A Study of Bounded Variation Sequence Spaces

In the theory of classes of sequence, a wonderful application of Hahn-Banach extension theorem gave rise to the concept of Banach limit, i.e., the limit functional defined on c can be extended to the whole space l ∞ and this extended functional is called as the Banach limit. After that, in 1948 Lorentz used this concept of a week limit to introduce a new type of convergence, named as the almost convergence. Later on, Raimi generalized the concept of almost convergence known as σ (cid:1) convergence and the sequence space BV σ was introduced and studied by Mursaleen. The main aim of this chapter is to study some new double sequence spaces of invariant means defined by ideal, modulus function and Orlicz function. Further-more, we also study several properties relevant to topological structures and inclusion relations between these spaces.


Introduction
The concept of convergence of a sequence of real numbers has been extended to statistical convergence independently by Fast [1] and Schoenberg [2]. There has been an effort to introduce several generalizations and variants of statistical convergence in different spaces. One such very important generalization of this notion was introduced by Kostyrko et al. [3] by using an ideal I of subsets of the set of natural numbers, which they called I-convergence. After that the idea of I-convergence for double sequence was introduced by Das et al. [4] in 2008.
Throughout a double sequence is defined by x ¼ x ij À Á and we denote 2 ω showing the space of all real or complex double sequences.
Let X be a nonempty set then a family I ⊂ 2 X is said to be an ideal in X if Ø ∈ I, I is additive, i.e., for all A, B ∈ I ) A ∪ B ∈ I and I is hereditary, i.e., for all A ∈ I, B ⊆ A ) B ∈ I. A nonempty family of sets F ⊂ 2 X is said to be a filter on X if for all A, B ∈ F implies A ∩ B ∈ F and for all A ∈ F with A ⊆ B implies B ∈ F . An ideal I ⊂ 2 X is said to be nontrivial if I 6 ¼ 2 X , this non trivial ideal is said to be admissible if I ⊇ x f g : x ∈ X f gand is said to be maximal if there cannot exist any nontrivial ideal J 6 ¼ I containing I as a subset. For each ideal I there is a filter F I ð Þ called as filter associate with ideal I, that is A double sequence x ¼ x ij À Á ∈ 2 ω is said to be I-convergent [5][6][7][8] to a number L if for every ϵ>0, we have i; j ð Þ∈ N Â N : jx ij À Lj ≥ ϵ È É ∈ I: In this case, we write I À lim x ij ¼ L: A double sequence x ¼ x ij À Á ∈ 2 ω is said to be I-Cauchy if for every ϵ>0 there exists numbers m ¼ m ϵ ð Þ, n ¼ n ϵ ð Þ such that i; j ð Þ∈ N Â N : jx ij À x mn j ≥ ϵ È É ∈ I: A continuous linear functional ϕ on l ∞ is said to be an invariant mean [9,10] or σ-mean if and only if: where σ be an injective mapping of the set of the positive integers into itself having no finite orbits. If where m ≥ 0, k > 0: where σ m k ð Þ denote the mth-iterate of σ k ð Þ at k. In this case σ is the translation mapping, that is, σ k ð Þ ¼ k þ 1, σÀ mean is called a Banach limit [11] and V σ , the set of bounded sequences of all whose invariant means are equal, is the set of almost convergent sequences. The special case of (3) in which σ k ð Þ ¼ k þ 1 was given by Lorentz [12] and the general result can be proved in a similar way. It is familiar that a Banach limit extends the limit functional on c in the sense that Definition 1.1 A sequence x ∈ l ∞ is of σ-bounded variation if and only if: (i) ∑|ϕ m, k x ð Þ| converges uniformly in k, (ii) lim m!∞ t m, k x ð Þ, which must exist, should take the same value for all k.
We denote by BV σ , the space of all sequences of σ-bounded variation: is a Banach space normed by A function M : 0; ∞ ½ Þ! 0; ∞ ½ Þis said to be an Orlicz function [13,14] if it satisfies the following conditions: (i) M is continuous, convex and non-decreasing, [iii] sequence algebra if x ij y ij ∈ E whenever x ij À Á , y ij ∈ X: [iv] convergence free if y ij ∈ X whenever x ij À Á ∈ X and x ij ¼ 0 implies y ij ¼ 0, for all i; j ð Þ∈ N Â N.
: i; j ð Þ : n 1 < n 2 < n 3 < :… and k 1 < k 2 È < k 3 < :…g ⊆ N Â N and X be a double sequence space. A K-step space of X is a sequence space A canonical preimage of a sequence x n i k j ∈ X is a sequence b nk ð Þ∈ X defined as follows: b nk ¼ a nk , for n, k ∈ K 0, otherwise: & A sequence space X is said to be monotone if it contains the canonical preimages of all its step spaces.
The following subspaces l p ð Þ, l ∞ p ð Þ, c p ð Þ and c 0 p ð Þ where p ¼ p k À Á is a sequence of positive real numbers. These subspaces were first introduced and discussed by Maddox [16]. The following inequalities will be used throughout the section. Let p ¼ p ij be a double sequence of positive real numbers [19]. For any complex λ with 0 < p ij ≤ sup ij p ij ¼ G < ∞, we have λ j j p ij ≤ max 1; λ j j G : , then for the sequences a ij À Á and b ij À Á in the complex plane, we have

Bounded variation sequence spaces defined by Orlicz function
In this section, we define and study the concepts of I-convergence for double sequences defined by Orlicz function and present some basic results on the above definitions [8,20].
3 A Study of Bounded Variation Sequence Spaces DOI: http://dx.doi.org/10.5772/intechopen.81907 Now, we read some theorems based on these sequence spaces. These theorems are of general importance as indispensable tools in various theoretical and practical problems.
Theorem 2.1 Let M 1 , M 2 be two Orlicz functions with Δ 2 condition, then be an arbitrary element, so there exists ρ > 0 such that Let ϵ > 0 and choose δ with 0 < δ < 1 such that Now, since M 1 is an Orlicz function so we have For y ij >δ, we have y ij < y ij δ < 1 þ y ij δ . Now, since M 1 is non-decreasing and convex, it follows that, Since M 1 satisfies the Δ 2 -condition, so we have This implies that, Hence, we have Therefore from (12) and (16), we have The other cases can be proved in similar way.
Let ϵ>0 be given. Then there exist ρ>0, such that and Therefore from Eqs. (17) and (18), we get Then for any given ϵ>0, we have Hence, we have Using the definition of convergence free sequence space, let us give another theorem which will be of particular importance in our future work: consider the double sequence x ij À Á , y ij which defined as follows: To gain a good understanding of these double sequence spaces and related concepts, let us finally look at this theorem on inclusions: Theorem 2.3 Let M be an Orlicz function. Then Proof. For this let us consider Now taking the limit on both sides we get For this let us consider Now taking the supremum on both sides, we get sup ij M |ϕ mnij x ð Þ| ρ < ∞:

Paranorm bounded variation sequence spaces
In this section we study double sequence spaces by using the double sequences of strictly positive real numbers p ¼ p ij with the help of BV σ space and an Orlicz function M. We study some of its properties and prove some inclusion relations related to these new spaces. For m, n ≥ 0, we have We also denote We can now state and proof the theorems based on these double sequence spaces which are as follows: Therefore g x þ y ð Þ≤ g x ð Þ þ g y ð Þ: in the sense that Then, since the inequality holds by subadditivity of g, the sequence g x ij À Á is bounded. Therefore,

Applied Mathematics
Let P 0 ¼ x ij À Á : x ij ¼ 0 or 1; for i; j ∈ M and x ij ¼ 0; otherwise È É : Since A is infinite, so P 0 is uncountable. Consider the class of open balls Let C 1 be an open cover of 2 M I BV σ M; p ð Þcontaining B 1 . Since B 1 is uncountable, so C 1 cannot be reduced to a countable subcover for 2 M I BV σ M; p ð Þ. Thus 2 M I BV σ M; p ð Þis not separable. We shall now introduce a theorem which improves our work.

Theorem 3.3 Let p ij
and q ij be two double sequences of positive real numbers.
>0 and x ij À Á ∈ 2 0 M I BV σ M; q ð Þ : Then, there exists β > 0 such that p ij > β q ij for sufficiently large i; j ð Þ∈ K: : For a given ϵ > 0, there exist ρ > 0 such that Therefore, x ij À Á ∈ 2 0 M I BV σ M; p ð Þ . The converse part of the result follows obviously.  For a given ϵ>0, we have , then B c ∈ I. We choose k 0 ∈ B c , then for each p, q, r, s ≥ k 0 , we have Then a pq À Á is a Cauchy double sequence in C. So, there exists a scalar a ∈ C such that a pq À Á ! a, as p, q ! ∞: (2) For the next step, let 0 < δ < 1 be given. Then, we show that if then U c ∈ I: Since x pq ð Þ ! x, then there exists p 0 , q 0 ∈ N such that, where P c ∈ I. The number p 0 ; q 0 À Á can be so chosen that together with (25), we have We have Then we have a subset S ⊆ N Â N such that S c ∈ I, where Hence the result 2 M I BV σ M; p ð Þ⊂ 2 l ∞ M; p ð Þfollows.
Since the inclusions 2 are strict so in view of Theorem (3.3), we have the following result. The above theorem is interesting and itself will have various applications in our future work.

Bounded variation sequence spaces defined by modulus function
In this section, we study some new double sequence spaces of invariant means defined by ideal and modulus function. Furthermore, we also study several properties relevant to topological structures and inclusion relations between these spaces. The following classes of double sequence spaces are as follows: We also denote We shall now consider important theorems of these double sequence spaces by using modulus function. Proof. Suppose x ¼ x ij À Á and y ¼ y ij ∈ 2 BV I σ f ð Þ be any two arbitrary elements.
Let α, β are scalars. Now, since x ij À Á , y ij ∈ 2 BV I σ f ð Þ. Then this implies that there exists some positive numbers L 1 , L 2 ∈ C and such that the sets Now, assume be such that B c 1 , B c 2 ∈ I: Since f is a modulus function, we have Thus α x ij À Á þ β y ij ∈ 2 BV I σ f ð Þ: As x ij À Á and y ij are two arbitrary element then α x ij À Á þ β y ij ∈ 2 BV I σ f ð Þ for all x ij À Á , y ij ∈ 2 BV I σ f ð Þ and for all scalars α, β. Hence 2 BV I σ f ð Þ is linear space. The proof for other spaces will follow similarly. ▪ We may go a step further and define another theorem on ideal convergence which basically depends upon the set in the filter associated with the same ideal.
which holds for all i; j ð Þ∈ B ϵ : Hence Conversely, suppose that Then, being f a modulus function and by using basic triangular inequality, we have If we fix ϵ>0 then, we have C ϵ ∈ F I ð Þ as well as C ϵ 2 ∈ F I ð Þ. Hence C ϵ ∩ C ϵ 2 ∈ F I ð Þ. This implies that This shows that where the diam J denotes the length of interval J. In this way, by induction we get the sequence of closed intervals with the property that diam I k ≤ Then there exists a ξ ∈ ∩ I k where k ∈ N such that showing that x ¼ x ij À Á ∈ 2 M I BV σ f ð Þ is I-convergent. Hence the result holds. As the reader knows about solid and monotone sequence space now turn to theorem on solid and monotone double sequence spaces of invariant mean defined by ideal and modulus function. Proof. We consider 2 0 BV I σ f ð Þ À Á and for 2 0 M I BV σ f ð Þ the proof shall be similar.
Let x ¼ x ij À Á ∈ 2 0 BV I σ f ð Þ À Á be an arbitrary element, then the set Let α ij À Á be a sequence of scalars with |α ij | ≤ 1 for all i, j ∈ N: Now, since f is a modulus function. Then the result follows from (2.18)