Open access peer-reviewed chapter

A Study of Bounded Variation Sequence Spaces

By Vakeel Ahmad Khan, Hira Fatima and Mobeen Ahmad

Submitted: September 19th 2018Reviewed: October 9th 2018Published: April 23rd 2019

DOI: 10.5772/intechopen.81907

Downloaded: 624


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 − 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. Furthermore, we also study several properties relevant to topological structures and inclusion relations between these spaces.


  • invariant mean
  • bounded variation
  • ideal
  • filter
  • I-convergence
  • Orlicz function
  • modulus function
  • paranorm

1. 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=xijand we denote 2ωshowing the space of all real or complex double sequences.

Let Xbe a nonempty set then a family I2Xis said to be an idealin Xif ØI, Iis additive, i.e., for all A,BIABIand Iis hereditary, i.e., for all AI,BABI. A nonempty family of sets F2Xis said to be a filteron Xif for all A,BFimplies ABFand for all AFwith ABimplies BF. An ideal I2Xis said to be nontrivialif I2X, this non trivial ideal is said to be admissible if Ix:xXand is said to be maximalif there cannot exist any nontrivial ideal JIcontaining Ias a subset. For each ideal Ithere is a filter FIcalled as filter associate with ideal I, that is


A double sequence x=xij2ωis said to be I-convergent[5, 6, 7, 8] to a number Lif for every ϵ>0, we have ijN×N:xijLϵI.In this case, we write Ilimxij=L.A double sequence x=xij2ωis said to be I-Cauchyif for every ϵ>0there exists numbers m=mϵ,n=nϵsuch that ijN×N:xijxmnϵI.

A continuous linear functional ϕon lis said to be an invariant mean[9, 10] or σ-mean if and only if:

1. ϕx0where the sequence x=xkhas xk0for all k,

2. ϕe=1where e=1,1,1,1,

3. ϕxσn=ϕxfor all xl,

where σbe an injective mapping of the set of the positive integers into itself having no finite orbits.

If x=xk, write Tx=Txk=xσk, so we have


where m0,k>0.


where σmkdenote the mth-iterate of σkat k. In this case σis the translation mapping, that is, σk=k+1,σmean is called a Banach limit [11] and Vσ, 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 σk=k+1was 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 cin the sense that


Definition 1.1A sequence xlis of σ-bounded variationif and only if:

(i) ϕm,kxconverges uniformly in k,

(ii) limmtm,kx,which must exist, should take the same value for all k.

We denote by BVσ, the space of all sequences of σ-bounded variation:


is a Banach space normed by


A function M:00is said to be an Orlicz function[13, 14] if it satisfies the following conditions:

(i) M is continuous, convex and non-decreasing,

(ii) M0=0,Mx>0and Mxas x.

Remark 1.1If the convexity of an Orlicz function is replaced by Mx+yMx+My, then this function is called Modulus function[15, 16, 17]. If Mis an Orlicz function, then MλXλMxfor all λwith 0<λ<1.An Orlicz function Mis said to satisfy Δ2-condition for all values of uif there exists a constant K>0such that MLuKLMufor all values of L>1[18].

Definition 1.2A double sequence space Xis said to be:

[i] solidor normalif xijXimplies that αijxijXfor all sequence of scalars αijwith αij<1for all ijN×N.

[ii] symmetricif xπijXwhenever xijX, where πijis a permutation on N×N.

[iii] sequence algebraif xijyijEwhenever xij,yijX.

[iv] convergence freeif yijXwhenever xijXand xij=0implies yij=0, for all ijN×N.

Definition 1.3Let K=nikj:ij:n1<n2<n3<.andk1<k2<k3<.N×Nand Xbe a double sequence space. A K-step space of Xis a sequence space


A canonical preimage of a sequence xnikjXis a sequence bnkXdefined as follows:


A sequence space Xis said to be monotoneif it contains the canonical preimages of all its step spaces.

The following subspaces lp,lp,cpandc0pwhere p=pkis 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=pijbe a double sequence of positive real numbers [19]. For any complex λwith 0<pijsupijpij=G<, we have


Let D=max12G1andH=max1supijpij, then for the sequences aijand bijin the complex plane, we have


2. 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].


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 2.1Let M1,M2be two Orlicz functions with Δ2condition, then


(b) χM1χM2χM1+M2for χ=2BVσI,20BVσI.

Proof.(a) Let x=xij20BVσIM2be an arbitrary element, so there exists ρ>0such that


Let ϵ>0and choose δwith 0<δ<1such that M1t<εfor 0<tδ.

Write yij=M2ϕmnijxρ. Consider,


Now, since M1is an Orlicz function so we have M1λxλM1x,0<λ<1. Therefore, we have


For yij>δ, we have yij<yijδ<1+yijδ. Now, since M1is non-decreasing and convex, it follows that,


Since M1satisfies the Δ2-condition, so we have


This implies that,


Hence, we have


Therefore from (12) and (16), we have


This implies that x=xij20BVσIM1M2. Hence χM2χM1M2for χ=20BVσI.The other cases can be proved in similar way.

(b) Let x=xij20BVσIM120BVσIM2. Let ϵ>0be given. Then there exist ρ>0,such that






from Eqs. (17) and (18), we get


so we have x=xij20BVσIM1+M2.

Hence, 20BVσIM120BVσIM220BVσIM1+M2.For χ=2BVσIthe inclusion are similar.


Proof.For this let Mx=x, for all x=xijX. Let us suppose that x=xij20BVσI. Then for any given ϵ>0, we have


Now let A1=ij:ϕmnijx<ϵI,be such that A1cI. Consider for ρ>0,


This implies that IlimMϕmnijxρ=0, which shows that x=xij20BVσIM.

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:

Theorem 2.2The spaces 20BVσIMand 2BVσIMare not convergence free.

Example 2.1To show this let I=Ifand Mx=x, for all x=0. Now consider the double sequence xij,yijwhich defined as follows:


Then we have xijbelong to both 20BVσIMand 2BVσIM,but yijdoes not belong to 20BVσIMand 2BVσIM. Hence, the spaces 20BVσIMand 2BVσIMare not convergence free.

To gain a good understanding of these double sequence spaces and related concepts, let us finally look at this theorem on inclusions:

Theorem 2.3Let M be an Orlicz function. Then


Proof.For this let us consider x=xij20BVσIM.It is obvious that it must belong to 2BVσIM.Now consider


Now taking the limit on both sides we get


Hence x=xij2BVσIM.Now it remains to show that


For this let us consider x=xij2BVσIMthis implies that there exist ρ>0s.t


Now consider,


Now taking the supremum on both sides, we get


Hence, x=xij2BVσIM.

3. Paranorm bounded variation sequence spaces

In this section we study double sequence spaces by using the double sequences of strictly positive real numbers p=pijwith the help of BVσspace 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:

Theorem 3.1Let p=pij2lthen the classes of double sequence 2MBVσIMpand 20MBVσIMpare paranormed spaces, paranormed by


where H=max1supijpij.

Proof.P1It is clear that gx=0if and only if x=0.

P2gx=gxis obvious.

P3Let x=xij,y=yij2MBVσIMp. Now for ρ1,ρ2>0, we denote


Let us take ρ3=ρ1+ρ2. Then by using the convexity of M, we have


which in terms give us




Therefore gx+ygx+gy.

P4Let λijbe a double sequence of scalars with λijλijand xij,L2MBVσIMpsuch that


in the sense that


Then, since the inequality


holds by subadditivity of g, the sequence gxijis bounded.



as ij. That implies that the scalar multiplication is continuous. Hence 2MBVσIMpis a paranormed space. For another space 20MBVσIMp, the result is similar.

We shall see about the separability of these new defined double sequence spaces in the next theorem.

Theorem 3.2The spaces 2MBVσIMpand 20MBVσIMpare not separable.

Example 3.1By counter example, we prove the above result for the space 2MBVσIMp.

Let A be an infinite subset of increasing natural numbers, i.e., AN×Nsuch that AI.


Let P0=xij:xij=0or1forijMandxij=0otherwise.

Since A is infinite, so P0is uncountable. Consider the class of open balls


Let C1be an open cover of 2MBVσIMpcontaining B1.

Since B1is uncountable, so C1cannot be reduced to a countable subcover for 2MBVσIMp. Thus 2MBVσIMpis not separable.

We shall now introduce a theorem which improves our work.

Theorem 3.3Let pijand qijbe two double sequences of positive real numbers. Then 20MBVσIMp20MBVσIMqif and only if limi,jKinfpijqij>0, where KcN×Nsuch that KI.

Proof.Let limi,jKinfpijqij>0and xij20MBVσIMq.Then, there exists β>0such that pij>βqijfor sufficiently large ijK.

Since xij20MBVσIMq.For a given ϵ>0,there exist ρ>0such that


Let G0=KcB0.Then for all sufficiently large ijG0.

ij:Mϕmnijxρpijϵ ij:MϕmnijxρβqijϵI.

Therefore, xij20MBVσIMp. The converse part of the result follows obviously.

Remark 3.1Let pijand qijbe two double sequences of positive real numbers. Then 20MBVσIMq20MBVσIMpif and only if limi,jKinfqijpij>0and 20MBVσIMq=20MBVσIMpif and only if limi,jKinfpijqij>0and limi,jKinfqijpij>0, where KcN×Nsuch that KI.

Theorem 3.4The set 2MBVσIMpis closed subspace of 2lMp.

Proof.Let xijpqbe a Cauchy double sequence in 2MBVσIMpsuch that xpqx. We show that x2MBVσIMp.Since, xijpq2MBVσIMp, then there exists apq, and ρ>0such that


We need to show that

(1) apqconverges to a.

(2) If U=ij:Mϕmnijxpqaρpij<ϵ, then UcI.

Since xijpqbe a Cauchy double sequence in 2MBVσIMpthen for a given ϵ>0there exists k0Nsuch that


For a given ϵ>0, we have


Then Bpqrsc,Bpqc,BrscI. Let Bc=BpqrscBpqcBrsc,

where B=ij:Mapqarsρpij<ϵ,then BcI. We choose k0Bc, then for each p,q,r,sk0, we have


Then apqis a Cauchy double sequence in C. So, there exists a scalar aCsuch that apqa,asp,q.

(2) For the next step, let 0<δ<1be given. Then, we show that if


then UcI.Since xpqx, then there exists p0,q0Nsuch that,


where D=max12G1,G=supijpij0andH=max1supijpijimplies PcI. The number p0q0can be so chosen that together with (25), we have


such that QcI.Since xijpq2MBVσIMp.

We have


Then we have a subset SN×Nsuch that ScI, where


Let Uc=PcQcSc,where


Therefore, for ijUc, we have


Hence the result 2MBVσIMp2lMpfollows.

Since the inclusions 2MBVσIMp2lMpand 20MBVσIMp2lMpare 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.

4. 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.

Theorem 4.1For any modulus functionf, the classes of double sequence20BVσIf, 2BVσIf,20MBVσIfand2MBVσIfare linear spaces.

Proof.Suppose x=xijand y=yij2BVσIfbe any two arbitrary elements. Let α,βare scalars. Now, since xij,yij2BVσIf. Then this implies that there exists some positive numbers L1,L2Cand such that the sets


Now, assume


be such that B1c,B2cI.Since fis a modulus function, we have


This implies that ij:m,n=0fϕmnijαx+βyαL1+βL2ϵI.Thus αxij+βyij2BVσIf.As xijand yijare two arbitrary element then αxij+βyij2BVσIffor all xij,yij2BVσIfand for all scalars α,β. Hence 2BVσIfis 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.

Theorem 4.2A sequence x=xij2MBVσIfI-convergent if and only if for every ϵ>0, there exists Mε,NεNsuch that


Proof.Let x=xij2MBVσIf. Suppose Ilimx=L. Then, the set


Fix Mε,NεBε.Then we have


which holds for all ijBϵ.



Conversely, suppose that


Then, being fa modulus function and by using basic triangular inequality, we have


Then, the set


Let Jϵ=m,n=0f(ϕmnijxMϵ,Nϵ)ϵm,n=0f(ϕmnijxMϵ,Nϵ)+ϵ.

If we fix ϵ>0then, we have CϵFIas well as Cϵ2FI.

Hence CϵCϵ2FI. This implies that


That is


This shows that


where the diamJdenotes the length of interval J. In this way, by induction we get the sequence of closed intervals


with the property that diamIk12diamIk1for k=2,3,4and ij:m,n=0 fϕmnijxijIkFIfor k=1,2,3,4.

Then there exists a ξIkwhere kNsuch that


showing that x=xij2MBVσIfis 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.

Theorem 4.3For any modulus function f, the spaces 20BVσIfand 20MBVσIfare solid and monotone.

Proof.We consider 20BVσIfand for 20MBVσIfthe proof shall be similar.

Let x=xij20BVσIfbe an arbitrary element, then the set


Let αijbe a sequence of scalars with αij1for all i,jN.

Now, since fis a modulus function. Then the result follows from (2.18) and the inequality




implies that


Thus we have αijxij20BVσIf.Hence 20BVσIfis solid. Therefore 20BVσIfis monotone. Since every solid sequence space is monotone.

Remark 4.1The space 2BVσIfand 2MBVσIfare neither solid nor monotone in general.

Example 4.1Here we give counter example for establishment of this result. Let X=2BVσIand 2MBVσI. Let us consider I=Ifand fx=x, for all x=xijand xij0. Consider, the K-step space XKfof Xfdefined as follows:

Let x=xijXfand y=yijXKfbe such that


Consider the sequence xijdefined by xij=1for all i,jN.

Then, x=xij2BVσIfand 2MBVσIf, but K-step space preimage does not belong to BVσIfand 2MBVσIf. Thus, 2BVσIfand 2MBVσIfare not monotone and hence they are not solid.

After discussing about solid and monotone sequence space now we come to the concept of sequence algebra which will help to understand our further work.

Theorem 4.4For any modulus function f, the spaces 20BVσIfand 2BVσIfare sequence algebra.

Proof.Let x=xij,y=yij20BVσIfbe any two arbitrary elements.

Then, the sets






Thus, we have xij.yij20BVσIf.Hence 20BVσIfis sequence algebra. And for 2BVσIfthe result can be proved similarly.

Remark 4.2If Iis not maximal and IIfthen the spaces 2BVσIfand 20BVσIfare not symmetric.

Example 4.2Let AIbe an infinite set and fx=xfor all x=xijand xij0.If


Then, it is clearly seen that xij20BVσIf2BVσIf.

Let KN×Nbe such that KIandKcI.Let ϕ:KAandψ:KcAcbe a bijective maps (as all four sets are infinite). Then, the mapping π:N×NN×Ndefined by


is a permutation on N×N.

But xπij2BVσIfand hence xπij20BVσIfshowing that 2BVσIfand 20BVσIfare not symmetric double sequence spaces.

5. Conclusion

In this chapter, we study different forms of BVσdouble sequence spaces of invariant means with the help of ideal, operators and some functions such as Orlicz function and modulus function. The chapter shows the potential of the new theoretical tools to deal with the convergence problems of sequences in sigma bounded variation, occurring in many branches of science, engineering and applied mathematics.


The authors express their gratitude to our Chairman Prof. Mohammad Imdad for his advices and continuous support. We are grateful to the learned referees for careful reading of our manuscript and considering the same for publication.


Conflict of interest

The authors declare that they have no competing interests.

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite and reference

Link to this chapter Copy to clipboard

Cite this chapter Copy to clipboard

Vakeel Ahmad Khan, Hira Fatima and Mobeen Ahmad (April 23rd 2019). A Study of Bounded Variation Sequence Spaces, Applied Mathematics, Bruno Carpentieri, IntechOpen, DOI: 10.5772/intechopen.81907. Available from:

chapter statistics

624total chapter downloads

More statistics for editors and authors

Login to your personal dashboard for more detailed statistics on your publications.

Access personal reporting

Related Content

This Book

Next chapter

Simple Approach to Special Polynomials: Laguerre, Hermite, Legendre, Tchebycheff, and Gegenbauer

By Vicente Aboites and Miguel Ramírez

Related Book

First chapter

Text Mining for Industrial Machine Predictive Maintenance with Multiple Data Sources

By Giancarlo Nota and Alberto Postiglione

We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.

More About Us