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Spaces of continuous and bounded functions over the field of geometric complex numbers
Journal of Inequalities and Applications volume 2013, Article number: 363 (2013)
Following Grossman and Katz (Non-Newtonian Calculus, 1972), we construct the sets and of geometric complex-valued bounded and continuous functions, where A denotes the compact subset of the complex plane ℂ. We show that the sets and of complex-valued bounded and continuous functions form a vector space with respect to the addition and scalar multiplication in the sense of multiplicative calculus. Finally, we prove that and are complete metric spaces.
MSC:26A06, 11U10, 08A05.
Grossman and Katz , introduced the non-Newtonian calculus consisting of the branches of geometric, anageometric and biogeometric calculus, etc. Bashirov et al.  gave results with applications to the well-known properties of derivative and integral in the multiplicative calculus. Uzer  extended the multiplicative calculus to the complex-valued functions, was interested in the statements of some fundamental theorems and concepts of multiplicative complex calculus, and demonstrated some analogies between the multiplicative complex calculus and the classical calculus by theoretical and numerical examples. Recently, Çakmak and Başar  introduced the field of non-Newtonian real numbers and gave the triangle and Minkowski’s inequalities in the sense of non-Newtonian calculus. They defined the complete metric spaces , , , and of all bounded, convergent, null and p-absolutely summable sequences in the sense of non-Newtonian calculus over the field . Quite recently, Tekin and Başar  have introduced the spaces , , , and over the non-Newtonian complex field and obtained the corresponding results for these spaces, where .
Following , the main purpose of this paper is the investigation of the space of functions defined by the multiplicative calculus. Following Türkmen and Başar , first we define the set of multiplicative complex numbers by
where denotes the set of multiplicative real numbers and
It is easy to see that . It is clear from the definition of complex exp function that for all . Since α-generator is a bijective function, it maps all complex numbers without zero to the set of values.
We suppose throughout that the A is a compact subset of the complex plane ℂ and denotes the geometric complex field introduced by Türkmen and Başar .
We will consider the sets and in the following forms:
For and , we define the operations addition (⊞) and scalar multiplication (⊡) by
2 Multiplicative complex field and related properties
Theorem 2.1 The set is a vector space with respect to the algebraic operations addition (⊞) and scalar multiplication (⊡).
Proof Let , and . Then, since , there exist positive numbers and such that and for all . Therefore, one can see by the triangle inequality that
This means that .
Since the equality holds for , by using this fact, we observe that
That is, .
(V1) Addition is commutative, that is,
(V2) Addition is associative, i.e.,
(V3) An identity element exists for addition. Indeed, since
the identity element is the function 0 such that for all .
(V4) The inverse element of any exists such that
which yields that
i.e., the inverse element of with respect to ⊞ is .
(V5) Scalar multiplication distributes to the addition over the field. Indeed, since
scalar multiplication distributes to the addition over the field.
(V6) Scalar multiplication distributes to vector addition, i.e.,
(V7) Compatibility of scalar multiplication with field multiplication holds:
(V8) e is the identity element of scalar multiplication. It is easy to see that
which says that the identity element of scalar multiplication is e.
From (V1)-(V8) vector space axioms are satisfied. Hence is a vector space over with the algebraic operations addition (⊞) and scalar multiplication (⊡). □
Theorem 2.2 The set is a subspace of the space with addition (⊞) and scalar multiplication (⊡).
Proof First, we should show that and .
Since for all , , that is, the set is not empty.
Suppose that . Then there is such that for for all . Since A is compact, is a bounded geometric sequence. So, has at least one convergent subsequence , say , as . Since A is closed , hence f is continuous at the point . Therefore, for , there exists at least such that for all . Now, we choose . Thus, we have which leads to . This contradicts the fact . Hence, and the inclusion holds.
Let , and . Then we have
Therefore, the algebraic operations ⊞ and ⊡ are closed on .
Axioms (V1)-(V8) on can be fulfilled in the same way as in the proof of Theorem 2.1. □
3 Geometric metric spaces
For each , we define by
Theorem 3.1 is a complete metric space.
Proof Let and . Now, we check the metric axioms. Let for , where , and are the complex-valued bounded functions.
(GM1) Non-negative property holds: , .
⇒: From the definition of supremum, we can write
and from the definition of geometric absolute value (see ),
Using (3.2) and (3.3), we have
⇐: Conversely, we get
(GM3) Symmetry property holds. From the definition of the relation , we have
(GM4) The triangle inequality holds. Firstly we will get
Therefore, one can easily see that
which leads us to the desired inequality
Properties (GM1)-(GM4) imply that is a metric space.
Suppose that is a Cauchy sequence in the metric space . Then, for every , there is an such that
for all . Hence, is a Cauchy sequence of geometric complex numbers for each fixed . Since is complete by Theorem 4.5 of Türkmen and Başar , the sequence is convergent, say for , as . By letting with , we derive from (3.4) that . Therefore we have for all and for all . That is to say, for every , there exists at least such that for all and for all . This means that the sequence converges uniformly to f as .
Additionally, since there exists a such that
for all and for all , . That is to say, an arbitrary Cauchy sequence in the metric space is convergent. This completes the proof. □
It is obvious that is an induced metric from the norm , that is,
So, we have the following as a direct consequence of Theorem 3.1.
Corollary 3.2 is a Banach space, where is defined by (3.5) .
Theorem 3.3 is a complete metric space, where is defined on the space by
Theorem 3.3 leads to the following result.
Corollary 3.4 is a Banach space, where is defined on the space by
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The author would like to thank Professor Feyzi Başar, Department of Mathematics, Faculty of Arts and Sciences, Fatih University, The Hadımköy Campus, Büyükçekmece, 34500-İstanbul, Turkey, for his careful reading and constructive criticism of an earlier version of this paper, which improved the presentation and its readability. The author is very grateful to the anonymous referee for many helpful suggestions and comments on the paper. Finally, the author should express his gratitude to Ph.D. students Sebiha Tekin and Ahmet Faruk Çakmak for their valuable help for revising the manuscript in the light of reviewer’s suggestions. The main results of this paper were presented in part at the conference First International Conference on Analysis and Applied Mathematics (ICAAM 2012) held October 18-21, 2012 in Gümüşhane, Turkey at the University of Gümüşhane.
The author declares that he has no competing interests.