Arithmetic Convergent Sequence Space Defined By Modulus Function

abstract: The aim of this article is to introduce the sequence spaces AC(f) and AS(f) using arithmetic convergence and modulus function, and study algebraic and topological properties of this space, and certain inclusion results.


Introduction
Throughout, N, R and C will denote the set of natural, real and complex numbers, respectively and x = (x k ) denotes a sequence whose k th term is x k .Similarly w, c, ℓ ∞ , ℓ 1 denotes the space of all, convergent, bounded, absolutely summable sequences of complex terms, respectively.
For a sequence x = (x k ) defined on N and n ∈ N, the notation k|n x k means the finite sum of all the numbers x k as k ranges over the integers that divide n including 1 and n.In general for integers k and n we write k|n to mean 'k divides n' or 'n is a multiple of k'.We use the symbol < m, n > to denote the greatest common divisor of two integers m and n.
In [17], Ruckle was introduced the notions arithmetic summability and arithmetic convergence as follows: (i) A sequence x = (x k ) defined on N is called arithmetically summable if for each ε > 0 there is an integer n such that for every integer m we have x k < ε.
Taja Yaying and Bipan Hazarika (ii) A sequence y = (y k ) is called arithmetically convergent if for each ε > 0 there is an integer n such that for every integer m we have |y m − y <m,n> | < ε.
The notion of a modulus function was introduced by Nakano [12] in the year 1953 (also see [11,13]).We recall that a modulus f is a function so that in view of (iv), f is continuous everywhere on [0, ∞).A modulus function may be unbounded (for example, f (x) = x p , 0 < p ≤ 1) or bounded (for example, f (x) = x 1+x ).It is easy to see that f 1 + f 2 is a modulus function when f 1 and f 2 are modulus functions, and that the function f i (i is a positive integer), the composition of a modulus function f with itself i times, is also a modulus function.Ruckle [14] used the idea of a modulus function f to construct a class of FK spaces The space X(f ) is closely related to the space l 1 which is an X(f ) space with f (x) = x for all real x ≥ 0. Thus Ruckle [14] proved that, for any modulus f, X(f ) ⊂ l 1 .
The space X(f ) is a Banach space with respect to the norm Spaces of the type X(f ) are a special case of the spaces structured by Gramsch in [6].From the point of view of local convexity, spaces of the type X(f ) are quite pathological.Therefore symmetric sequence spaces, which are locally convex have been frequently studied by Garling [4,5], Köthe [10] and Ruckle [15,16].After then Kolk [8,9] gave an extension of X(f ) by considering a sequence of modulii F = (f k ) and defined the sequence space Also we refer to readers [1,2,3] for different types of sequence spaces defined by modulus function.

Definitions and Preliminaries
Now we give the following definition for establishing certain results of this article.

Arithmetic convergent sequence space AC(f )
In this section we study certain algebraic and topological properties of arithmetic convergent sequence space AC(f ) defined by modulus function f.We define Thus AC(f ) is a linear space.✷ Theorem 3.2.The sequence space AC(f ) is a sequence algebra.
Proof: Let (x m ), (y m ) be sequences AC(f ).Then by definition, for ε > 0 and an integer n, we have Then we have f (|x m y m − x <m,n> y <m,n> |) < ε for all m.(see [18]) Hence the sequence (x m y m ) ∈ AC(f ).This completes the proof.✷ Theorem 3.3.Let f 1 and f 2 be two modulus functions, then AC(f 1 ) ⊂ AC(f 1 f 2 ).
We choose δ with 0 < δ < 1 such that f 1 (x) < ε for 0 ≤ x ≤ δ.Let us denote For y m > δ we use the fact that where [x] denotes the integral part of x.
Since f 1 is a modulus function, by definition of modulus function and Lemma 2.2, we have for Thus lim The sequence space of all arithmetic convergent sequences, AC is subset of AC(f ) .i.e.AC ⊂ AC(f ).
Theorem 3.5.Let f 1 and f 2 be two modulus functions.Then we have From the above equations we can easily see that for ε > 0 and integer n Thus (x m ) ∈ AC(f 1 + f 2 ).Hence the result.✷ Arithmetic Convergent Sequence Space Defined By Modulus Function 133 4. Arithmetic summable sequence space AS(f ) In this section we study certain algebraic and topological properties of arithmetic summable sequence space AS(f ) defined by modulus function f .We define Thus AS(f ) is a linear space.✷ Theorem 4.2.Let f 1 and f 2 be two modulus functions, then AS(f 1 ) ⊂ AS(f 1 f 2 ).
Proof: The proof of the theorem is analogous to the proof of the Theorem 3.3 ✷ Corollary 4.3.The sequence space of all arithmetic summable sequences, AS is subset of AS(f ) i.e.AS ⊂ AS(f ).
Theorem 4.4.Let f 1 and f 2 be two modulus functions.Then AS(f From the above equations we can easily see that for ε > 0 and integer n x k   < ε for all m. Thus (x m ) ∈ AS(f 1 + f 2 ).Hence the result.✷ The next result gives an important inclusion property between l 1 and AS(f ).
Proof: The result can be easily obtained from the inclusion l 1 ⊂ AS given by Proposition 16 in [17] and by using the Corollary 4.3.✷ The following theorem established a relation between AS(f ) and AC(f ).✷

Theorem 4 . 1 .
The sequence space AS(f ) is a linear space.Proof: Let (x m ), (y m ) ∈ AS(f ).By definition of AS(f ), for ε > 0 and an integer nfor all m.Since f is a modulus function, for ε > 0 and an integer n and scalars α and β, f  k|m (αx k + βy k ) − k|<m,n>

Theorem 4 . 6 .⇒
If the sequence (x m ) ∈ AS(f ) then the sequence (y m ) defined by y m = k|m x k is in AC(f ).Proof: Let (x m ) ∈ AS(f ).Then by definition we have f f (|y m − y <m,n> |) < ε ⇒ y m ∈ AC(f ).