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# Spaces of continuous and bounded functions over the field of geometric complex numbers

- Zafer Cakir
^{1}Email author

**2013**:363

https://doi.org/10.1186/1029-242X-2013-363

© Cakir; licensee Springer 2013

**Received:**1 January 2013**Accepted:**19 July 2013**Published:**5 August 2013

## Abstract

Following Grossman and Katz (Non-Newtonian Calculus, 1972), we construct the sets $B(A)$ and $C(A)$ of geometric complex-valued bounded and continuous functions, where *A* denotes the compact subset of the complex plane ℂ. We show that the sets $B(A)$ and $C(A)$ 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 $B(A)$ and $C(A)$ are complete metric spaces.

**MSC:**26A06, 11U10, 08A05.

## Keywords

- non-Newtonian calculus
- algebraic structures with respect to non-Newtonian calculus
- non-Newtonian function space

## 1 Introduction

Grossman and Katz [1], introduced the non-Newtonian calculus consisting of the branches of geometric, anageometric and biogeometric calculus, *etc.* Bashirov *et al.* [2] gave results with applications to the well-known properties of derivative and integral in the multiplicative calculus. Uzer [3] 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 [4] introduced the field $\mathbb{R}(N)$ 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 $\omega (N)$, ${\ell}_{\mathrm{\infty}}(N)$, $c(N)$, ${c}_{0}(N)$ and ${\ell}_{p}(N)$ of all bounded, convergent, null and *p*-absolutely summable sequences in the sense of non-Newtonian calculus over the field $\mathbb{R}(N)$. Quite recently, Tekin and Başar [5] have introduced the spaces ${\omega}^{\ast}$, ${\ell}_{\mathrm{\infty}}^{\ast}$, ${c}^{\ast}$, ${c}_{0}^{\ast}$ and ${\ell}_{p}^{\ast}$ over the non-Newtonian complex field ${\mathbb{C}}^{\ast}$ and obtained the corresponding results for these spaces, where $p\phantom{\rule{0.2em}{0ex}}\ddot{>}\phantom{\rule{0.2em}{0ex}}\ddot{0}$.

Following Bashirov *et al.* [2, 6, 7] and Uzer [3], Türkmen and Başar [8] obtained corresponding results for multiplicative complex numbers and the concept of multiplicative metric.

*the set*$\mathbb{C}(G)$

*of multiplicative complex numbers*by

It is easy to see that $\mathbb{C}(G)=\mathbb{C}\setminus \{0\}$. It is clear from the definition of complex exp function that $\alpha (z)={e}^{z}\ne 0$ for all $z\in {\mathbb{C}}_{\mathrm{str}}$. 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 $(\mathbb{C}(G),\oplus ,\odot )$ denotes the geometric complex field introduced by Türkmen and Başar [8].

## 2 Multiplicative complex field and related properties

## 3 Geometric metric spaces

**Theorem 3.1** $(B(A),{d}_{G})$ *is a complete metric space*.

*Proof* Let $x\in A$ and $f,g,h\in B(A)$. Now, we check the metric axioms. Let ${e}^{{f}_{1}(y)}=f(x),{e}^{{g}_{1}(y)}=g(x),{e}^{{h}_{1}(y)}=h(x)\in B(A)$ for $x,y\in A$, where ${f}_{1}$, ${g}_{1}$ and ${h}_{1}$ are the complex-valued bounded functions.

(GM1) Non-negative property holds: $\dot{0}=1$, $\mathrm{\forall}f,g\in B(A)\u27f9{d}_{G}(f,g)\ge \dot{0}$.

(GM2) ${d}_{G}(f,g)=\dot{0}\iff f(x)\ominus g(x)=\dot{0}$.

Properties (GM1)-(GM4) imply that $(B(A),{d}_{G})$ is a metric space.

for all $m,n>{n}_{0}$. Hence, $\{{f}_{n}(x)\}$ is a Cauchy sequence of geometric complex numbers for each fixed $x\in A$. Since $\mathbb{C}(G)$ is complete by Theorem 4.5 of Türkmen and Başar [8], the sequence $\{{f}_{n}(x)\}$ is convergent, say ${f}_{n}(x)\stackrel{G}{\to}f(x)$ for $x\in A$, as $n\to \mathrm{\infty}$. By letting $m\to \mathrm{\infty}$ with $n>{n}_{0}$, we derive from (3.4) that ${sup}_{x\in A}{|{f}_{n}(x)-f(x)|}_{G}\le \u03f5$. Therefore we have ${|{f}_{n}(x)-f(x)|}_{G}\le \u03f5$ for all $n>{n}_{0}$ and for all $x\in A$. That is to say, for every $\u03f5>1$, there exists at least ${n}_{0}={n}_{0}(\u03f5)\in \mathbb{N}$ such that ${|{f}_{n}(x)-f(x)|}_{G}\le \u03f5$ for all $n>{n}_{0}$ and for all $x\in A$. This means that the sequence $({f}_{n})$ converges uniformly to *f* as $n\to \mathrm{\infty}$.

for all $x\in A$ and for all $n\in \mathbb{N}$, $f\in B(A)$. That is to say, an arbitrary Cauchy sequence in the metric space $(B(A),{d}_{G})$ is convergent. This completes the proof. □

So, we have the following as a direct consequence of Theorem 3.1.

**Corollary 3.2** $(B(A),{\parallel \cdot \parallel}_{G})$ *is a Banach space*, *where* ${\parallel \cdot \parallel}_{G}$ *is defined by* (3.5) .

**Theorem 3.3**$(C(A),{d}_{G}^{\prime})$

*is a complete metric space*,

*where*${d}_{G}^{\prime}$

*is defined on the space*$C(A)$

*by*

Theorem 3.3 leads to the following result.

**Corollary 3.4**$(C(A),{\parallel \cdot \parallel}_{G})$

*is a Banach space*,

*where*${\parallel \cdot \parallel}_{G}$

*is defined on the space*$C(A)$

*by*

## Declarations

### Acknowledgements

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.

## Authors’ Affiliations

## References

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.