- Open Access
Spaces of continuous and bounded functions over the field of geometric complex numbers
© Cakir; licensee Springer 2013
- Received: 1 January 2013
- Accepted: 19 July 2013
- Published: 5 August 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.
- non-Newtonian calculus
- algebraic structures with respect to non-Newtonian calculus
- non-Newtonian function space
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 .
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 .
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: , .
Properties (GM1)-(GM4) imply that is a metric space.
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 .
for all and for all , . That is to say, an arbitrary Cauchy sequence in the metric space is convergent. This completes the proof. □
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 leads to the following result.
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.
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