# Certain Sequence Spaces over the Non-Newtonian Complex Field

Authors: Sebiha Tekin; Feyzi Başar
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

It is known from functional analysis that in classical calculus, the setsω, ℓ∞, c, c0 and ℓp of all bounded, convergent, null and p-absolutely summable sequences are Banach spaces with their natural norms and they are complete according to the metric reduced from their norm, where 0<p<∞. In this study, our main goal is to construct the spaces ω*, ℓ∞*, c*, c0* and ℓp* over the non-Newtonian complex field ℂ* and to obtain
the corresponding results for these spaces, where 0¨<¨p<¨∞.


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## Body

## 1. Preliminaries, Background, and Notations

Acomplete ordered field is a system consisting of a set X, four binary operations +˙, -˙, ×˙, /˙ for X, and an ordering relation <˙ for X, all of which behave with respect to the set X exactly as +, −, ×, /, < behave with respect to the set ℝ of real numbers. We call X the realm of the complete ordered field, [1, page 32]. A complete ordered field is called arithmetic if its realm is a subset of ℝ. A bijective function with domain ℝ and range a subset of ℝ is called a generator. For example, the identity function I, exponential function, and the function x3 are generators.Bashirov et al. [2] have recently emphasized on the multiplicative calculus and gave the results with applications corresponding to the well-known properties of derivative and integral in the classical calculus. Quite recently, Uzer [3] has extended the multiplicative calculus to the complex valued functions and gave the statements of some fundamental theorems and concepts of multiplicative complex calculus, and demonstrated some analogies between the multiplicative complex calculus and classical calculus by theoretical and numerical examples. Bashirov and Rıza [4] have studied the multiplicative differentiation for complex-valued functions and established the multiplicative Cauchy-Riemann conditions. Bashirov et al. [5] have investigated various problems from different fields which can be modelled more efficiently using multiplicative calculus, in place of Newtonian calculus.Letα be a generator with range A. An arithmetic with range A, and operations and ordering relation defined as follows, is called α-arithmetic. Let y,z∈A. Then, we define the operations α-addition (+˙), α-subtraction (-˙), α-multiplication (×˙), α-division (/˙), and α-ordering (≤˙) as follows:
(1)α-addition:y+˙z=α{α-1(y)+α-1(z)}α-subtraction:y-˙z=α{α-1(y)-α-1(z)}α-multiplication:y×˙z=α{α-1(y)×α-1(z)}α-division(z≠0˙):y/˙z=α{α-1(y)α-1(z)}α-ordering:y≤˙z⟺α-1(y)≤α-1(z).
With the above new operations, (A,+˙,-˙,×˙,/˙,≤˙) is an α-arithmetic. In other words, one can easily show that (A,+˙,-˙,×˙,/˙,≤˙) is a complete ordered field. As was seen, α-generator generates α-arithmetic. For example, the identity function generates classical arithmetic, and exponential function generates geometric arithmetic. Each generator generates exactly one arithmetic and each arithmetic is generated by exactly one generator. We denote α-zero by 0˙  and α-one by 1˙ which are obtained from α(0) and α(1), respectively. 0˙ and numbers obtained by successive α-addition of 1˙ to 0˙ with numbers obtained by successive α-subtraction of 1˙ from 0˙ are called α-integers. Thus, α-integers are given as follows:
(2)…,α(-2),α(-1),α(0),α(1),α(2),…
Thus, we have for all n∈ℤ that n˙=α(n). Let x∈A. If x>˙0˙ then x is called α-positive and if x<˙0˙ then x is called α-negative. The α-absolute value|˙x|˙ of x∈A is defined by
(3)|˙x|˙={x,x>˙0˙,0˙,x=0,˙0˙-˙x,x<˙0˙.
For any elements r and s in A with r<˙s, the set of all elements x in A such that r≤˙x≤˙s is called an α-interval, is denoted by [˙r,s]˙, has α-extent of s-˙r, and has the α-interior consisting of all elements x in A such that r<˙x<˙s. Let (un) be an infinite sequence of the elements in A. Then there is at most one element u in A such that every α-interval with u in its α-interior contains all but finitely many terms of (un). If there is such a number u, then (un) is said to be α-convergent to u, which is called the α-limit of (un). In other words,
(4)limn→∞un=u(α-convergent)⟺∀ε>˙0˙,∃n0∈ℕ∋|˙un-˙u|˙<˙εfddvvv∀n≥n0and  someu∈A.Letα and β be two arbitrarily selected generators and let *-(“star”)  also be the ordered pair of arithmetics (α-arithmetic,β-arithmetic).(B,+¨,-¨,×¨,/¨,≤¨) is a β-arithmetic. Definitions given for α-arithmetic are also valid for β-arithmetic. For example, β-convergence is defined by means of β-intervals and their β-interiors.In the*-calculus,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. The operators of the *-calculus are applied only to functions with arguments in A and values in B. The *-limitof a function f at an element a in A is, if it exists, the unique number b in B such that for every sequence (an) of arguments of f distinct from a, if (an) is α-convergent to a, then {f(an)}β-convergesto b and is denoted by limx→a*f(x)=b. That is,
(5)lim*x→af(x)=b⟺∀ϵ>¨0¨,∃δ>˙0˙∋|¨f(x)-¨b|¨<¨ϵlimf(x)*=ffjjjjjjjjj∀x∈Awith|˙x-˙a|˙<˙δ.
A function f is *-continuous at a point a in A 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)
ιis one to one.(ii)
ι is from A onto B.(iii)
For any numbersu and v in A,
(6)ι(u+˙v)=ι(u)+¨ι(v)ι(u-˙v)=ι(u)-¨ι(v)ι(u×˙v)=ι(u)×¨ι(v)ι(u/˙v)=ι(u)/¨ι(v),v≠0˙u≤˙v⟺ι(u)≤¨ι(v).It turns out that ι(x)=β{α-1(x)} for every x in A, 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. Non-Newtonian Complex Field and Some Inequalities

Leta˙∈(A,+˙,-˙,×˙,/˙,≤˙) and b¨∈(B,+¨,-¨,×¨,/¨,≤¨) be arbitrarily chosen elements from corresponding arithmetics. Then the ordered pair (a˙,b¨) is called as a *-point. The set of all *-points is called the set of non-Newtonian complex numbers and is denoted by ℂ*; that is,
(7)ℂ*:={z*=a˙⊕(i*⊙b˙):a,b∈ℝ,i*=(0˙,1¨)}={z*=(a˙,b¨):a˙∈A⊆ℝ,b¨∈B⊆ℝ}.
The binary operations addition (⊕) and multiplication (⊙) of non-Newtonian complex numbers z1*=(a˙1,b¨1) and z2*=(a˙2,b¨2) are defined, as follows:
(8)⊕:ℂ*×ℂ*⟶ℂ*(z1*,z2*)↦z1*⊕z2*=(a˙1+˙a˙2,b¨1+¨b¨2),⊙:ℂ*×ℂ*⟶ℂ*(z1*,z2*)↦z1*⊙z2*=(α(a˙¯1a˙¯2-b¨¯1b¨¯2),(z1*,z2*)↦z1*ez2*=uuhgβ(a˙¯1b¨¯2+b¨¯1a˙¯2)),
where a˙1,a˙2∈A and b¨1,b¨2∈B with
(9)a˙¯1=α-1(a˙1)=α-1(α(a1))=a1∈ℝ,b¨¯1=β-1(b¨1)=β-1(β(b1))=b1∈ℝ.
Then we have the following.Theorem 1.
(
ℂ
*
,
⊕
,
⊙
) is a field.Proof.
A straightforward calculation leads to the following statements:(i)(ℂ*,⊕) is an Abelian group;(ii)(ℂ*∖{θ*},⊙) is an Abelian group;(iii) the operation⊙ is distributive over the operation ⊕which conclude that (ℂ*,⊕,⊙) is a field.Letb¨∈B⊆ℝ. Then the number b¨×¨b¨ is called the β-square of b¨ and is denoted by b2¨. Let b¨ be a non-negative number in B. Then, β[β-1(b¨)] is called the β-square root of b¨ and is denoted by b¨... The *-distanced* between two arbitrarily elements z1*=(a˙1,b¨1) and z2*=(a˙2,b¨2) of the set ℂ* is defined by
(10)d*:ℂ*×ℂ*⟶[¨0¨,∞)¨=B′⊂B(z1*,z2*)⟶d*(z1*,z2*)=[ι(a˙1-˙a˙2)]2¨+¨(b¨1-¨b¨2)2¨..=β[(a1-a2)2+(b1-b2)2].
Up to now, we know that ℂ* is a field and the distance between two points in ℂ* is computed by the function d*, defined by (10). Now, we define the *-norm and next derive some required inequalities in the sense of non-Newtonian complex calculus.Letz*∈ℂ* be an arbitrary element. d*(z*,θ*) is called *-norm of z* and is denoted by ∥¨·∥¨. In other words,
(11)∥¨z*∥¨=d*(z*,θ*)=[ι(a˙-˙0˙)]2¨+¨(b¨-¨0¨)2¨..=β(a2+b2),
where z*=(a˙,b¨) and θ*=(0˙,0¨). Moreover, since for all z1*,z2*∈ℂ* we have d*(z1*,z2*)=∥¨z1*⊖z2*∥¨, d* is the induced metric from the norm ∥¨·∥¨.Lemma 2 (*-Triangle inequality).
Letz1*,z2*∈ℂ*. Then,
(12)∥¨z1*⊕z2*∥¨≤¨∥¨z1*∥¨+¨∥¨z2*∥¨.Proof.
Letz1*,z2*∈ℂ*. Then, a straightforward calculation gives that
(13)∥¨z1*⊕z2*∥¨=[ι(a˙1+˙a˙2)]2¨+¨(b¨1+¨b¨2)2¨..=β[(a1+a2)2+(b1+b2)2]≤¨β(a12+b12+a22+b22)=β{β-1[β(a12+b12)]+β-1[β(a22+b22)]}=β[β-1(∥¨z1*∥¨)+β-1(∥¨z2*∥¨)]=∥¨z1*∥¨+¨∥¨z2*∥¨.
Hence, the inequality (12) holds.Lemma 3.
∥
¨
z
1
*
⊙
z
2
*
∥
¨
=
∥
¨
z
1
*
∥
¨
×
¨
∥
¨
z
2
*
∥
¨ for all z1*,z2*∈ℂ*.Proof.
Letz1*,z2*∈ℂ*. In this case, one can observe by a routine verification that
(14)∥¨z1*⊙z2*∥¨={ι[α(a˙¯1a˙¯2-b¨¯1b¨¯2)]}2¨+¨[β(a˙¯1b¨¯2+b¨¯1a˙¯2)]2¨..=β[(a1a2-b1b2)2+(a1b2+b1a2)2]=β(a12+b12a22+b22)=β{β-1[β(a12+b12)]×β-1[β(a22+b22)]}=β[β-1(∥¨z1*∥¨)×β-1(∥¨z2*∥¨)]=∥¨z1*∥¨×¨∥¨z2*∥¨,
as required.Lemma 4.
Letz1*,z2*∈ℂ*. Then, the following inequality holds:
(15)∥¨z1*⊕z2*∥¨/¨(1¨+¨∥¨z1*⊕z2*∥¨)≤¨∥¨z1*∥¨/¨(1¨+¨∥¨z1*∥¨)+¨∥¨z2*∥¨/¨(1¨+¨∥¨z2*∥¨).Proof.
Letz1*,z2*∈ℂ*. Then, one can see that
(16)∥¨z1*⊕z2*∥¨/¨(1¨+¨∥¨z1*⊕z2*∥¨)=β[(a1+a2)2+(b1+b2)2]/¨{1¨+¨β[(a1+a2)2+(b1+b2)2]}=β[(a1+a2)2+(b1+b2)21+(a1+a2)2+(b1+b2)2]≤¨β[(a12+b121+a12+b12)+(a22+b221+a22+b22)]=β{β-1[β(a12+b12)]β-1{β[β-1(1¨)+β-1[β(a12+b12)]]}=βjghg+β-1[β(a22+b22)]β-1{β{β-1(1¨)+β-1[β(a22+b22)]}}}=β{[β-1(∥¨z1*∥¨)β-1(1¨+¨∥¨z1*∥¨)]+[β-1(∥¨z2*∥¨)β-1(1¨+¨∥¨z2*∥¨)]}=β{β-1{β[β-1(∥¨z1*∥¨)β-1(1¨+¨∥¨z1*∥¨)]}=βhhhg+β-1{β[β-1(∥¨z2*∥¨)β-1(1¨+¨∥¨z2*∥¨)]}}=β{β-1[∥¨z1*∥¨/¨(1¨+¨∥¨z1*∥¨)]+β-1[∥¨z2*∥¨/¨(1¨+¨∥¨z2*∥¨)]}=∥¨z1*∥¨/¨(1¨+¨∥¨z1*∥¨)+¨∥¨z2*∥¨/¨(1¨+¨∥¨z2*∥¨).
This means that the inequality (15) holds.Lemma 5 (*-Minkowski inequality).
Letp≥¨1¨ and zk*,tk*∈ℂ* for all k∈{1,2,3,…,n}. Then,
(17)(∑‍k=0n..∥¨zk*⊕tk*∥¨p)1/¨p≤¨(∑‍k=0n..∥¨zk*∥¨p)1/¨p+¨(∑‍k=0n..∥¨tk*∥¨p)1/¨p.Proof.
Letp≥¨1¨ and zk*,tk*∈ℂ* for all k∈{1,2,3,…,n}. Then,
(18)∑‍k=0n..∥¨zk*⊕tk*∥¨p=∥¨z0*⊕t0*∥¨p+¨∥¨z1*⊕t1*∥¨p+¨⋯+¨∥¨zn*⊕tn*∥¨p=β[β-1(∥¨z0*⊕t0*∥¨p)+β-1(∥¨z1*⊕t1*∥¨p)ghhhhxss+⋯+β-1(∥¨zn*⊕tn*∥¨p)].
Let us take zk*=(x˙k,y¨k),zk=(xk,yk),tk*=(u˙k,v¨k),tk=(uk,vk). Then, since the equality
(19)∥¨zk*⊕tk*∥¨=[ι(x˙k+˙u˙k)]2¨+¨(y¨k+¨v¨k)2¨..=β[(xk+uk)2+(yk+vk)2]=β(|zk+tk|)
holds for every fixed k, we obtain
(20)∥¨zk*⊕tk*∥¨p=∥¨zk*⊕tk*∥¨×¨⋯×¨∥¨zk*⊕tk*∥¨︸p-times=β[β-1(∥¨zk*⊕tk*∥¨)×⋯×β-1(∥¨zk*⊕tk*∥¨)]=β{β-1[β(|zk+tk|)]×⋯×β-1[β(|zk+tk|)]}=β(|zk+tk|β-1(p))
by (18) and Minkowski inequality in the complex field leads us to
(21)∑k=0n..∥¨zk*⊕tk*∥¨p=β{β-1[β(|z0⊕t0|β-1(p))]+β-1[β(|z1⊕t1|β-1(p))]hgfffffg+⋯+β-1[β(|zn⊕tn|β-1(p))]}=β(∑k=0n|zk+tk|β-1(p))≤¨β{[(∑k=0n|zk|β-1(p))1/β-1(p)+(∑k=0n|tk|β-1(p))1β-1(p)]β-1(p)},(22)β-1{[(∑k=0n..∥¨zk*∥¨p)1/¨p+¨(∑k=0n..∥¨tk*∥¨p)1/¨p]p}=β-1{[(∑k=0n..∥¨zk*∥¨p)1/¨p+¨(∑k=0n..∥¨tk*∥¨p)1/¨p]hhjjjjjjgg×¨⋯×¨[(∑k=0n..∥¨zk*∥¨p)1/¨p+¨(∑k=0n..∥¨tk*∥¨p)1/¨p]}=β-1[(∑k=0n..∥¨zk*∥¨p)1/¨p+¨(∑k=0n..∥¨tk*∥¨p)1/¨p]hhjg×⋯×β-1[(∑k=0n..∥¨zk*∥¨p)1/¨p+¨(∑k=0n..∥¨tk*∥¨p)1/¨p]={β-1[(∑k=0n..∥¨zk*∥¨p)1/¨p]+β-1[(∑k=0n..∥¨tk*∥¨p)1/¨p]}ggjjff×⋯×{β-1[(∑k=0n..∥¨zk*∥¨p)1/¨p]ggggggggggffd+β-1[(∑k=0n..∥¨tk*∥¨p)1/¨p]}={β-1[(∑k=0n..∥¨zk*∥¨p)1/¨p]+β-1[(∑k=0n..∥¨tk*∥¨p)1/¨p]}β-1(p).
On the other hand, let us prove(23)β[(∑k=0n‍|zk|β-1(p))1/β-1(p)]=(∑‍k=0n..∥¨zk*∥¨p)1/¨p.
Indeed,
(24){β[(∑k=0n‍|zk|β-1(p))1/β-1(p)]}p=β[(∑k=0n‍|zk|β-1(p))1/β-1(p)]×¨⋯×¨β[(∑k=0n‍|zk|β-1(p))1/β-1(p)]︸p-times=β[(∑k=0n‍|zk|β-1(p))1/β-1(p)×⋯×(∑k=0n‍|zk|β-1(p))1/β-1(p)]=β{[(∑k=0n‍|zk|β-1(p))1/β-1(p)]β-1(p)}=β(∑k=0n‍|zk|β-1(p))=∑‍k=0n..∥¨zk*∥¨p.
Substituting the relation (24) in (22) we obtain,
(25)β-1{[(∑‍k=0n..∥¨zk*∥¨p)1/¨p+¨(∑‍k=0n..∥¨tk*∥¨p)1/¨p]p}=[(∑k=0n‍|zk|β-1(p))1/β-1(p)+(∑k=0n‍|tk|β-1(p))1/β-1(p)]β-1(p).
By using this equality in (21), we get
(26)(∑‍k=0n..∥¨zk*⊕tk*∥¨p)1/¨p≤¨(∑‍k=0n..∥¨zk*∥¨p)1/¨p+¨(∑‍k=0n..∥¨tk*∥¨p)1/¨p,
as desired.Theorem 6.
(
ℂ
*
,
d
*
) is a complete metric space, where d* is defined by (10).Proof.
First, we show thatd*, defined by (10), is a metric on ℂ*.
It is immediate forz1*,z2*∈ℂ* that
(27)d*(z1*,z2*)=[ι(a˙1-˙a˙2)]2¨+¨(b¨1-¨b¨2)2¨..=β[(a1-a2)2+(b1-b2)2]≥¨0¨.(i)
Now we show thatd*(z1*,z2*)=0¨ if and only if z1*=z2* for z1*,z2*∈ℂ*. Indeed,
(28)d*(z1*,z2*)=0¨⟺[ι(a˙1-˙a˙2)]2¨+¨(b¨1-¨b¨2)2¨..=0¨⟺β[(a1-a2)2+(b1-b2)2]=β(0)⟺(a1-a2)2+(b1-b2)2=0⟺a1-a2=0,b1-b2=0⟺a1=a2,b1=b2⟺a˙1=a˙2,b¨1=b¨2⟺z1*=(a˙1,b¨1)=(a˙2,b¨2)=z2*.(ii)
One can easily establish for allz1*,z2*∈ℂ* that
(29)d*(z1*,z2*)=[ι(a˙1-˙a˙2)]2¨+¨(b¨1-¨b¨2)2¨..=β[(a1-a2)2+(b1-b2)2]=β[(a2-a1)2+(b2-b1)2]=d*(z2*,z1*).(iii)
We show that the inequalityd*(z1*,z2*)+¨d*(z2*,z3*)≥¨d*(z1*,z3*) holds for all z1*,z2*,z3*∈ℂ*. In fact,
(30)d*(z1*,z2*)+¨d*(z2*,z3*)=[ι(a˙1-˙a˙2)]2¨+¨(b¨1-¨b¨2)2¨..+¨[ι(a˙2-˙a˙3)]2¨+¨(b¨2-¨b¨3)2¨..=β[(a1-a2)2+(b1-b2)2]+¨β[(a2-a3)2+(b2-b3)2]=β[(a1-a2)2+(b1-b2)2+(a2-a3)2+(b2-b3)2]≥¨β[(a1-a3)2+(b1-b3)2]=[ι(a˙1-˙a˙3)]2¨+¨(b¨1-¨b¨3)2¨..=d*(z1*,z3*).
Therefore,d* is a metric over ℂ*.
Now, we can show that the metric space(ℂ*,d*) is complete. Let (zn*)n∈ℕ be an arbitrary Cauchy sequence in ℂ*. In this case, for all ε>¨0¨ there exists an n0∈ℕ such that d*(zm*,zn*)<¨ε for all m,n≥n0. Let zm*=(a˙m,b¨m)∈ℂ* and m∈ℕ. Then,
(31)d*(zm*,zn*)=[ι(a˙m-˙a˙n)]2¨+¨(b¨m-¨b¨n)2¨..=β[(am-an)2+(bm-bn)2]<¨ε=β(ε′).
Thus we obtain that
(32)(am-an)2+(bm-bn)2<ε′.
On the other hand, since the following inequalities:
(33)|am-an|=(am-an)2≤(am-an)2+(bm-bn)2,|bm-bn|=(bm-bn)2≤(am-an)2+(bm-bn)2
hold we therefore have by (32) that |am-an|<ε′ and |bm-bn|<ε′. This means that (an) and (bn) are Cauchy sequences with real terms. Since ℝ is complete, it is clear that for every ε′>0 there exists an n1∈ℕ such that |an-a|<ε′/2 for all n≥n1 and for every ε′>0 there exists an n2∈ℕ such that |bn-b|<ε′/2 for all n≥n2.
Define,z*∈ℂ* by z*=(a˙,b¨). Then, we have
(34)d*(zn*,z*)=[ι(a˙n-˙a˙)]2¨+¨[b¨n-¨b¨]2¨..=β[(an-a)2+(bn-b)2]≤¨β[(an-a)2+(bn-b)2]=β(|an-a|+|bn-b|)≤¨β(ε′2+ε′2)=β(ε′)=ε.
Hence, (ℂ*,d*) is a complete metric space.Sinceℂ* is a complete metric space with the metric d* defined by (10) induced by the norm ∥¨·∥¨, as a direct consequence of Theorem 6, we have the following.Corollary 7.
ℂ
* is a Banach space with the norm ∥¨·∥¨ defined by
(35)∥¨z*∥¨=[ι(a˙)]2¨+¨(b¨)2¨..;z*=(a˙,b¨)∈ℂ*.


## 3. Sequence Spaces over Non-Newtonian Complex Field

In this section, we define the setsω*, ℓ∞*, c*, c0*, and ℓp* of all, bounded, convergent, null, and absolutely p-summable sequences over the non-Newtonian complex field ℂ* which correspond to the sets ω, ℓ∞, c, c0, and ℓp over the complex field ℂ, respectively. That is to say that
(36)ω*:={z*=(zk*):zk*∈ℂ*∀k∈ℕ},ℓ∞*:={z*=(zk*)∈ω*:supk∈ℕ∥¨zk*∥¨<¨∞},c*:={z*=(zk*)∈ω*:∃l*∈ℂ*∋lim*k→∞zk*=l*},c0*:={z*=(zk*)∈ω*:lim*k→∞zk*=θ*},ℓp*:={z*=(zk*)∈ω*:∑k..∥¨zk*∥¨p<¨∞}.
For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞. One can easily see that the set ω* forms a vector space over ℂ* with respect to the algebraic operations addition (+) and scalar multiplication (×) defined on ω*, as follows:
(37)+:ω*×ω*⟶ω*hh(z*,t*)↦z*+t*=(zk*⊕tk*);hhyhhz*=(zk*),t*=(tk*)∈ω*,×:ℂ*×ω*⟶ω*(α,z*)↦α×z*=(α⊙zk*);hhhjz*=(zk*)∈ω*,α∈ℂ*.Theorem 8.
Define the functiondω* by
(38)dω*:ω*×ω*⟶B′⊆B(z*,t*)↦dω*(z*,t*)=∑‍k..μk×¨{d*(zk*,tk*)/¨[1¨+¨d*(zk*,tk*)]},
where (μk)∈B′⊆B such that ∑‍¨kμk is convergent with μk>¨0¨ for all k∈ℕ. Then, (ω*,dω*) is a metric space.Proof.
We show thatdω* satisfies the metric axioms on the space ω* of all non-Newtonian complex valued sequences.
(i) First we show thatdω*(z*,t*)≥¨0¨ for all z*,t*∈ω*.
Because(ℂ*,d*) is a metric space, we have d*(zk*,tk*)≥¨0¨; zk*,tk*∈ℂ* and 1¨+¨d*(zk*,tk*)≥¨0¨. Hence, we obtain that d*(zk*,tk*)/¨[1¨+¨d*(zk*,tk*)]≥¨0¨. Moreover, since μk>¨0¨ for all k∈ℕ, we have
(39)μk×¨{d*(zk*,tk*)/¨[1¨+¨d*(zk*,tk*)]}≥¨0¨.
This means that
(40)dω*(z*,t*)=∑‍k..μk×¨{d*(zk*,tk*)/¨[1¨+¨d*(zk*,tk*)]}≥¨0¨.
(ii) We show thatdω*(z*,t*)=0¨ iff z*=t*. In this situation, one can see that
(41)dω*(z*,t*)=∑‍k..μk×¨{d*(zk*,tk*)/¨[1¨+¨d*(zk*,tk*)]}=0¨⟺μk×¨{d*(zk*,tk*)/¨[1¨+¨d*(zk*,tk*)]}=0¨,sswdqdqedededededdeedkkkjkjjjjgfftcdsa∀k∈ℕ⟺d*(zk*,tk*)/¨[1¨+¨d*(zk*,tk*)]=0¨,dsqwqwwwwedeiiiiydddwμk>¨0¨,∀k∈ℕ⟺d*(zk*,tk*)=0¨;d*(zk*,tk*)≥¨0¨,∀k∈ℕ⟺zk*=tk*;d*metric,∀k∈ℕ⟺z*=(zk*)=(tk*)=t*.
(iii) We show thatdω*(z*,t*)=dω*(t*,z*) for z*=(zk*),t*=(tk*)∈ω*. First, we know that d* is a metric over ℂ*. Thus,
(42)dω*(z*,t*)=∑‍k..μk×¨{d*(zk*,tk*)/¨[1¨+¨d*(zk*,tk*)]}=∑‍k..μk×¨{d*(tk*,zk*)/¨[1¨+¨d*(tk*,zk*)]}=dω*(t*,z*).
(iv) We show thatdω*(z*,t*)+¨dω*(t*,u*)≥¨dω*(z*,u*) holds for z*=(zk*),t*=(tk*),u*=(uk*)∈ω*. Again, using the fact that (ℂ*,d*) is a metric space, it is easy to see by Lemma 4 that
(43)dω*(z*,u*)=∑‍k..μk×¨{d*(zk*,uk*)/¨[1¨+¨d*(zk*,uk*)]}≤¨∑‍k..μk×¨{/¨[d*(zk*,tk*)+¨d*(tk*,uk*)]gfsfffsfffffss/¨{1¨+¨[d*(zk*,tk*)+¨d*(tk*,uk*)]}}≤¨∑‍k..μk×¨{{d*(zk*,tk*)/¨[1¨+¨d*(zk*,tk*)]}≤∑ddddkkkkk+¨{d*(tk*,uk*)/¨[1¨+¨d*(tk*,uk*)]}}=∑‍k..μk×¨{d*(zk*,tk*)/¨[1¨+¨d*(zk*,tk*)]}hh+¨∑k..μk×¨{d*(tk*,uk*)/¨[1¨+¨d*(tk*,uk*)]}=dω*(z*,t*)+¨dω*(t*,u*),
as required.Theorem 9.
The setℓ∞* is a sequence space.Proof.
It is trivial that the inclusionℓ∞*⊂ω* holds.
(i) We show thatz*+t*∈ℓ∞* for z*=(zk*),t*=(tk*)∈ℓ∞*. Indeed, combining the hypothesis
(44)supk∈ℕ∥¨zk*∥¨<¨∞,supk∈ℕ∥¨tk*∥¨<¨∞
with the fact ∥¨zk*⊕tk*∥¨≤¨∥¨zk*∥¨+¨∥¨tk*∥¨ obtained from Lemma 2, we can easily derive that
(45)supk∈ℕ∥¨zk*⊕tk*∥¨≤¨supk∈ℕ∥¨zk*∥¨+¨supk∈ℕ∥¨tk*∥¨<¨∞.
Hence,z*+t*∈ℓ∞*.
(ii) We show thatα×z*∈ℓ∞* for any α∈ℂ* and for z*=(zk*)∈ℓ∞*.
Since∥¨α⊙zk*∥¨=∥¨α∥¨×¨∥¨zk*∥¨ by Lemma 3 and supk∈ℕ∥¨zk*∥¨<¨∞, it is immediate that
(46)supk∈ℕ∥¨α⊙zk*∥¨=∥¨α∥¨×¨supk∈ℕ∥¨zk*∥¨<¨∞
which means that α×z*∈ℓ∞*.
Therefore, we have proved thatℓ∞* is a subspace of the space ω*.Theorem 10.
Define the relationd∞* by
(47)d∞*:ℓ∞*×ℓ∞*⟶B′⊆B(z*,t*)↦d∞*(z*,t*)=supk∈ℕ∥¨zk*⊖tk*∥¨.
Then,(ℓ∞*,d∞*) is a complete metric space.Proof.
One can easily show by a routine verification thatd∞* satisfies the metric axioms on the space ℓ∞*. So, we omit the details.
Now, we prove the second part of the theorem. Let(zm*) be a Cauchy sequence in ℓ∞*, where zm*=(zk*m)k∈ℕ. Then, there exists a positive integer k0 such that d∞*(zm*,zr*)=supk∈ℕ∥¨zk*m⊖zk*r∥¨<¨ε for all m,r∈ℕ with m,r>k0. For any fixed k, if m,r>k0 then
(48)∥¨zk*m⊖zk*r∥¨<¨ε.
In this case for any fixed k, (zk*0,zk*1,…) is a Cauchy sequence of non-Newtonian complex numbers and since ℂ* is complete, it converges to a zk*∈ℂ*. Define z*=(z0*,z1*,…) with infinitely many limits z0*,z1*,…. Let us show z*∈ℓ∞* and zm*→z*, as m→∞. Indeed, by (48), by letting r→∞, for m>k0 we obtain that
(49)∥¨zk*m⊖zk*∥¨≤¨ε.
On the other hand, since zm*=(zk*m)k∈ℕ∈ℓ∞*, there exists tm∈B⊆ℝ such that ∥¨zk*m∥¨≤¨tm for all k∈ℕ. Hence, by triangle inequality (12), the inequality
(50)∥¨zk*∥¨≤¨∥¨zk*⊖zk*m∥¨+¨∥¨zk*m∥¨≤¨ε+¨tm
holds for all k∈ℕ which is independent of k. Hence, z*=(zk*)k∈ℕ∈ℓ∞*. By (49), since m>k0, we obtain d∞*(zm*,z*)=supk∈ℕ∥¨zk*m⊖zk*∥¨≤¨ε. Therefore, the sequence (zm*) converges to z* which means that ℓ∞* is complete.Since it is known by Theorem10 that ℓ∞ is a complete metric space with the metric d∞* induced by the norm ∥¨·∥¨∞, defined by
(51)∥¨z*∥¨∞=supk∈ℕ∥¨zk*∥¨;z*=(zk*)∈ℓ∞*,
we have the following.Corollary 11.
ℓ
∞
* is a Banach space with the norm ∥¨·∥¨∞ defined by (51).Now, we give the following lemma required in proving the fact thatℓp* is a sequence space in the case 0¨<¨p<¨1¨.Lemma 12.
Let0¨<¨p<¨1¨. Then, the inequality (a¨+¨b¨)p<¨a¨p+¨b¨p holds for all a¨,b¨>¨0¨.Proof.
Let0¨<¨p<¨1¨ and a¨,b¨>¨0¨. Then, one can easily see that
(52)(a¨+¨b¨)p=(a¨+¨b¨)×¨⋯×¨(a¨+¨b¨)︸p-times=β{a+b}×¨⋯×¨β{a+b}=β{β-1[β(a+b)]×⋯×β-1[β(a+b)]︸β-1(p)-times}=β{(a+b)β-1(p)}<¨β{aβ-1(p)+bβ-1(p)}=β{β-1(a¨p)+β-1(b¨p)}=a¨p+¨b¨p,
as desired.Theorem 13.
The setsc*,c0*,and ℓp* are sequence spaces, where 0¨<¨p<¨∞.Proof.
It is not hard to establish by the similar way thatc0* and ℓp* are the sequence spaces. So, to avoid the repetition of the similar statements, we consider only the set c*.
It is obvious that the inclusionc*⊂ω* strictly holds.
(i) Letz*=(zk*),t*=(tk*)∈c*. Then, there exist l1*,l2*∈ℂ* such that limk→∞*zk*=l1* and limk→∞*tk*=l2*. Thus, there exist k1,k2∈ℕ such that
(53)∀ε>¨0¨,∥¨zk*⊖l1*∥¨≤¨ε/¨2¨∀k≥k1,∀ε>¨0¨,∥¨tk*⊖l2*∥¨≤¨ε/¨2¨∀k≥k2.
Thus if we setk0=max{k1,k2}, by (53) we obtain for all k≥k0 that
(54)∥¨(zk*⊕tk*)⊖(l1*⊕l2*)∥¨=∥¨(zk*⊖l1*)⊕(tk*⊖l2*)∥¨≤¨∥¨zk*⊖l1*∥¨+¨∥¨tk*⊖l2*∥¨≤¨ε/¨2¨+¨ε/¨2¨=ε
which means that
(55)lim*k→∞(zk*⊕tk*)=l1*⊕l2*=lim*k→∞zk*⊕lim*k→∞tk*.
Therefore,z*+t*∈c*.
(ii) Letz*=(zk*)∈c* and α∈ℂ*∖{θ*}. Since z*∈c* there exists an l*∈ℂ* such that limk→∞*zk*=l*, we have
(56)∀ε>¨0¨,∃k0∈ℕsuch that∥¨zk*⊖l*∥¨≤¨ε/¨∥¨α∥¨∀k≥k0.
Thus, fork≥k0, we have
(57)∥¨(α⊙zk*)⊖(α⊙ℓ*)∥¨=∥¨α⊙(zk*⊖ℓ*)∥¨=∥¨α∥¨×¨∥¨zk*⊖ℓ*∥¨≤¨∥¨α∥¨×¨ε/¨∥¨α∥¨=ε
which implies that limk→∞*(α⊙zk*)=α⊙l*=α⊙limk→∞*zk*Hence, α×z*∈c*.
That is to say thatc* is a subspace of ω*.Theorem 14.
(
c
*
,
d
∞
*
), (c0*,d∞*), and (ℓp*,dp*) are complete metric spaces, where dp* is defined as follows:
(58)dp*(z*,t*)={∑‍k..∥¨zk*⊖tk*∥¨p,0¨<¨p<¨1¨(∑‍k..∥¨zk*⊖tk*∥¨p)1/¨p,p≥¨1¨;z*=(zk*),t*=(tk*)∈ℓp*.Proof.
We consider only the spaceℓp* with p≥¨1¨.
(i) Forz*=(zk*),t*=(tk*)∈ℓp*, we establish that the two sided implication dp*(z*,t*)=0¨⇔z*=t* holds. In fact,
(59)dp*(z*,t*)=(∑‍k..∥¨zk*⊖tk*∥¨p)1/¨p=0¨⟺∑‍k..∥¨zk*⊖tk*∥¨p=0¨⟺∀k∈ℕ,∥¨zk*⊖tk*∥¨p=0¨⟺∀k∈ℕ,∥¨zk*⊖tk*∥¨=0¨⟺∀k∈ℕ,zk*=tk*⟺z*=(zk*)=(tk*)=t*.
(ii) Forz*=(zk*),t*=(tk*)∈ℓp*, we show that dp*(z*,t*)=dp*(t*,z*). In this situation, since ∥¨zk*⊖tk*∥¨=∥¨tk*⊖zk*∥¨ holds for every fixed k∈ℕ it is immediate that
(60)dp*(z*,t*)=(∑k..∥¨zk*⊖tk*∥¨p)1/¨p=(∑k..∥¨tk*⊖zk*∥¨p)1/¨p=dp*(t*,z*).
(iii)  By Minkowski inequality in Lemma5, we have for z*=(zk*),t*=(tk*),u*=(uk*)∈ℓp* that
(61)dp*(z*,t*)=(∑k..∥¨zk*⊖tk*∥¨p)1/¨p=[∑k..∥¨(zk*⊖uk*)⊕(uk*⊖tk*)∥¨p]1/¨p≤¨(∑k..∥¨zk*⊖uk*∥¨p)1/¨p+¨(∑k..∥¨uk*⊖tk*∥¨p)1/¨p=dp*(z*,u*)+¨dp*(u*,t*).
Hence, triangle inequality is satisfied bydp* on the space ℓp*. Therefore, the function dp* is a metric over the space ℓp*.
Now we show that the metric space(ℓp*,dp*) is complete. Let (zm*)m∈ℕ be an arbitrary Cauchy sequence in the space ℓp*, where zm*=(z1*m,z2*m,…). Then, for any ε>¨0¨ there exists an n0∈ℕ such that
(62)dp*(zm*,zn*)=(∑k..∥¨zk*m⊖zk*n∥¨p)1/¨p<¨ε
for all m,n≥n0. Hence, for m,n≥n0 and every fixed k∈ℕ, we obtain
(63)∥¨zk*m⊖zk*n∥¨<¨ε.
If we set k fixed then, it follows by (63) that (zk*0,zk*1,…) is a Cauchy sequence. Since ℂ* is complete, this sequence converges to a point say zk*. Let us define the sequence z*=(z0*,z1*,…) with these limits and show z*∈ℓp* and zm*→z*, as m→∞. Indeed, by (62), we obtain the inequality for all m,n∈ℕ with m,n≥n0 that
(64)∑k=0j..∥¨zk*m⊖zk*n∥¨p<¨εp∀j∈ℕ
and thus we have by letting n→∞ and m>n0 that
(65)∑k=0j..∥¨zk*m⊖zk*∥¨p<¨εp∀j∈ℕ
which gives as j→∞ and for all m>n0 that
(66)[dp*(zm*,z*)]p=∑k..∥¨zk*m⊖zk*∥¨p<¨εp.
Setting zk*=zk*m⊕(zk*⊖zk*m) and applying Lemma 5 we obtain by (66) and the fact (zk*m)∈ℓp* that
(67)(∑k..∥¨zk*∥¨p)1/¨p=(∑k..∥¨zk*m⊕(zk*⊖zk*m)∥¨p)1/¨p≤¨(∑k..∥¨zk*m∥¨p)1/¨p+¨(∑k..∥¨zk*⊖zk*m∥¨p)1/¨p<¨∞
which means that z*=(zk*)∈ℓp*. Therefore, we see from (66) that zm*→z*. Since the arbitrary Cauchy sequence (zm*)=(zk*m)k,m∈ℕ∈ℓp* is convergent, the space ℓp* is complete.Corollary 15.
c
* and c0* are the Banach spaces equipped with the norm ∥¨·∥¨∞ defined in (51).
Since it is known by Theorem14 that ℓp* is a complete metric space with the metric dp* induced in the case p≥¨1¨ by the norm ∥¨·∥¨p and in the case 0¨<¨p<¨1¨ by the p-norm ∥¨·∥¨^p, defined for  z*=(zk*)∈ℓp* by
(68)∥¨z*∥¨p=(∑k..∥¨zk*∥¨p)1/¨p∥¨z*∥¨^p=∑k..∥¨zk*∥¨p,
we have the following.Corollary 16.
The spaceℓp* is a Banach space with the norm ∥¨·∥¨p and p-norm ∥¨·∥¨^p defined by (68).


## 4. Conclusion

We present some important inequalities such as triangle, Minkowski, and some other inequalities in the sense of non-Newtonian complex calculus which are frequently used. We state the classical sequence spaces over the non-Newtonian complex fieldℂ* and try to understand their structure of being non-Newtonian complex vector space. There are lots of techniques that have been developed in the sense of non-Newtonian complex calculus. If the non-Newtonian complex calculus is employed instead of the classical calculus in the formulations, then many of the complicated phenomena in physics or engineering may be analyzed more easily.As an alternative to the classical (additive) calculus, Grossman and Katz [1] introduced some new kind of calculus named as non-Newtonian calculus, geometric calculus, anageometric calculus, and bigeometric calculus. Türkmen and Başar [6, 7] have recently studied the classical sequence spaces and related topics, in the sense of geometric calculus. Quite recently, Çakmak and Başar [8] have also worked on the same subject by using non-Newtonian calculus. In the present paper, we use the non-Newtonian complex calculus instead of non-Newtonian real calculus and geometric calculus. It is trivial that in the special cases of the generators α and β, the non-Newtonian complex calculus gives the special kind of the following calculus:(i)
ifα=β=I, the identity function, then the non-Newtonian complex calculus is reduced to the classical calculus;(ii)
ifα=I and β=exp, then the non-Newtonian complex calculus is reduced to the geometric calculus;(iii)
ifα=exp and β=I, then the non-Newtonian complex calculus is reduced to the anageometric calculus;(iv)
ifα=β=exp, then the non-Newtonian complex calculus is reduced to the bigeometric calculus.Since our results are obtained by using the non-Newtonian complex calculus, they are much more general and comprehensive than those of Türkmen and Başar [6, 7], Çakmak and Başar [8].Quite recently, Talo and Başar have studied the certain sets of sequences of fuzzy numbers and introduced the classical setsℓ∞(F), c(F), c0(F), and ℓp(F) consisting of the bounded, convergent, null, and absolutely p-summable sequences of fuzzy numbers in [9]. Nextly, they have defined the alpha-, beta-, and gamma-duals of a set of sequences of fuzzy numbers and gave the duals of the classical sets of sequences of fuzzy numbers together with the characterization of the classes of infinite matrices of fuzzy numbers transforming one of the classical set into another one. Following Bashirov et al. [2] and Uzer [3], we give the corresponding results for multiplicative calculus to the results derived for the sets of fuzzy valued sequences in Talo and Başar [9], as a beginning. As a natural continuation of this paper, we should record from now on that it is meaningful to define the alpha-, beta-, and gamma-duals of a set of sequences over the non-Newtonian complex field ℂ* and to determine the duals of classical spaces ℓ∞*, c*, c0*, and ℓp*. Further, one can obtain the similar results by using another type of calculus instead of non-Newtonian complex calculus.



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*Source: 739319-2013-06-26.xml*