{λ⋆(p)}β⊂{λ⋆(p)}γ⊂ω⋆; in particular, {λ⋆(p)}ζ is a sequence space over C⋆.(b)
{l1⋆(p)}ζ=l∞⋆(p) and {l∞⋆(p)}ζ=l1⋆(p).(c)
Ifλ⋆(p)⊂μ⋆(p)⊂ω⋆, then {μ⋆(p)}ζ⊂{λ⋆(p)}ζ.



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*Source: 416906-2015-10-01.xml*