Certain Sequence Spaces over the Non-Newtonian Complex Field

Abstract

It is known from functional analysis that in classical calculus, the sets 𝜔, ℓ_∞, 𝑐, 𝑐₀ and ℓ_𝑝 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^∗, 𝑐^∗₀ and ℓ^∗_𝑝 over the non-Newtonian complex field ℂ^∗ and to obtain the corresponding results for these spaces, where 0̈<̈𝑝<̈∞.

1. Preliminaries, Background, and Notations

A complete 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.

Let 𝛼 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: 𝛼-addition∶𝑦+̇𝑧=𝛼𝛼⁻¹𝑦+𝛼⁻¹(𝑧𝛼)𝛼-subtraction∶𝑦−̇𝑧=𝛼⁻¹𝑦−𝛼⁻¹(𝑧̇𝛼)𝛼-multiplication∶𝑦×𝑧=𝛼⁻¹𝑦×𝛼⁻¹(𝑧0̇̇𝛼)𝛼-division𝑧≠∶𝑦/𝑧=𝛼⁻¹𝑦𝛼⁻¹(𝑧)𝛼-ordering∶𝑦≤̇𝑧⟺𝛼⁻¹𝑦≤𝛼⁻¹(𝑧). (1) 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 0̇  and 𝛼-one by 1̇ which are obtained from 𝛼(0) and 𝛼(1), respectively. 0̇ and numbers obtained by successive 𝛼-addition of 1̇ to 0̇ with numbers obtained by successive 𝛼-subtraction of 1̇ from 0̇ are called 𝛼-𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠. Thus, 𝛼-integers are given as follows: …,𝛼(−2),𝛼(−1),𝛼(0),𝛼(1),𝛼(2),… (2) Thus, we have for all 𝑛∈ℤ that 𝑛̇=𝛼(𝑛). Let 𝑥∈𝐴. If 𝑥>̇0̇ then 𝑥 is called 𝛼-positive and if 𝑥<̇0̇ then 𝑥 is called 𝛼-𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒. The 𝛼-absolute value |𝑥̇|̇ of 𝑥∈𝐴 is defined by ̇̇>̇̇̇̇̇<̇̇|𝑥|=𝑥,𝑥0,0,𝑥=0,0−̇𝑥,𝑥0. (3) 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 [𝑟̇,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 (𝑢_𝑛) 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 (𝑢_𝑛). If there is such a number 𝑢, then (𝑢_𝑛) is said to be 𝛼-𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡 to 𝑢, which is called the 𝛼-limit of (𝑢_𝑛). In other words, lim_𝑛→∞𝑢_𝑛>̇̇=𝑢𝛼-convergent⟺∀𝜀0,∃𝑛₀∈ℕ∋|𝑢̇_𝑛−̇𝑢|̇<̇𝜀∀𝑛≥𝑛₀andsome𝑢∈𝐴. (4)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 (𝑎_𝑛) of arguments of 𝑓 distinct from 𝑎, if (𝑎_𝑛) is 𝛼-convergent to 𝑎, then {𝑓(𝑎_𝑛)}𝛽-convergesto 𝑏 and is denoted by lim^∗_𝑥→𝑎𝑓(𝑥)=𝑏. That is, lim^∗_𝑥→𝑎𝑓(𝑥)>̈̈>̇̇̈(𝑥)|̈̈̇|̇̇=𝑏⟺∀𝜖0,∃𝛿0∋|𝑓−̈𝑏<𝜖∀𝑥∈𝐴with|𝑥−̇𝑎<𝛿. (5) A function 𝑓 is ∗-𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 at a point 𝑎 in 𝐴 if and only if 𝑎 is an argument of 𝑓 and lim^∗_𝑥→𝑎𝑓(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 from 𝛼-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 numbers 𝑢 and 𝑣 in 𝐴, 𝜄(𝑢̇)(𝑢)̈(𝑣)𝜄()(𝑢)(𝑣)𝜄(𝑢̇)(𝑢)̈(𝑣)𝜄𝑢̇(𝑢)̈(𝑣)0̇𝑢̇(𝑢)̈(𝑣).+𝑣=𝜄+𝜄𝑢−̇𝑣=𝜄−̈𝜄×𝑣=𝜄×𝜄/𝑣=𝜄/𝜄,𝑣≠≤𝑣⟺𝜄≤𝜄 (6)It turns out that 𝜄(𝑥)=𝛽{𝛼⁻¹(𝑥)} for every 𝑥 in 𝐴, and that 𝜄(𝑛)̇=𝑛̈ 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

Let 𝑎̇∈(𝐴,+̇,−̇,×̇,/,̇≤̇) and 𝑏̈∈(𝐵,+̈,−̈,×̈,/,̈≤̈) be arbitrarily chosen elements from corresponding arithmetics. Then the ordered pair (𝑎,̇𝑏̈) 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, ℂ^∗∶=𝑧^∗=𝑎̇⊕𝑖^∗⊙𝑏̇∶𝑎,𝑏∈ℝ,𝑖^∗=̇1̈=𝑧0,_∗=𝑎,̇𝑏̈∶𝑎̇∈𝐴⊆ℝ,𝑏̈∈𝐵⊆ℝ. (7) The binary operations addition (⊕) and multiplication (⊙) of non-Newtonian complex numbers 𝑧^∗₁=(𝑎̇₁,𝑏̈₁) and 𝑧^∗₂=(𝑎̇₂,𝑏̈₂) are defined, as follows: ⊕∶ℂ^∗×ℂ^∗⟶ℂ^∗𝑧^∗₁,𝑧^∗₂↦𝑧^∗₁⊕𝑧^∗₂=𝑎̇₁+̇𝑎̇₂,𝑏̈₁+̈𝑏̈₂,⊙∶ℂ^∗×ℂ^∗⟶ℂ^∗𝑧^∗₁,𝑧^∗₂↦𝑧^∗₁⊙𝑧^∗₂=𝛼𝑎₁̇𝑎₂̇−𝑏̈₁𝑏̈₂,𝛽𝑎₁̇𝑏̈₂+𝑏̈₁𝑎₂̇, (8) where 𝑎̇₁,𝑎̇₂∈𝐴 and 𝑏̈₁,𝑏̈₂∈𝐵 with 𝑎₁̇=𝛼⁻¹𝑎̇₁=𝛼⁻¹𝛼𝑎₁=𝑎₁∈ℝ,𝑏̈₁=𝛽⁻¹𝑏̈₁=𝛽⁻¹𝛽𝑏₁=𝑏₁∈ℝ. (9) 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.

Let 𝑏̈∈𝐵⊆ℝ. Then the number 𝑏̈×̈𝑏̈ is called the 𝛽-square of 𝑏̈ and is denoted by 𝑏^̈2. Let 𝑏̈ be a non-negative number in 𝐵. Then, 𝛽[𝛽⁻¹(𝑏̈)] is called the 𝛽-square root of 𝑏̈ and is denoted by 𝑏̈^... The ∗-distance 𝑑^∗ between two arbitrarily elements 𝑧^∗₁=(𝑎̇₁,𝑏̈₁) and 𝑧^∗₂=(𝑎̇₂,𝑏̈₂) of the set ℂ^∗ is defined by 𝑑^∗∶ℂ^∗×ℂ^∗⟶[0̈̈,∞)̈=𝐵^′𝑧⊂𝐵_∗1,𝑧^∗₂⟶𝑑^∗𝑧^∗₁,𝑧^∗₂=^..𝜄𝑎̇₁−̇𝑎̇₂^̈2+̈𝑏̈₁−̈𝑏̈₂^̈2=𝛽𝑎₁−𝑎₂²+𝑏₁−𝑏₂². (10) 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.

Let 𝑧^∗∈ℂ^∗ be an arbitrary element. 𝑑^∗(𝑧^∗,𝜃^∗) is called ∗-norm of 𝑧^∗ and is denoted by ‖̈⋅‖̈. In other words, ‖𝑧̈^∗‖̈=𝑑^∗𝑧^∗,𝜃^∗=^..𝜄𝑎̇−̇0̇^̈2+̈𝑏̈−̈0̈^̈2=𝛽𝑎²+𝑏², (11) where 𝑧^∗=(𝑎,̇𝑏̈) and 𝜃^∗=(0̇,0̈). Moreover, since for all 𝑧^∗₁,𝑧^∗₂∈ℂ^∗ we have 𝑑^∗(𝑧^∗₁,𝑧^∗₂)=‖𝑧̈^∗₁⊖𝑧^∗₂‖̈, 𝑑^∗ is the induced metric from the norm ‖̈⋅‖̈.

Lemma 2 (∗-Triangle inequality). Let 𝑧^∗₁,𝑧^∗₂∈ℂ^∗. Then, ‖𝑧̈^∗₁⊕𝑧^∗₂‖̈≤̈‖𝑧̈^∗₁‖̈+̈‖𝑧̈^∗₂‖.̈ (12)

Proof. Let 𝑧^∗₁,𝑧^∗₂∈ℂ^∗. Then, a straightforward calculation gives that ‖𝑧̈^∗₁⊕𝑧^∗₂‖̈=^..𝜄𝑎̇₁+̇𝑎̇₂^̈2+̈𝑏̈₁+̈𝑏̈₂^̈2=𝛽𝑎₁+𝑎₂²+𝑏₁+𝑏₂²≤̈𝛽𝑎²₁+𝑏²₁+𝑎²₂+𝑏²₂=𝛽𝛽⁻¹𝛽𝑎²₁+𝑏²₁+𝛽⁻¹𝛽𝑎²₂+𝑏²₂=𝛽𝛽⁻¹‖𝑧̈^∗₁‖̈+𝛽⁻¹‖𝑧̈^∗₂‖̈=‖𝑧̈^∗₁‖̈+̈‖𝑧̈^∗₂‖.̈ (13) Hence, the inequality (12) holds.

Lemma 3. ‖𝑧̈^∗₁⊙𝑧^∗₂‖̈=‖𝑧̈^∗₁‖̈×̈‖𝑧̈^∗₂‖̈ for all 𝑧^∗₁,𝑧^∗₂∈ℂ^∗.

Proof. Let 𝑧^∗₁,𝑧^∗₂∈ℂ^∗. In this case, one can observe by a routine verification that ‖𝑧̈^∗₁⊙𝑧^∗₂‖̈=^..𝜄𝛼𝑎₁̇𝑎₂̇−𝑏̈₁𝑏̈₂^̈2+̈𝛽𝑎₁̇𝑏̈₂+𝑏̈₁𝑎̇₂^̈2=𝛽𝑎₁𝑎₂−𝑏₁𝑏₂²+𝑎₁𝑏₂+𝑏₁𝑎₂²=𝛽𝑎²₁+𝑏²₁𝑎²₂+𝑏²₂=𝛽𝛽⁻¹𝛽𝑎²₁+𝑏²₁×𝛽⁻¹𝛽𝑎²₂+𝑏²₂=𝛽𝛽⁻¹‖𝑧̈^∗₁‖̈×𝛽⁻¹‖𝑧̈^∗₂‖̈=‖𝑧̈^∗₁‖̈×̈‖𝑧̈^∗₂‖,̈ (14) as required.

Lemma 4. Let 𝑧^∗₁,𝑧^∗₂∈ℂ^∗. Then, the following inequality holds: ‖𝑧̈^∗₁⊕𝑧^∗₂‖̈/̈1̈+̈‖𝑧̈^∗₁⊕𝑧^∗₂‖̈≤̈‖𝑧̈^∗₁‖̈/̈1̈+̈‖𝑧̈^∗₁‖̈+̈‖𝑧̈^∗₂‖̈/̈1̈+̈‖𝑧̈^∗₂‖̈. (15)

Proof. Let 𝑧^∗₁,𝑧^∗₂∈ℂ^∗. Then, one can see that ‖𝑧̈^∗₁⊕𝑧^∗₂‖̈/̈1̈+̈‖𝑧̈^∗₁⊕𝑧^∗₂‖̈=𝛽𝑎₁+𝑎₂²+𝑏₁+𝑏₂²/̈1̈+̈𝛽𝑎₁+𝑎₂²+𝑏₁+𝑏₂²=𝛽𝑎₁+𝑎₂²+𝑏₁+𝑏₂²1+𝑎₁+𝑎₂²+𝑏₁+𝑏₂²≤̈𝛽𝑎²₁+𝑏²₁1+𝑎²₁+𝑏²₁+𝑎²₂+𝑏²₂1+𝑎²₂+𝑏²₂𝛽=𝛽⁻¹𝛽𝑎²₁+𝑏²₁𝛽⁻¹𝛽𝛽⁻¹1̈+𝛽⁻¹𝛽𝑎²₁+𝑏²₁+𝛽⁻¹𝛽𝑎²₂+𝑏²₂𝛽⁻¹𝛽𝛽⁻¹1̈+𝛽⁻¹𝛽𝑎²₂+𝑏²₂𝛽=𝛽⁻¹‖𝑧̈^∗₁‖̈𝛽⁻¹1̈+̈‖𝑧̈^∗₁‖̈+𝛽⁻¹‖𝑧̈^∗₂‖̈𝛽⁻¹1̈+̈‖𝑧̈^∗₂‖̈=𝛽𝛽⁻¹𝛽𝛽⁻¹‖𝑧̈^∗₁‖̈𝛽⁻¹1̈+̈‖𝑧̈^∗₁‖̈+𝛽⁻¹𝛽𝛽⁻¹‖𝑧̈^∗₂‖̈𝛽⁻¹1̈+̈‖𝑧̈^∗₂‖̈=𝛽𝛽⁻¹‖𝑧̈^∗₁‖̈/̈1̈+̈‖𝑧̈^∗₁‖̈+𝛽⁻¹‖𝑧̈^∗₂‖̈/̈1̈+̈‖𝑧̈^∗₂‖̈=‖𝑧̈^∗₁‖̈/̈1̈+̈‖𝑧̈^∗₁‖̈+̈‖𝑧̈^∗₂‖̈/̈1̈+̈‖𝑧̈^∗₂‖̈. (16) This means that the inequality (15) holds.

Lemma 5 (∗-Minkowski inequality). Let 𝑝≥̈1̈ and 𝑧^∗_𝑘,𝑡^∗_𝑘∈ℂ^∗ for all 𝑘∈{1,2,3,…,𝑛}. Then, ‖𝑧̈^𝑛.._𝑘=0^∗_𝑘⊕𝑡^∗_𝑘‖̈^𝑝1̈/𝑝≤̈^𝑛.._𝑘=0‖𝑧̈^∗_𝑘‖̈^𝑝1̈/𝑝+̈^𝑛.._𝑘=0‖𝑡̈^∗_𝑘‖̈^𝑝1̈/𝑝. (17)

Proof. Let 𝑝≥̈1̈ and 𝑧^∗_𝑘,𝑡^∗_𝑘∈ℂ^∗ for all 𝑘∈{1,2,3,…,𝑛}. Then, ‖𝑧̈^𝑛.._𝑘=0^∗_𝑘⊕𝑡^∗_𝑘‖^𝑝̈=‖𝑧̈^∗₀⊕𝑡^∗₀‖^𝑝̈+̈‖𝑧̈^∗₁⊕𝑡^∗₁‖^𝑝̈+̈⋯+̈‖𝑧̈^∗_𝑛⊕𝑡^∗_𝑛‖^𝑝̈=𝛽𝛽⁻¹‖𝑧̈^∗₀⊕𝑡^∗₀‖^𝑝̈+𝛽⁻¹‖𝑧̈^∗₁⊕𝑡^∗₁‖^𝑝̈+⋯+𝛽⁻¹‖𝑧̈^∗_𝑛⊕𝑡^∗_𝑛‖^𝑝̈. (18) Let us take 𝑧^∗_𝑘=(𝑥̇_𝑘,𝑦̈_𝑘),𝑧_𝑘=(𝑥_𝑘,𝑦_𝑘),𝑡^∗_𝑘=(𝑢̇_𝑘,𝑣̈_𝑘),𝑡_𝑘=(𝑢_𝑘,𝑣_𝑘). Then, since the equality ‖𝑧̈^∗_𝑘⊕𝑡^∗_𝑘‖̈=^..𝜄𝑥̇_𝑘+̇𝑢̇_𝑘^̈2+̈𝑦̈_𝑘+̈𝑣̈_𝑘^̈2=𝛽𝑥_𝑘+𝑢_𝑘²+𝑦_𝑘+𝑣_𝑘²=𝛽𝑧_𝑘+𝑡_𝑘 (19) holds for every fixed 𝑘, we obtain ‖𝑧̈^∗_𝑘⊕𝑡^∗_𝑘‖^𝑝̈=‖𝑧̈^∗_𝑘⊕𝑡^∗_𝑘‖̈×̈⋯×̈‖𝑧̈^∗_𝑘⊕𝑡^∗_𝑘‖_𝑝-times̈=𝛽𝛽⁻¹‖𝑧̈^∗_𝑘⊕𝑡^∗_𝑘‖̈×⋯×𝛽⁻¹‖𝑧̈^∗_𝑘⊕𝑡^∗_𝑘‖̈=𝛽𝛽⁻¹𝛽𝑧_𝑘+𝑡_𝑘×⋯×𝛽⁻¹𝛽𝑧_𝑘+𝑡_𝑘=𝛽𝑧_𝑘+𝑡_𝑘^{𝛽−1(𝑝)} (20) by (18) and Minkowski inequality in the complex field leads us to ‖𝑧̈^𝑛.._𝑘=0^∗_𝑘⊕𝑡^∗_𝑘‖^𝑝̈=𝛽𝛽⁻¹𝛽𝑧₀⊕𝑡₀^{𝛽−1(𝑝)}+𝛽⁻¹𝛽𝑧₁⊕𝑡₁^{𝛽−1(𝑝)}+⋯+𝛽⁻¹𝛽𝑧_𝑛⊕𝑡_𝑛^{𝛽−1(𝑝)}=𝛽^𝑛_𝑘=0𝑧_𝑘+𝑡_𝑘^{𝛽−1(𝑝)}≤̈𝛽^𝑛_𝑘=0𝑧_𝑘^{𝛽−1(𝑝)1/𝛽−1(𝑝)}+^𝑛_𝑘=0𝑡_𝑘^{𝛽−1(𝑝)}1𝛽⁻¹(𝑝)^{𝛽−1(𝑝)},𝛽^−1𝑛.._𝑘=0‖𝑧̈^∗_𝑘‖̈^𝑝1̈/𝑝+̈^𝑛.._𝑘=0‖𝑡̈^∗_𝑘‖̈^{𝑝1̈/𝑝𝑝}=𝛽^−1𝑛.._𝑘=0‖𝑧̈^∗_𝑘‖̈^𝑝1̈/𝑝+̈^𝑛.._𝑘=0‖𝑡̈^∗_𝑘‖̈^𝑝1̈/𝑝×̈⋯×̈^𝑛.._𝑘=0‖𝑧̈^∗_𝑘‖̈^𝑝1̈/𝑝+̈^𝑛.._𝑘=0‖𝑡̈^∗_𝑘‖̈^𝑝1̈/𝑝=𝛽^−1𝑛.._𝑘=0‖𝑧̈^∗_𝑘‖̈^𝑝1̈/𝑝+̈^𝑛.._𝑘=0‖𝑡̈^∗_𝑘‖̈^𝑝1̈/𝑝×⋯×𝛽^−1𝑛.._𝑘=0‖𝑧̈^∗_𝑘‖̈^𝑝1̈/𝑝+̈^𝑛.._𝑘=0‖𝑡̈^∗_𝑘‖̈^𝑝1̈/𝑝=𝛽^−1𝑛.._𝑘=0‖𝑧̈^∗_𝑘‖̈^𝑝1̈/𝑝+𝛽^−1𝑛.._𝑘=0‖𝑡̈^∗_𝑘‖̈^𝑝1̈/𝑝×⋯×𝛽^−1𝑛.._𝑘=0‖𝑧̈^∗_𝑘‖̈^𝑝1̈/𝑝+𝛽^−1𝑛.._𝑘=0‖𝑡̈^∗_𝑘‖̈^𝑝1̈/𝑝=𝛽^−1𝑛.._𝑘=0‖𝑧̈^∗_𝑘‖̈^𝑝1̈/𝑝+𝛽^−1𝑛.._𝑘=0‖𝑡̈^∗_𝑘‖̈^{𝑝1̈/𝑝𝛽−1(𝑝)}. (21)(22)
On the other hand, let us prove 𝛽^𝑛_𝑘=0𝑧_𝑘^{𝛽−1(𝑝)1/𝛽−1(𝑝)}=^𝑛.._𝑘=0‖𝑧̈^∗_𝑘‖̈^𝑝1̈/𝑝. (23) Indeed, 𝛽^𝑛_𝑘=0𝑧_𝑘^{𝛽−1(𝑝)1/𝛽−1(𝑝)𝑝}=𝛽^𝑛_𝑘=0𝑧_𝑘^{𝛽−1(𝑝)1/𝛽−1(𝑝)}×̈⋯×̈𝛽^𝑛_𝑘=0𝑧_𝑘^{𝛽−1(𝑝)1/𝛽−1(𝑝)}_𝑝-times=𝛽^𝑛_𝑘=0𝑧_𝑘^{𝛽−1(𝑝)1/𝛽−1(𝑝)}×⋯×^𝑛_𝑘=0𝑧_𝑘^{𝛽−1(𝑝)1/𝛽−1(𝑝)}=𝛽^𝑛_𝑘=0𝑧_𝑘^{𝛽−1(𝑝)1/𝛽−1(𝑝)𝛽−1(𝑝)}=𝛽^𝑛_𝑘=0𝑧_𝑘^{𝛽−1(𝑝)}=^𝑛.._𝑘=0‖𝑧̈^∗_𝑘‖^𝑝̈. (24) Substituting the relation (24) in (22) we obtain, 𝛽^−1𝑛.._𝑘=0‖𝑧̈^∗_𝑘‖̈^𝑝1̈/𝑝+̈^𝑛.._𝑘=0‖𝑡̈^∗_𝑘‖̈^{𝑝1̈/𝑝𝑝}=^𝑛_𝑘=0|𝑧_𝑘|^{𝛽−1(𝑝)1/𝛽−1(𝑝)}+^𝑛_𝑘=0|𝑡_𝑘|^{𝛽−1(𝑝)1/𝛽−1(𝑝)𝛽−1(𝑝)}. (25) By using this equality in (21), we get ‖𝑧̈^𝑛.._𝑘=0^∗_𝑘⊕𝑡^∗_𝑘‖̈^𝑝1̈/𝑝≤̈^𝑛.._𝑘=0‖𝑧̈^∗_𝑘‖̈^𝑝1̈/𝑝+̈^𝑛.._𝑘=0‖𝑡̈^∗_𝑘‖̈^𝑝1̈/𝑝, (26) as desired.

Theorem 6. (ℂ^∗,𝑑^∗) is a complete metric space, where 𝑑^∗ is defined by (10).

Proof. First, we show that 𝑑^∗, defined by (10), is a metric on ℂ^∗.
It is immediate for 𝑧^∗₁,𝑧^∗₂∈ℂ^∗ that 𝑑^∗𝑧^∗₁,𝑧^∗₂=^..𝜄𝑎̇₁−̇𝑎̇₂^̈2+̈𝑏̈₁−̈𝑏̈₂^̈2=𝛽𝑎₁−𝑎₂²+𝑏₁−𝑏₂²≥̈0̈. (27)(i)Now we show that 𝑑^∗(𝑧^∗₁,𝑧^∗₂)=0̈ if and only if 𝑧^∗₁=𝑧^∗₂ for 𝑧^∗₁,𝑧^∗₂∈ℂ^∗. Indeed, 𝑑^∗𝑧^∗₁,𝑧^∗₂=0̈⟺^..𝜄𝑎̇₁−̇𝑎̇₂^̈2+̈𝑏̈₁−̈𝑏̈₂^̈2=0̈⟺𝛽𝑎₁−𝑎₂²+𝑏₁−𝑏₂²(0)⟺𝑎=𝛽₁−𝑎₂²+𝑏₁−𝑏₂²=0⟺𝑎₁−𝑎₂=0,𝑏₁−𝑏₂=0⟺𝑎₁=𝑎₂,𝑏₁=𝑏₂⟺𝑎̇₁=𝑎̇₂,𝑏̈₁=𝑏̈₂⟺𝑧^∗₁=𝑎̇₁,𝑏̈₁=𝑎̇₂,𝑏̈₂=𝑧^∗₂. (28)(ii)One can easily establish for all 𝑧^∗₁,𝑧^∗₂∈ℂ^∗ that 𝑑^∗𝑧^∗₁,𝑧^∗₂=^..𝜄𝑎̇₁−̇𝑎̇₂^̈2+̈𝑏̈₁−̈𝑏̈₂^̈2=𝛽𝑎₁−𝑎₂²+𝑏₁−𝑏₂²=𝛽𝑎₂−𝑎₁²+𝑏₂−𝑏₁²=𝑑^∗𝑧^∗₂,𝑧^∗₁. (29)(iii)We show that the inequality 𝑑^∗(𝑧^∗₁,𝑧^∗₂)+̈𝑑^∗(𝑧^∗₂,𝑧^∗₃)≥̈𝑑^∗(𝑧^∗₁,𝑧^∗₃) holds for all 𝑧^∗₁,𝑧^∗₂,𝑧^∗₃∈ℂ^∗. In fact, 𝑑^∗𝑧^∗₁,𝑧^∗₂+̈𝑑^∗𝑧^∗₂,𝑧^∗₃=^..𝜄𝑎̇₁−̇𝑎̇₂^̈2+̈𝑏̈₁−̈𝑏̈₂^̈2+̈^..𝜄𝑎̇₂−̇𝑎̇₃^̈2+̈𝑏̈₂−̈𝑏̈₃^̈2=𝛽𝑎₁−𝑎₂²+𝑏₁−𝑏₂²+̈𝛽𝑎₂−𝑎₃²+𝑏₂−𝑏₃²=𝛽𝑎₁−𝑎₂²+𝑏₁−𝑏₂²+𝑎₂−𝑎₃²+𝑏₂−𝑏₃²≥̈𝛽𝑎₁−𝑎₃²+𝑏₁−𝑏₃²=^..𝜄𝑎̇₁−̇𝑎̇₃^̈2+̈𝑏̈₁−̈𝑏̈₃^̈2=𝑑^∗𝑧^∗₁,𝑧^∗₃. (30)
Therefore, 𝑑^∗ is a metric over ℂ^∗.
Now, we can show that the metric space (ℂ^∗,𝑑^∗) is complete. Let (𝑧^∗_𝑛)_𝑛∈ℕ be an arbitrary Cauchy sequence in ℂ^∗. In this case, for all 𝜀>̈0̈ there exists an 𝑛₀∈ℕ such that 𝑑^∗(𝑧^∗_𝑚,𝑧^∗_𝑛)<̈𝜀 for all 𝑚,𝑛≥𝑛₀. Let 𝑧^∗_𝑚=(𝑎̇_𝑚,𝑏̈_𝑚)∈ℂ^∗ and 𝑚∈ℕ. Then, 𝑑^∗𝑧^∗_𝑚,𝑧^∗_𝑛=^..𝜄𝑎̇_𝑚−̇𝑎̇_𝑛^̈2+̈𝑏̈_𝑚−̈𝑏̈_𝑛^̈2=𝛽𝑎_𝑚−𝑎_𝑛²+𝑏_𝑚−𝑏_𝑛²<̈𝜀=𝛽𝜀^′. (31) Thus we obtain that 𝑎_𝑚−𝑎_𝑛²+𝑏_𝑚−𝑏_𝑛²<𝜀^′. (32) On the other hand, since the following inequalities: 𝑎_𝑚−𝑎_𝑛=𝑎_𝑚−𝑎_𝑛²≤𝑎_𝑚−𝑎_𝑛²+𝑏_𝑚−𝑏_𝑛²,𝑏_𝑚−𝑏_𝑛=𝑏_𝑚−𝑏_𝑛²≤𝑎_𝑚−𝑎_𝑛²+𝑏_𝑚−𝑏_𝑛² (33) hold we therefore have by (32) that |𝑎_𝑚−𝑎_𝑛|<𝜀^′ and |𝑏_𝑚−𝑏_𝑛|<𝜀^′. This means that (𝑎_𝑛) and (𝑏_𝑛) 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 |𝑎_𝑛−𝑎|<𝜀^′/2 for all 𝑛≥𝑛₁ and for every 𝜀^′>0 there exists an 𝑛₂∈ℕ such that |𝑏_𝑛−𝑏|<𝜀^′/2 for all 𝑛≥𝑛₂.
Define, 𝑧^∗∈ℂ^∗ by 𝑧^∗=(𝑎,̇𝑏̈). Then, we have 𝑑^∗𝑧^∗_𝑛,𝑧^∗=^..𝜄𝑎̇_𝑛−̇𝑎̇^̈2+̈𝑏̈_𝑛−̈𝑏̈^̈2=𝛽𝑎_𝑛−𝑎²+𝑏_𝑛−𝑏²≤̈𝛽𝑎_𝑛−𝑎²+𝑏_𝑛−𝑏²=𝛽𝑎_𝑛−𝑎+𝑏_𝑛̈𝜀−𝑏≤𝛽^′2+𝜀^′2=𝛽𝜀^′=𝜀. (34) Hence, (ℂ^∗,𝑑^∗) is a complete metric space.

Since ℂ^∗ 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 ‖𝑧̈^∗‖̈=^..[𝜄(𝑎)̇]^̈2+̈𝑏̈^̈2;𝑧^∗=𝑎,̇𝑏̈∈ℂ^∗. (35)

3. Sequence Spaces over Non-Newtonian Complex Field

In this section, we define the sets 𝜔^∗, ℓ^∗_∞, 𝑐^∗, 𝑐^∗₀, and ℓ^∗_𝑝 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 ℓ_𝑝 over the complex field ℂ, respectively. That is to say that 𝜔^∗∶=𝑧^∗=𝑧^∗_𝑘∶𝑧^∗_𝑘∈ℂ^∗,ℓ∀𝑘∈ℕ_∗∞∶=𝑧^∗=𝑧^∗_𝑘∈𝜔^∗∶sup_𝑘∈ℕ‖𝑧̈^∗_𝑘‖̈̈,𝑐<∞_∗∶=𝑧^∗=𝑧^∗_𝑘∈𝜔^∗∶∃𝑙^∗∈ℂ^∗∋lim^∗_𝑘→∞𝑧^∗_𝑘=𝑙^∗,𝑐^∗₀∶=𝑧^∗=𝑧^∗_𝑘∈𝜔^∗∶lim^∗_𝑘→∞𝑧^∗_𝑘=𝜃^∗,ℓ^∗_𝑝∶=𝑧^∗=𝑧^∗_𝑘∈𝜔^∗∶^.._𝑘‖𝑧̈^∗_𝑘‖^𝑝̈<̈∞. (36) 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: +∶𝜔^∗×𝜔^∗⟶𝜔^∗ℎℎ𝑧^∗,𝑡^∗↦𝑧^∗+𝑡^∗=𝑧^∗_𝑘⊕𝑡^∗_𝑘;𝑧^∗=𝑧^∗_𝑘,𝑡^∗=𝑡^∗_𝑘∈𝜔^∗,×∶ℂ^∗×𝜔^∗⟶𝜔^∗𝛼,𝑧^∗↦𝛼×𝑧^∗=𝛼⊙𝑧^∗_𝑘;𝑧^∗=𝑧^∗_𝑘∈𝜔^∗,𝛼∈ℂ^∗. (37)

Theorem 8. Define the function 𝑑_𝜔∗ by 𝑑_𝜔∗∶𝜔^∗×𝜔^∗⟶𝐵^′𝑧⊆𝐵_∗,𝑡^∗↦𝑑_𝜔∗𝑧^∗,𝑡^∗=^.._𝑘𝜇_𝑘×̈𝑑^∗𝑧^∗_𝑘,𝑡^∗_𝑘/̈1̈+̈𝑑^∗𝑧^∗_𝑘,𝑡^∗_𝑘, (38) where (𝜇_𝑘)∈𝐵^′⊆𝐵 such that ∑̈_𝑘𝜇_𝑘 is convergent with 𝜇_𝑘>̈0̈ for all 𝑘∈ℕ. Then, (𝜔^∗,𝑑_𝜔∗) is a metric space.

Proof. We show that 𝑑_𝜔∗ satisfies the metric axioms on the space 𝜔^∗ of all non-Newtonian complex valued sequences.
(i) First we show that 𝑑_𝜔∗(𝑧^∗,𝑡^∗)≥̈0̈ for all 𝑧^∗,𝑡^∗∈𝜔^∗.
Because (ℂ^∗,𝑑^∗) is a metric space, we have 𝑑^∗(𝑧^∗_𝑘,𝑡^∗_𝑘)≥̈0̈; 𝑧^∗_𝑘,𝑡^∗_𝑘∈ℂ^∗ and 1̈+̈𝑑^∗(𝑧^∗_𝑘,𝑡^∗_𝑘)≥̈0̈. Hence, we obtain that 𝑑^∗(𝑧^∗_𝑘,𝑡^∗_𝑘)/̈[1̈+̈𝑑^∗(𝑧^∗_𝑘,𝑡^∗_𝑘)]≥̈0̈. Moreover, since 𝜇_𝑘>̈0̈ for all 𝑘∈ℕ, we have 𝜇_𝑘×̈𝑑^∗𝑧^∗_𝑘,𝑡^∗_𝑘/̈1̈+̈𝑑^∗𝑧^∗_𝑘,𝑡^∗_𝑘≥̈0̈. (39) This means that 𝑑_𝜔∗𝑧^∗,𝑡^∗=^.._𝑘𝜇_𝑘×̈𝑑^∗𝑧^∗_𝑘,𝑡^∗_𝑘/̈1̈+̈𝑑^∗𝑧^∗_𝑘,𝑡^∗_𝑘≥̈0̈. (40)
(ii) We show that 𝑑_𝜔∗(𝑧^∗,𝑡^∗)=0̈ iff 𝑧^∗=𝑡^∗. In this situation, one can see that 𝑑_𝜔∗𝑧^∗,𝑡^∗=^.._𝑘𝜇_𝑘×̈𝑑^∗𝑧^∗_𝑘,𝑡^∗_𝑘/̈1̈+̈𝑑^∗𝑧^∗_𝑘,𝑡^∗_𝑘=0̈⟺𝜇_𝑘×̈𝑑^∗𝑧^∗_𝑘,𝑡^∗_𝑘/̈1̈+̈𝑑^∗𝑧^∗_𝑘,𝑡^∗_𝑘=0̈,∀𝑘∈ℕ⟺𝑑^∗𝑧^∗_𝑘,𝑡^∗_𝑘/̈1̈+̈𝑑^∗𝑧^∗_𝑘,𝑡^∗_𝑘=̈𝜇0,_𝑘>̈0̈,∀𝑘∈ℕ⟺𝑑^∗𝑧^∗_𝑘,𝑡^∗_𝑘=0̈;𝑑^∗𝑧^∗_𝑘,𝑡^∗_𝑘≥̈0̈,∀𝑘∈ℕ⟺𝑧^∗_𝑘=𝑡^∗_𝑘;𝑑^∗metric,∀𝑘∈ℕ⟺𝑧^∗=𝑧^∗_𝑘=𝑡^∗_𝑘=𝑡^∗. (41)
(iii) We show that 𝑑_𝜔∗(𝑧^∗,𝑡^∗)=𝑑_𝜔∗(𝑡^∗,𝑧^∗) for 𝑧^∗=(𝑧^∗_𝑘),𝑡^∗=(𝑡^∗_𝑘)∈𝜔^∗. First, we know that 𝑑^∗ is a metric over ℂ^∗. Thus, 𝑑_𝜔∗𝑧^∗,𝑡^∗=^.._𝑘𝜇_𝑘×̈𝑑^∗𝑧^∗_𝑘,𝑡^∗_𝑘/̈1̈+̈𝑑^∗𝑧^∗_𝑘,𝑡^∗_𝑘=^.._𝑘𝜇_𝑘×̈𝑑^∗𝑡^∗_𝑘,𝑧^∗_𝑘/̈1̈+̈𝑑^∗𝑡^∗_𝑘,𝑧^∗_𝑘=𝑑_𝜔∗𝑡^∗,𝑧^∗. (42)
(iv) We show that 𝑑_𝜔∗(𝑧^∗,𝑡^∗)+̈𝑑_𝜔∗(𝑡^∗,𝑢^∗)≥̈𝑑_𝜔∗(𝑧^∗,𝑢^∗) holds for 𝑧^∗=(𝑧^∗_𝑘),𝑡^∗=(𝑡^∗_𝑘),𝑢^∗=(𝑢^∗_𝑘)∈𝜔^∗. Again, using the fact that (ℂ^∗,𝑑^∗) is a metric space, it is easy to see by Lemma 4 that 𝑑_𝜔∗𝑧^∗,𝑢^∗=^.._𝑘𝜇_𝑘×̈𝑑^∗𝑧^∗_𝑘,𝑢^∗_𝑘/̈1̈+̈𝑑^∗𝑧^∗_𝑘,𝑢^∗_𝑘≤̈^.._𝑘𝜇_𝑘×̈𝑑^∗𝑧^∗_𝑘,𝑡^∗_𝑘+̈𝑑^∗𝑡^∗_𝑘,𝑢^∗_𝑘/̈1̈+̈𝑑^∗𝑧^∗_𝑘,𝑡^∗_𝑘+̈𝑑^∗𝑡^∗_𝑘,𝑢^∗_𝑘≤̈^.._𝑘𝜇_𝑘×̈𝑑^∗𝑧^∗_𝑘,𝑡^∗_𝑘/̈1̈+̈𝑑^∗𝑧^∗_𝑘,𝑡^∗_𝑘+̈𝑑^∗𝑡^∗_𝑘,𝑢^∗_𝑘/̈1̈+̈𝑑^∗𝑡^∗_𝑘,𝑢^∗_𝑘=^.._𝑘𝜇_𝑘×̈𝑑^∗𝑧^∗_𝑘,𝑡^∗_𝑘/̈1̈+̈𝑑^∗𝑧^∗_𝑘,𝑡^∗_𝑘+̈^.._𝑘𝜇_𝑘×̈𝑑^∗𝑡^∗_𝑘,𝑢^∗_𝑘/̈1̈+̈𝑑^∗𝑡^∗_𝑘,𝑢^∗_𝑘=𝑑_𝜔∗𝑧^∗,𝑡^∗+̈𝑑_𝜔∗𝑡^∗,𝑢^∗, (43) as required.

Theorem 9. The set ℓ^∗_∞ is a sequence space.

Proof. It is trivial that the inclusion ℓ^∗_∞⊂𝜔^∗ holds.
(i) We show that 𝑧^∗+𝑡^∗∈ℓ^∗_∞ for 𝑧^∗=(𝑧^∗_𝑘),𝑡^∗=(𝑡^∗_𝑘)∈ℓ^∗_∞. Indeed, combining the hypothesis sup_𝑘∈ℕ‖𝑧̈^∗_𝑘‖̈<̈∞,sup_𝑘∈ℕ‖𝑡̈^∗_𝑘‖̈<̈∞ (44) with the fact ‖𝑧̈^∗_𝑘⊕𝑡^∗_𝑘‖̈≤̈‖𝑧̈^∗_𝑘‖̈+̈‖𝑡̈^∗_𝑘‖̈ obtained from Lemma 2, we can easily derive that sup_𝑘∈ℕ‖𝑧̈^∗_𝑘⊕𝑡^∗_𝑘‖̈≤̈sup_𝑘∈ℕ‖𝑧̈^∗_𝑘‖̈+̈sup_𝑘∈ℕ‖𝑡̈^∗_𝑘‖̈<̈∞. (45)
Hence, 𝑧^∗+𝑡^∗∈ℓ^∗_∞.
(ii) We show that 𝛼×𝑧^∗∈ℓ^∗_∞ for any 𝛼∈ℂ^∗ and for 𝑧^∗=(𝑧^∗_𝑘)∈ℓ^∗_∞.
Since ‖𝛼̈⊙𝑧^∗_𝑘‖̈=‖𝛼̈‖̈×̈‖𝑧̈^∗_𝑘‖̈ by Lemma 3 and sup_𝑘∈ℕ‖𝑧̈^∗_𝑘‖̈<̈∞, it is immediate that sup_𝑘∈ℕ‖𝛼̈⊙𝑧^∗_𝑘‖̈=‖𝛼̈‖̈×̈sup_𝑘∈ℕ‖𝑧̈^∗_𝑘‖̈<̈∞ (46) which means that 𝛼×𝑧^∗∈ℓ^∗_∞.
Therefore, we have proved that ℓ^∗_∞ is a subspace of the space 𝜔^∗.

Theorem 10. Define the relation 𝑑^∗_∞ by 𝑑^∗_∞∶ℓ^∗_∞×ℓ^∗_∞⟶𝐵^′𝑧⊆𝐵_∗,𝑡^∗↦𝑑^∗_∞𝑧^∗,𝑡^∗=sup_𝑘∈ℕ‖𝑧̈^∗_𝑘⊖𝑡^∗_𝑘‖.̈ (47)
Then, (ℓ^∗_∞,𝑑^∗_∞) is a complete metric space.

Proof. One can easily show by a routine verification that 𝑑^∗_∞ satisfies the metric axioms on the space ℓ^∗_∞. So, we omit the details.
Now, we prove the second part of the theorem. Let (𝑧^∗_𝑚) be a Cauchy sequence in ℓ^∗_∞, where 𝑧^∗_𝑚=(𝑧^∗_𝑘^𝑚)_𝑘∈ℕ. Then, there exists a positive integer 𝑘₀ such that 𝑑^∗_∞(𝑧^∗_𝑚,𝑧^∗_𝑟)=sup_𝑘∈ℕ‖𝑧̈^∗_𝑘^𝑚⊖𝑧^∗_𝑘^𝑟‖̈<̈𝜀 for all 𝑚,𝑟∈ℕ with 𝑚,𝑟>𝑘₀. For any fixed 𝑘, if 𝑚,𝑟>𝑘₀ then ‖𝑧̈^∗_𝑘^𝑚⊖𝑧^∗_𝑘^𝑟‖̈<̈𝜀. (48) In this case for any fixed 𝑘, (𝑧^∗_𝑘⁰,𝑧^∗_𝑘¹,…) is a Cauchy sequence of non-Newtonian complex numbers and since ℂ^∗ is complete, it converges to a 𝑧^∗_𝑘∈ℂ^∗. Define 𝑧^∗=(𝑧^∗₀,𝑧^∗₁,…) with infinitely many limits 𝑧^∗₀,𝑧^∗₁,…. Let us show 𝑧^∗∈ℓ^∗_∞ and 𝑧^∗_𝑚→𝑧^∗, as 𝑚→∞. Indeed, by (48), by letting 𝑟→∞, for 𝑚>𝑘₀ we obtain that ‖𝑧̈^∗_𝑘^𝑚⊖𝑧^∗_𝑘‖̈≤̈𝜀. (49) On the other hand, since 𝑧^∗_𝑚=(𝑧^∗_𝑘^𝑚)_𝑘∈ℕ∈ℓ^∗_∞, there exists 𝑡_𝑚∈𝐵⊆ℝ such that ‖𝑧̈^∗_𝑘^𝑚‖̈≤̈𝑡_𝑚 for all 𝑘∈ℕ. Hence, by triangle inequality (12), the inequality ‖𝑧̈^∗_𝑘‖̈≤̈‖𝑧̈^∗_𝑘⊖𝑧^∗_𝑘^𝑚‖̈+̈‖𝑧̈^∗_𝑘^𝑚‖̈≤̈𝜀+̈𝑡_𝑚 (50) holds for all 𝑘∈ℕ which is independent of 𝑘. Hence, 𝑧^∗=(𝑧^∗_𝑘)_𝑘∈ℕ∈ℓ^∗_∞. By (49), since 𝑚>𝑘₀, we obtain 𝑑^∗_∞(𝑧^∗_𝑚,𝑧^∗)=sup_𝑘∈ℕ‖𝑧̈^∗_𝑘^𝑚⊖𝑧^∗_𝑘‖̈≤̈𝜀. Therefore, the sequence (𝑧^∗_𝑚) converges to 𝑧^∗ which means that ℓ^∗_∞ is complete.

Since it is known by Theorem 10 that ℓ_∞ is a complete metric space with the metric 𝑑^∗_∞ induced by the norm ‖̈⋅‖_∞̈, defined by ‖𝑧̈^∗‖_∞̈=sup_𝑘∈ℕ‖𝑧̈^∗_𝑘‖;̈𝑧^∗=𝑧^∗_𝑘∈ℓ^∗_∞, (51) 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 that ℓ^∗_𝑝 is a sequence space in the case 0̈<̈𝑝<̈1̈.

Lemma 12. Let 0̈<̈𝑝<̈1̈. Then, the inequality (𝑎̈+̈𝑏̈)^𝑝<̈𝑎̈^𝑝+̈𝑏̈^𝑝 holds for all 𝑎,̈𝑏̈>̈0̈.

Proof. Let 0̈<̈𝑝<̈1̈ and 𝑎,̈𝑏̈>̈0̈. Then, one can easily see that 𝑎̈+̈𝑏̈^𝑝=𝑎̈+̈𝑏̈×̈⋯×_𝑝-times̈𝑎̈+̈𝑏̈{}̈̈{}𝛽=𝛽𝑎+𝑏×⋯×𝛽𝑎+𝑏=𝛽⁻¹𝛽(𝑎+𝑏)×⋯×𝛽⁻¹𝛽(𝑎+𝑏)_{𝛽−1𝑝-times}=𝛽(𝑎+𝑏)^{𝛽−1(𝑝)}<̈𝛽𝑎^{𝛽−1(𝑝)}+𝑏^{𝛽−1(𝑝)}=𝛽𝛽⁻¹𝑎̈^𝑝+𝛽⁻¹𝑏̈^𝑝=𝑎̈^𝑝+̈𝑏̈^𝑝, (52)
as desired.

Theorem 13. The sets 𝑐^∗,𝑐^∗₀,and ℓ^∗_𝑝 are sequence spaces, where 0̈<̈𝑝<̈∞.

Proof. It is not hard to establish by the similar way that 𝑐^∗₀ and ℓ^∗_𝑝 are the sequence spaces. So, to avoid the repetition of the similar statements, we consider only the set 𝑐^∗.
It is obvious that the inclusion 𝑐^∗⊂𝜔^∗ strictly holds.
(i) Let 𝑧^∗=(𝑧^∗_𝑘),𝑡^∗=(𝑡^∗_𝑘)∈𝑐^∗. Then, there exist 𝑙^∗₁,𝑙^∗₂∈ℂ^∗ such that lim^∗_𝑘→∞𝑧^∗_𝑘=𝑙^∗₁ and lim^∗_𝑘→∞𝑡^∗_𝑘=𝑙^∗₂. Thus, there exist 𝑘₁,𝑘₂∈ℕ such that ∀𝜀>̈0̈,‖𝑧̈^∗_𝑘⊖𝑙^∗₁‖̈≤̈𝜀/2̈̈∀𝑘≥𝑘₁,∀𝜀>̈0̈,‖𝑡̈^∗_𝑘⊖𝑙^∗₂‖̈≤̈𝜀/2̈̈∀𝑘≥𝑘₂. (53)
Thus if we set 𝑘₀=max{𝑘₁,𝑘₂}, by (53) we obtain for all 𝑘≥𝑘₀ that ‖̈𝑧^∗_𝑘⊕𝑡^∗_𝑘⊖𝑙^∗₁⊕𝑙^∗₂‖̈=‖̈𝑧^∗_𝑘⊖𝑙^∗₁⊕𝑡^∗_𝑘⊖𝑙^∗₂‖̈≤̈‖𝑧̈^∗_𝑘⊖𝑙^∗₁‖̈+̈‖𝑡̈^∗_𝑘⊖𝑙^∗₂‖̈≤̈𝜀/̈2̈+̈𝜀/̈2̈=𝜀 (54) which means that lim^∗_𝑘→∞𝑧^∗_𝑘⊕𝑡^∗_𝑘=𝑙^∗₁⊕𝑙^∗₂=lim^∗_𝑘→∞𝑧^∗_𝑘⊕lim^∗_𝑘→∞𝑡^∗_𝑘. (55)
Therefore, 𝑧^∗+𝑡^∗∈𝑐^∗.
(ii) Let 𝑧^∗=(𝑧^∗_𝑘)∈𝑐^∗ and 𝛼∈ℂ^∗⧵{𝜃^∗}. Since 𝑧^∗∈𝑐^∗ there exists an 𝑙^∗∈ℂ^∗ such that lim^∗_𝑘→∞𝑧^∗_𝑘=𝑙^∗, we have ∀𝜀>̈0̈,∃𝑘₀∈ℕsuchthat‖𝑧̈^∗_𝑘⊖𝑙^∗‖̈≤̈𝜀/̈‖𝛼̈‖̈∀𝑘≥𝑘₀. (56)
Thus, for 𝑘≥𝑘₀, we have ‖̈𝛼⊙𝑧^∗_𝑘⊖𝛼⊙ℓ^∗‖̈=‖𝛼̈⊙𝑧^∗_𝑘⊖ℓ^∗‖̈=‖𝛼̈‖̈×̈‖𝑧̈^∗_𝑘⊖ℓ^∗‖̈≤̈‖𝛼̈‖̈×̈𝜀/̈‖𝛼̈‖̈=𝜀 (57) which implies that lim^∗_𝑘→∞(𝛼⊙𝑧^∗_𝑘)=𝛼⊙𝑙^∗=𝛼⊙lim^∗_𝑘→∞𝑧^∗_𝑘Hence, 𝛼×𝑧^∗∈𝑐^∗.
That is to say that 𝑐^∗ is a subspace of 𝜔^∗.

Theorem 14. (𝑐^∗,𝑑^∗_∞), (𝑐^∗₀,𝑑^∗_∞), and (ℓ^∗_𝑝,𝑑^∗_𝑝) are complete metric spaces, where 𝑑^∗_𝑝 is defined as follows: 𝑑^∗_𝑝𝑧^∗,𝑡^∗=^..∑_𝑘‖𝑧̈^∗_𝑘⊖𝑡^∗_𝑘‖^𝑝̈,0̈<̈𝑝_..<̈1̈∑_𝑘‖𝑧̈^∗_𝑘⊖𝑡^∗_𝑘‖̈^𝑝1̈/𝑝≥̈̈𝑧,𝑝1;_∗=𝑧^∗_𝑘,𝑡^∗=𝑡^∗_𝑘∈ℓ^∗_𝑝. (58)

Proof. We consider only the space ℓ^∗_𝑝 with 𝑝≥̈1̈.
(i) For 𝑧^∗=(𝑧^∗_𝑘),𝑡^∗=(𝑡^∗_𝑘)∈ℓ^∗_𝑝, we establish that the two sided implication 𝑑^∗_𝑝(𝑧^∗,𝑡^∗)=0̈⇔𝑧^∗=𝑡^∗ holds. In fact, 𝑑^∗_𝑝𝑧^∗,𝑡^∗=^.._𝑘‖𝑧̈^∗_𝑘⊖𝑡^∗_𝑘‖̈^𝑝1̈/𝑝=0̈⟺^.._𝑘‖𝑧̈^∗_𝑘⊖𝑡^∗_𝑘‖^𝑝̈=0̈⟺∀𝑘∈ℕ,‖𝑧̈^∗_𝑘⊖𝑡^∗_𝑘‖^𝑝̈=0̈⟺∀𝑘∈ℕ,‖𝑧̈^∗_𝑘⊖𝑡^∗_𝑘‖̈=0̈⟺∀𝑘∈ℕ,𝑧^∗_𝑘=𝑡^∗_𝑘⟺𝑧^∗=𝑧^∗_𝑘=𝑡^∗_𝑘=𝑡^∗. (59)
(ii) For 𝑧^∗=(𝑧^∗_𝑘),𝑡^∗=(𝑡^∗_𝑘)∈ℓ^∗_𝑝, we show that 𝑑^∗_𝑝(𝑧^∗,𝑡^∗)=𝑑^∗_𝑝(𝑡^∗,𝑧^∗). In this situation, since ‖𝑧̈^∗_𝑘⊖𝑡^∗_𝑘‖̈=‖𝑡̈^∗_𝑘⊖𝑧^∗_𝑘‖̈ holds for every fixed 𝑘∈ℕ it is immediate that 𝑑^∗_𝑝𝑧^∗,𝑡^∗=^.._𝑘‖𝑧̈^∗_𝑘⊖𝑡^∗_𝑘‖̈^𝑝1̈/𝑝=^.._𝑘‖𝑡̈^∗_𝑘⊖𝑧^∗_𝑘‖̈^𝑝1̈/𝑝=𝑑^∗_𝑝𝑡^∗,𝑧^∗. (60)
(iii)  By Minkowski inequality in Lemma 5, we have for 𝑧^∗=(𝑧^∗_𝑘),𝑡^∗=(𝑡^∗_𝑘),𝑢^∗=(𝑢^∗_𝑘)∈ℓ^∗_𝑝 that 𝑑^∗_𝑝𝑧^∗,𝑡^∗=^.._𝑘‖𝑧̈^∗_𝑘⊖𝑡^∗_𝑘‖̈^𝑝1̈/𝑝=^.._𝑘‖̈𝑧^∗_𝑘⊖𝑢^∗_𝑘⊕𝑢^∗_𝑘⊖𝑡^∗_𝑘‖̈^𝑝1̈/𝑝≤̈^.._𝑘‖𝑧̈^∗_𝑘⊖𝑢^∗_𝑘‖̈^𝑝1̈/𝑝+̈^.._𝑘‖𝑢̈^∗_𝑘⊖𝑡^∗_𝑘‖̈^𝑝1̈/𝑝=𝑑^∗_𝑝𝑧^∗,𝑢^∗+̈𝑑^∗_𝑝𝑢^∗,𝑡^∗. (61)
Hence, triangle inequality is satisfied by 𝑑^∗_𝑝 on the space ℓ^∗_𝑝. Therefore, the function 𝑑^∗_𝑝 is a metric over the space ℓ^∗_𝑝.
Now we show that the metric space (ℓ^∗_𝑝,𝑑^∗_𝑝) is complete. Let (𝑧^∗_𝑚)_𝑚∈ℕ be an arbitrary Cauchy sequence in the space ℓ^∗_𝑝, where 𝑧^∗_𝑚=(𝑧^∗₁^𝑚,𝑧^∗₂^𝑚,…). Then, for any 𝜀>̈0̈ there exists an 𝑛₀∈ℕ such that 𝑑^∗_𝑝𝑧^∗_𝑚,𝑧^∗_𝑛=^.._𝑘‖𝑧̈^∗_𝑘^𝑚⊖𝑧^∗_𝑘^𝑛‖̈^𝑝1̈/𝑝<̈𝜀 (62) for all 𝑚,𝑛≥𝑛₀. Hence, for 𝑚,𝑛≥𝑛₀ and every fixed 𝑘∈ℕ, we obtain ‖𝑧̈^∗_𝑘^𝑚⊖𝑧^∗_𝑘^𝑛‖̈<̈𝜀. (63) If we set 𝑘 fixed then, it follows by (63) that (𝑧^∗_𝑘⁰,𝑧^∗_𝑘¹,…) is a Cauchy sequence. Since ℂ^∗ is complete, this sequence converges to a point say 𝑧^∗_𝑘. Let us define the sequence 𝑧^∗=(𝑧^∗₀,𝑧^∗₁,…) with these limits and show 𝑧^∗∈ℓ^∗_𝑝 and 𝑧^∗_𝑚→𝑧^∗, as 𝑚→∞. Indeed, by (62), we obtain the inequality for all 𝑚,𝑛∈ℕ with 𝑚,𝑛≥𝑛₀ that ‖𝑧̈^𝑗.._𝑘=0^∗_𝑘^𝑚⊖𝑧^∗_𝑘^𝑛‖^𝑝̈<̈𝜀^𝑝∀𝑗∈ℕ (64) and thus we have by letting 𝑛→∞ and 𝑚>𝑛₀ that ‖𝑧̈^𝑗.._𝑘=0^∗_𝑘^𝑚⊖𝑧^∗_𝑘‖^𝑝̈<̈𝜀^𝑝∀𝑗∈ℕ (65) which gives as 𝑗→∞ and for all 𝑚>𝑛₀ that 𝑑^∗_𝑝𝑧^∗_𝑚,𝑧^∗𝑝=^.._𝑘‖𝑧̈^∗_𝑘^𝑚⊖𝑧^∗_𝑘‖^𝑝̈<̈𝜀^𝑝. (66) Setting 𝑧^∗_𝑘=𝑧^∗_𝑘^𝑚⊕(𝑧^∗_𝑘⊖𝑧^∗_𝑘^𝑚) and applying Lemma 5 we obtain by (66) and the fact (𝑧^∗_𝑘^𝑚)∈ℓ^∗_𝑝 that ‖𝑧̈^.._𝑘^∗_𝑘‖̈^𝑝1̈/𝑝=^.._𝑘‖𝑧̈^∗_𝑘^𝑚⊕𝑧^∗_𝑘⊖𝑧^∗_𝑘^𝑚‖̈^𝑝1̈/𝑝≤̈^.._𝑘‖𝑧̈^∗_𝑘^𝑚‖̈^𝑝1̈/𝑝+̈^.._𝑘‖𝑧̈^∗_𝑘⊖𝑧^∗_𝑘^𝑚‖̈^𝑝1̈/𝑝<̈∞ (67) which means that 𝑧^∗=(𝑧^∗_𝑘)∈ℓ^∗_𝑝. Therefore, we see from (66) that 𝑧^∗_𝑚→𝑧^∗. Since the arbitrary Cauchy sequence (𝑧^∗_𝑚)=(𝑧^∗_𝑘^𝑚)_{𝑘,𝑚∈ℕ}∈ℓ^∗_𝑝 is convergent, the space ℓ^∗_𝑝 is complete.

Corollary 15. 𝑐^∗ and 𝑐^∗₀ are the Banach spaces equipped with the norm ‖̈⋅‖_∞̈ defined in (51).
Since it is known by Theorem 14 that ℓ^∗_𝑝 is a complete metric space with the metric 𝑑^∗_𝑝 induced in the case 𝑝≥̈1̈ by the norm ‖̈⋅‖_𝑝̈ and in the case 0̈<̈𝑝<̈1̈ by the 𝑝-norm ‖̈⋅‖_𝑝̈̂, defined for 𝑧^∗=(𝑧^∗_𝑘)∈ℓ^∗_𝑝 by ‖𝑧̈^∗‖_𝑝̈=^.._𝑘‖𝑧̈^∗_𝑘‖̈^𝑝1̈/𝑝‖𝑧̈^∗‖_𝑝̈̂=^.._𝑘‖𝑧̈^∗_𝑘‖^𝑝̈, (68)
we have the following.

Corollary 16. The space ℓ^∗_𝑝 is a Banach space with the norm ‖̈⋅‖_𝑝̈ and 𝑝-norm ‖̈⋅‖_𝑝̈̂ 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 field ℂ^∗ 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)if 𝛼=𝛽=𝐼, the identity function, then the non-Newtonian complex calculus is reduced to the classical calculus;(ii)if 𝛼=𝐼 and 𝛽=exp, then the non-Newtonian complex calculus is reduced to the geometric calculus;(iii)if 𝛼=exp and 𝛽=𝐼, then the non-Newtonian complex calculus is reduced to the anageometric calculus;(iv)if 𝛼=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 sets ℓ_∞(𝐹), 𝑐(𝐹), 𝑐₀(𝐹), and ℓ_𝑝(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^∗, 𝑐^∗₀, and ℓ^∗_𝑝. Further, one can obtain the similar results by using another type of calculus instead of non-Newtonian complex calculus.

Acknowledgment

The authors are very grateful to the referees for many helpful suggestions and comments about the paper which improved the presentation and its readability.