1.Introduction

In the area of non-Newtonian calculus pioneering work was carried out by Grossman and Katz (^{9}) which we call as multiplicative calculus. The operations of multiplicative calculus are called as multiplicative derivative and multiplicative integral. We refer to M. Grossman (^{10}), Grossman et al.(^{11}) Jane Grossman (^{12}), Stanley (^{17}), Bashirov et al. (^{1}^{), (}^{2}) for different types of Non-Newtonian calculi and its applications. An extension of multiplicative calculus to functions of complex variables is handled by Bashirov and Riza (^{3}), Uzer (^{20}) Çakmak and Başar (^{5}), Tekin and Başar (^{18}), Türkmen and Başar (^{19}).

Nowadays geometric calculus is an alternative to the usual calculus of Newton and Leibniz. It provides differentiation and integration tools based on multiplication instead of addition. Almost all properties in Newtonian calculus has an analog in multiplicative calculus. Generally speaking multiplicative calculus is a methodology that allows one to have a different look at problems which can be investigated via calculus. In some cases, mainly problems of price elasticity, multiplicative growth etc. the use of multiplicative calculus is advocated instead of a traditional Newtonian one. To know better about Non-Newtonian calculus, we must have idea about different types of arithmetics and their generators.

**2.**α**-generator and geometric real field**

A *generator* is a one-to-one function whose domain is **R** (the set of real numbers) and range is a set A ⊂ **R**. Each generator generates exactly one arithmetic and each arithmetic is generated by exactly one generator. For example, the identity function generates classical arithmetic, and exponential function generates geometric arithmetic. As a generator, we choose the function α such that whose basic algebraic operations are defined as follows:

for *x,y* ∈ *A,* where *A* is a range of the function α

If we choose *exp* as an α*-generator* defined by α_{
(z) = ez
} for *z* ∈**
R
** then α

_{ -1(z) = ln z }and α

*- arithmetics*turns out to geometric arithmetic.

It is obvious that *ln(x) < ln(y) if x < y* for *x,y* ∈ **R**
^{
+
} . That is, *x < y* ⇔ _{
α-1(x) < α-1 (y
} ) So, without loss of generality, we use *x < y* instead of the geometric order *x < y*.

C. Türkmen and F. Başar (^{19}) defined the sets of geometric integers, geometric real numbers and geometric complex numbers **Z**
*(G)*, **R**
*(G)* and **C**
*(G),* respectively, as follows:

**Remark 2.1. (R**(*G*),⊕, ʘ)is a field with geometric zero 1 and geometric identity *e*, since

(1). **(R**(*G*)⊕) is a geometric additive Abelian group with geometric zero 1,

(2). **(R**(*G*) \ 1,ʘ) is a geometric multiplicative Abelian group with geometric identity *e*,

(3). ʘ is distributive over ⊕.

But **(C**(*G*), ⊕,ʘ) is not a field, however, geometric binary operation ʘ is not associative in **C**(*G*). For, we take _{
x = e1/4, y = e4
} and _{
z = e(1+iπ/2) = ie
} . Then

(*x* ʘ *y*) ʘ *z = e* ʘ *z = z = ie*

but *x* ʘ (*y* ʘ *z*) = *x* ʘ _{
e4 = e.
}

Let us define geometric positive real numbers and geometric negative real numbers as follows:

**R**
^{+}(*G*) = {*x* ∈ **R**(*G*) : *x* > 1}

**R**
^{-}(*G*) = {*x* ∈ **R**(*G*) : *x* < 1}.

Then for all *x,y* ∈ **R**(*G*)

2.1 Geometric Limit

According to Grossman and Katz (^{9}), geometric limit of a positive valued function defined in a positive interval is same to the ordinary limit. Here, we define Geometric limit of a function with the help of geometric arithmetic as follows:

A function *f*,which is positive in a given positive interval, is said to tend to the limit *l > 0* as *x* tends to α ∈ **R**, if, corresponding to any arbitrarily chosen number ∈ *> 1*, however small (but greater than 1), there exists a positive number *δ > 1*, such that

Thus, *f(x) Gl* means that for any given positive real number ϵ > 1, no matter however closer to 1,Ǝ a finite number δ > 1 such that *f(x)* ∈
, *lє*
for every *x*∈
*a*δ
It is to be noted that lengths of the open intervals
and
decreases as δ and є respectively decreases to 1. Therefore, as ϵ decreases to 1, *f(x)* becomes closer and closer to *l*, as well as *x* becomes closer and closer to *a* as δ decreases to 1. Hence, *l* is also the ordinary limit of *f(x).*
*i.e. f(x) Gl* ⇒ *f(x)→ l.* In other words, we can say that *G*-limit and ordinary limit are same for bipositive functions whose functional values as well as arguments are positive in the given interval. Only difference is that in geometric calculus we approach the limit geometrically, but in ordinary calculus we approach the limit linearly.

A function *f* is said to tend to limit *l* as *x* tends to *a* from the left, if for each є > 1 (however small), there exists δ > 1 such that |*f(x)*Ѳ *l*|^{
G
} < є when *a/δ < x < a*. In symbols, we then write

Similarly, a function *f* is said to tend to limit *l* as *x* tends to *a* from the right, if for each є > 1 (however small), there exists δ > 1 such that |*f(x)* Ѳ *l*|^{
G
} < *є* when *a < x < aδ*. In symbols, we then write

If *f(x)* is negative valued in a given interval, it will be said to tend to a limit *l* < 0 if for *є > 1, Ǝδ > 1* such that *f(x)* ∈
whenever *x* ∈

2.2. Geometric Continuity

A function *f* is said to be geometric continuous at *x = a* if

(i) *f(a)* i.e., the value of *f(x)* at *x = a*, is a definite number,

(ii) the Geometric-limit of the function *f(x)* as *xGa* exists and is equal to *f(a)*.

Alternatively, a function *f* is said to be Geometric-continuous at *x = a*, if for arbitrarily chosen є > 1, however small, there exists a number δ > 1 such that

|*f(x)* Ѳ *f(a)*|^{
G
} < є.

for all values of *x* for which,|*x* Ѳ *a*|^{
G
} < δ.

On comparing the above definitions of limits and continuity, we can conclude that a function *f* is geometric-continuous at *x = a* if

Let ℓ_{∞,}
*c* and _{
c0
} be the linear spaces of complex bounded, convergent and null sequences, respectively, normed by

It is easy to prove that

is a vector space over **R**(*G*) with respect to the algebraic operations ⊕ addition and ʘ multiplication

are classical sequence spaces over the field **R**(*G*)*.* Also they have shown that ℓ_{∞}(*G*), c(*G*) and c_{0}(*G*) are Banach spaces with the norm

Here, _{
Glim
} is the geometric-limit. For the convenience, we denote _{
ℓ∞(G), c(G), c0(G)
} respectively as

In 1981, Kizmaz (^{13}) introduced the notion of difference sequence spaces using forward difference operator Δ and studied the classical difference sequence spaces ℓ_{∞}(Δ), c(Δ), c_{0}(Δ). Following C. Türkmen and F. Başar (^{19}), we defined geometric sequence space in (^{4}) as follows:

In (^{4}), we introduced some theorems, definitions and basic results as follows:

**Theorem 2.1.** The space
is a normed linear space w.r.t. the norm

**Theorem 2.2.** The space
is a Banach space w.r.t. the norm

**Remark 2.2.** The spaces

are Banach spaces with respect to the norm Also these spaces are BK-spaces.

**Lemma 2.3.** The following conditions (a) and (b) are equivalent:

**Lemma 2.4.**

3. Main Results

Following Kizmaz (^{13}), generalized sequence spaces ℓ_{∞}(Δ^{
m
} ),*c*(Δ^{
m
} ) and *c*
_{0}(Δ^{
m
} ) were introduced by Et and Çolak (^{7}). Based on Et and Çolak (^{7}) Çolak and Et (^{6}). In (^{4}) we introduced the geometric difference sequence. Now we define the following geometric generalized difference sequence spaces

Where *m* ∈ **N** and

Then it can be easily proved that and are normed linear spaces with norm

**Note**: Throughout this paper often we write
instead of
and
instead of

**Definition 3.1 (Geometric Associative Algebra).** An associative algebra is a vector space *A* ⊂ **R**(G), equipped with a bilinear map(called multiplication)

ʘ : *A × A → A*

*(a,b)* → *a* ʘ *b*

which is associative, i.e.

(*a* ʘ *b*) ʘ *c = a* ʘ (*b* ʘ *c*) ∀*a, b, c* ∈ *A*.

An algebra is commutative if *a* ʘ *b = b* ʘ *a* for all *a, b* ∈ *A*. An algebra is unital if there exists a unique *e* ∈ *A* such that *e* ʘ *a = a* ʘ *e = a* for all *a* ∈ *A.* A subalgebra of the algebra *A* is a subspace *B* that is closed under multiplication, i.e. *a* ʘ *b* ∈ *A* for all *a, b* ∈ *B.*

**Definition 3.2 (Geometric Normed Algebra).** A normed algebra is a normed space *A* ⊂ ω(*G*) that is also an associative algebra, such that the norm is submultiplicative: || *a* ʘ *b*||^{
G
} ≤ || a||^{
G
} ʘ ||*b*||^{
G
} for all *a, b* ∈ *A*. A geometric algebra is a complete normed algebra, i.e., a normed algebra which is also a Banach space with respect to its norm.

It is to be noted that the submultiplicativity of the norm means that multiplication in normed algebras is jointly continuous, i.e. if _{
anGa
} and _{
bnGb
} then (_{
an
} ) is bounded and

**Definition 3.3 (Geometric Sequence Algebra).** A geometric sequence space *E(G)* is said to be sequence algebra if *x* ʘ *y* ∈ *E(G)* for _{
x = (xk), y = (yk)
} ∈ *E(G).* i.e. *E(G)* is closed under the geometric multiplication ʘ defined by

for any two sequences_{
x = (xk), y = (yk)
} ∈ *E*.

Since ω(*G*) is closed under geometric multiplication ʘ, hence, ω(*G*) is a sequence algebra. Also sequence algebra ω(*G*) is unital as ||_{
eG
} ||^{
G
} = e, where _{
eG = (e,e,e,…)
} ∈ ω(*G*).

**Definition 3.4 (Continuous Dual Space).** If *x* is a normed space, a linear map *f: X* → **R**(*G*) is called linear functional. *f* is called continuous linear functional or bounded linear functional if ||*f*||^{
G
} < ∞, where

||*f*||^{
G
} = sup{|*f*(*x*)|^{
G
}: ||*x*||^{
G
} ≤ *e* for all *x* ∈ *X*}.

Let _{
X*
} be the collection of all bounded linear functionals on *X.* If *f, g* ∈ _{
X*
} and *a* ∈ **R**(*G*), we define (α ⊙ *f* ⊕ *g*)(*x*) = α ⊙ *f* (*x*) ⊕ *g*(*x*); *X** is called the continuous dual space of *X*.

**Theorem 3.1.** The sequence spaces
and
are Banach spaces with the norm

**Proof.** Let (x_{n}) be a Cauchy sequence in
where x_{
n
} = (x_{i}
^{(n)}) = (x_{1}
^{(n)}, x_{2}
^{(n)}, x_{3}
^{(n)},…) for *n* Є **N** and _{
xk(n)
} is the *k*
^{th} coordinate of x_{n}. Then

Hence we obtain

|_{
xn(n)
} Ѳ _{
xk(l)
} |^{
G
} → 1

As *n,l* → ∞ and for each *k* Є **N**. Therefore (_{
Xk(n)
} )_{
= (xk(1), (xk(2), (xk(3),…)
} is a Cauchy sequence in **R**(*G*). Since **R**(*G*) is complete, (_{
Xk(n)
} ) is convergent.

Suppose
for each *k* Є **N**. Since (_{
xn
} ) is a Cauchy sequence, for each є > 1, there exists *N = N* (*є*) such that
for all *n,l* ≥ *N*. Hence from (3.1)

This implies
that is
as n → ∞, where _{
x = (xk)
} . Now we have to show that
We have

Therefore we obtain Hence is a Banach space.

It can be shown that and are closed subspaces of Therefore these sequence spaces are Banach spaces with the same norm defined for above.

Now we give some inclusion relations between these sequence spaces.

**Lemma 3.2.**

This inclusion is strict. For let

Then
as (m + 1)^{th} geometric difference of
is 1 (geometric zero). But
as *m*
^{th} geometric difference of
is a constant. Hence the inclusion is strict.

The proofs of (ii) and (iii) are similar to that of (i).

Proofs are similar to that of Lemma 3.2.

Furthermore, since the sequence spaces and are Banach spaces with continuous coordinates, that is, implies as n → ∞, these are also BK-spaces.

**Remark 3.1.** It can be easily proved that
is a sequence algebra. But in general,
and
are not sequence algebra. For let
Clearly
But

since *m*
^{th} geometric difference of
is constant.

Let us define the operator

Now let us define

**is a linear homomorphism**: Let
Then

This implies that
is a linear homomorphism. Hence
and
are equivalent as topological spaces (^{16}).
and
are norm preserving and

Let and denote the continuous duals of and respectively.

It can be shown that

is a linear isometry. So is equivalent to

In the same way, it can be shown that
and
are equivalent as topological space to _{
CG
} and _{
C0G
} , respectively. Also

**Lemma 3.4.** The following conditions (a) and (b) are equivalent:

**Proof.**

**4.** α-, β-, γ- **duals**

**Definition 4.1.** (^{8}^{), (}^{14}^{), (}^{15}^{), (}^{16}) If *x* is a sequence space, it is defined that

Then _{
Xα, Xβ
} and _{
Xγ
} are called α-dual (or Köthe-Toeplitz dual), β-dual (or generalized Köthe -Toeplitz dual) and γ- dual spaces of *X*, respectively. Then _{
Xα
} ⊂ _{
Xβ
} ⊂ _{
Xγ.
} If X ⊂ *Y*, then *Y*
^{†} ⊂ *X*
^{†} for † = α, β or γ. It is clear that *X* ⊂ (*X*
^{α})^{α} = *X*
^{αα}. If _{
X = Xαα
} then *x* is called *α-* space. *α-* space is also called a Köthe space or a perfect sequence space.

Then we defined and have proved that (^{4})

**Theorem 4.1.** (^{4})

Therefore *a* ∈ *U*
_{1}. This implies

Conversely, let
and ∉ _{
U2
} . Then we must have

Hence there exists a strictly increasing sequence (_{
ek(i)
} ) of geometric integers (see (^{(19)})), where *k (i)* is a strictly increasing sequence of positive integers such that

Let us define the sequence *x* by

**Proof.** In Theorem (4.7), putting m = 2 we obtain the result.