R E S E A R C H
Open Access
Spaces of continuous and bounded functions
over the field of geometric complex numbers
Zafer Cakir
**Correspondence:
[email protected] Department of Mathematical Engineering, Gümü¸shane University, Gümü¸shane, 29100, Turkey
Abstract
Following Grossman and Katz (Non-Newtonian Calculus, 1972), we construct the sets B(A) andC(A) of geometric complex-valued bounded and continuous functions,
whereAdenotes the compact subset of the complex planeC. We show that the sets
B(A) andC(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 thatB(A) andC(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 [], introduced the non-Newtonian calculus consisting of the branches of geometric, anageometric and biogeometric calculus,etc. Bashirovet al.[] gave re-sults 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 nu-merical examples. Recently, Çakmak and Başar [] introduced the fieldR(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 ω(N), ∞(N),
c(N), c(N) andp(N) of all bounded, convergent, null and p-absolutely summable
se-quences in the sense of non-Newtonian calculus over the field R(N). Quite recently, Tekin and Başar [] have introduced the spaces ω∗, ∗∞, c∗, c*
and ∗p over the
non-Newtonian complex fieldC∗ and obtained the corresponding results for these spaces, wherep>¨.¨
Following Bashirovet al.[, , ] and Uzer [], Türkmen and Başar [] obtained cor-responding results for multiplicative complex numbers and the concept of multiplicative metric.
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 setC(G)of multiplicative complex numbersby
C(G) :=w=u⊕igv:u,v∈R(G) andig=
√
eG
=ez=ex·eylnei:ex,ey∈R(G) andei=e√–
=ex+iy:x∈R; –π<y≤πandi=√– =ez:z∈Cstr
,
whereR(G) denotes the set of multiplicative real numbers and
Cstr:={z=x+iy:x∈R; –π<y≤π}.
It is easy to see thatC(G) =C\ {}. It is clear from the definition of complexexp func-tion thatα(z) =ez= for allz∈Cstr. Sinceα-generator is a bijective function, it maps all
complex numbers without zero to the set of values.
We suppose throughout that theAis a compact subset of the complex planeCand (C(G),⊕,) denotes the geometric complex field introduced by Türkmen and Başar [].
We will consider the setsB(A) andC(A) in the following forms:
B(A) :=f:A→C(G)| ∃K∈R+ ∀x∈A,f(x)G≤K,
C(A) :=f :A→C(G)|f is continuous onA.
Forf,g∈B(A) andλ∈C(G), we define the operations addition () and scalar multiplica-tion () by
: B(A)×B(A) −→ C(G)
(f,g) −→ (f g)(x) =f(x)⊕g(x),
: C(G)×B(A) −→ C(G)
(λ,f) −→ (λf)(x) =λf(x).
2 Multiplicative complex field and related properties
Theorem . The set B(A)is a vector space with respect to the algebraic operations addi-tion()and scalar multiplication().
Proof Letx∈A,f,g ∈B(A) andλ∈C(G). Then, since f,g∈B(A), there exist positive numbersK andK such that|f(x)|G≤K and|g(x)|G≤Kfor allx∈A. Therefore, one
can see by the triangle inequality that
(fg)(x)G=f(x)⊕g(x)G≤f(x)G⊕g(x)G
≤KK=K; K∈R+.
This means thatf g∈B(A).
Since the equality|αα|G=|α|G |α|Gholds forα,α∈C(G), by using this fact,
we observe that
(λf)(x)G=λf(x)G=|λ|Gf(x)G
=f(x)Gln|λ|G
(V) Addition is commutative, that is,
(f g)(x) =f(x)⊕g(x) =elnf(x)+lng(x)
=elng(x)+lnf(x)
=g(x)⊕f(x) = (gf)(x).
(V) Addition is associative,i.e.,
(f g)h(x) =f(x)⊕g(x)⊕h(x) =e[lnf(x)+lng(x)]+lnh(x)
=elnf(x)+[lng(x)+lnh(x)]
=f(x)⊕g(x)⊕h(x) =f(gh)(x).
(V) An identity element exists for addition. Indeed, since
[f ](x) =f(x)
⇒ f(x)⊕ ˙ =f(x)
⇒ elnf(x)eln˙=f(x) ⇒ f(x)· ˙ =f(x)
⇒ =˙ e= ∈B(A),
the identity element is the functionsuch that(x) = for allx∈A. (V) The inverse element of anyf ∈B(A)exists such that
(f g)(x) =f(x)⊕g(x) =elnf(x)elng(x)=f(x)·g(x) = ,
which yields that
g(x) =
f(x) for allf∈B(A),
i.e., the inverse element off ∈B(A)with respect toisg= /f.
(V) Scalar multiplication distributes to the addition over the field. Indeed, since
(λ⊕μ)f(x) = (λ⊕μ)f(x) =e(lnλ+lnμ)lnf(x)=elnλlnf(x)+lnμlnf(x)
=λf(x)⊕μf(x) = (λf)(x)(μf)(x),
(V) Scalar multiplication distributes to vector addition,i.e.,
λ(f g)(x) =λf(x)⊕g(x) =elnλ·[lnf(x)+lng(x)]
=elnλ·lnf(x)+lnλ·lng(x)
=λf(x)⊕λg(x) = (λf)(x)(λg)(x).
(V) Compatibility of scalar multiplication with field multiplication holds:
(λμ)f(x) = (λμ)f(x) =elnλ·lnμ·lnf(x)
=elnλ·(lnμ·lnf(x))
=λμf(x) =λ(μf)(x).
(V) eis the identity element of scalar multiplication. It is easy to see that
(ef)(x) =ef(x) =elnelnf(x)=f(x),
which says that the identity element of scalar multiplication ise.
From (V)-(V) vector space axioms are satisfied. HenceB(A) is a vector space overC(G) with the algebraic operations addition () and scalar multiplication ().
Theorem . The set C(A)is a subspace of the space B(A)with addition()and scalar multiplication().
Proof First, we should show thatC(A)=∅andC(A)⊂B(A).
Since (x) = for allx∈A, ∈C(A), that is, the setC(A) is not empty.
Suppose thatC(A)⊂B(A). Then there is f ∈C(A) such that|f(xn)|G≥nforxn∈A
for alln∈N. SinceAis compact, (xn) is a bounded geometric sequence. So, (xn) has at
least one convergent subsequence (xnk), sayxnk
G
→x, ask→ ∞. SinceAis closedx∈A,
hencef is continuous at the pointx. Therefore, for> , there exists at leastδ> such
that|f(x)f(x)|G<for all|xx|G<δ. Now, we choose=e. Thus, we have||f(x)|–
|f(x)||G≤ |f(x)f(x)|G<which leads to|f(x)|G<e+|f(x)|G. This contradicts the fact
|f(xn)|G≥n. Hence,f ∈B(A) and the inclusionC(A)⊂B(A) holds.
Letx∈A,f,g∈C(A) andλ,μ∈C(G). Then we have
(λf)(μg)(x) =λf(x)⊕μg(x)=elnλ·lnf(x)+lnμ·lng(x)
=elnf(x)lnλelng(x)lnμ
=f(x)lnλg(x)lnμ∈C(A).
Axioms (V)-(V) on C(A) can be fulfilled in the same way as in the proof of
Theo-rem ..
3 Geometric metric spaces
For eachf,g∈B(A), we definedGby
dG: B(A)×B(A) −→ R+(G)
(f,g) −→ dG(f,g) =sup x∈A
f(x)g(x)G. (.)
Theorem . (B(A),dG)is a complete metric space.
Proof Letx∈Aandf,g,h∈B(A). Now, we check the metric axioms. Letef(y)=f(x),eg(y)=
g(x),eh(y)=h(x)∈B(A) forx,y∈A, wheref
,gandhare the complex-valued bounded
functions.
(GM) Non-negative property holds: = ˙ ,∀f,g∈B(A)⇒dG(f,g)≥ ˙. (GM) dG(f,g) =˙ ⇐⇒ f(x)g(x) =˙.
⇒: From the definition of supremum, we can write
dG(f,g) =
⇒ sup
x∈A
f(x)g(x)G=
⇒ f(x)g(x)G≤, (.)
and from the definition of geometric absolute value (see []),
f(x)g(x)G≥ =e. (.)
Using (.) and (.), we have
f(x)g(x)G=
⇒ f(x)
g(x)
G
=
⇒ ef(y)
eg(y)
G
=
⇒ ef(y)–g(y)
G=
⇐: Conversely, we get
f(x) =g(x)
⇒ f(x)
g(x)
=
⇒ f(x)g(x)G=
⇒ sup
x∈A
f(x)g(x)
G=
⇒ dG(f,g) = .
(GM) Symmetry property holds. From the definition of the relationdG, we have
dG(f,g) =sup x∈A
f(x)g(x)G
=sup x∈A
fg((xx))
G
=sup y∈A
ef(y)
eg(y)
G
=sup y∈A
ef(y)–g(y)
G=sup y∈Ae
|f(y)–g(y)|
=sup y∈Ae
|g(y)–f(y)|=sup
y∈A
eg(y)–f(y)
G
=sup y∈A
eg(y)
ef(y) G =sup x∈A gf((xx))
G
=sup x∈A
g(x)f(x)G
=dG(g,f).
(GM) The triangle inequality holds. Firstly we will get
f(x)g(x)G =f(x)
g(x)
G
=e
f(y)
eg(y)
G
=ef(y)–g(y)
G
=e|f(y)–g(y)|=e|f(y)–h(y)+h(y)–g(y)|
≤e|f(y)–h(y)|+|h(y)–g(y)|=e|f(y)–h(y)|·e|h(y)–g(y)|
=ef(y)–h(y)
G·e
h(y)–g(y)
G=
ef(y)
eh(y)
G
·eh(y)
eg(y)
G
=f(x)
h(x)
G
·h(x)
g(x)
G
=f(x)h(x)G⊕h(x)g(x)G.
Therefore, one can easily see that
sup x∈A
f(x)g(x)G≤sup x∈A
f(x)h(x)
G⊕h(x)g(x)G
≤sup
x∈A
f(x)h(x)G⊕sup x∈A
which leads us to the desired inequality
dG(f,g)≤dG(f,h)⊕dG(h,g).
Properties (GM)-(GM) imply that (B(A),dG) is a metric space.
Suppose that (fn) is a Cauchy sequence in the metric space (B(A),dG). Then, for every ε> , there is ann=n(ε) such that
dG(fn,fm) =sup x∈A
fn(x)fm(x)G<ε (.)
for allm,n>n. Hence,{fn(x)}is a Cauchy sequence of geometric complex numbers for
each fixedx∈A. SinceC(G) is complete by Theorem . of Türkmen and Başar [], the sequence{fn(x)}is convergent, sayfn(x)
G
→f(x) forx∈A, asn→ ∞. By lettingm→ ∞
withn>n, we derive from (.) thatsupx∈A|fn(x) –f(x)|G≤. Therefore we have|fn(x) –
f(x)|G≤for alln>nand for allx∈A. That is to say, for every> , there exists at least n=n()∈Nsuch that|fn(x) –f(x)|G≤for alln>nand for allx∈A. This means that
the sequence (fn) converges uniformly tof asn→ ∞.
Additionally, since there exists aK> such that
f(x)G=f(x) –fn(x) +fn(x)G≤f(x) –fn(x)G⊕fn(x)G≤·K
for allx∈Aand for alln∈N,f ∈B(A). That is to say, an arbitrary Cauchy sequence in the metric space (B(A),dG) is convergent. This completes the proof.
It is obvious thatdGis an induced metric from the norm · G, that is,
fG=dG(f, ) =sup x∈A
f(x)(x)G; f∈B(A). (.)
So, we have the following as a direct consequence of Theorem ..
Corollary . (B(A), · G)is a Banach space,where · Gis defined by(.) .
Theorem . (C(A),dG)is a complete metric space,where dGis defined on the space C(A)
by
dG: C(A)×C(A) −→ R+(G)
(f,g) −→ dG(f,g) =max
x∈Af(x)g(x)G.
Theorem . leads to the following result.
Corollary . (C(A), · G)is a Banach space,where · G is defined on the space C(A)
by
Competing interests
The author declares that he has no competing interests.
Acknowledgements
The author would like to thank Professor Feyzi Ba¸sar, Department of Mathematics, Faculty of Arts and Sciences, Fatih University, The Hadımköy Campus, Büyükçekmece, 34500-˙Istanbul, 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 conferenceFirst International Conference on Analysis and Applied Mathematics (ICAAM 2012)held October 18-21, 2012 in Gümü¸shane, Turkey at the University of Gümü¸shane.
Received: 1 January 2013 Accepted: 19 July 2013 Published: 5 August 2013 References
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doi:10.1186/1029-242X-2013-363
