## Abstract

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.

### Keywords

- 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

Let **ideal** in **filter** on **nontrivial** if **maximal** if there cannot exist any nontrivial ideal

A double sequence **-convergent** [5, 6, 7, 8] to a number **-Cauchy** if for every

A continuous linear functional **invariant mean** [9, 10] or

1.

2.

3.

where

If

where

where *m*th-iterate of

**Definition 1.1** A sequence **-bounded variation** if and only if:

(i)

(ii)

We denote by

is a Banach space normed by

A function **Orlicz function** [13, 14] if it satisfies the following conditions:

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

(ii)

**Remark 1.1** If the convexity of an Orlicz function is replaced by **Modulus function** [15, 16, 17]. If

**Definition 1.2** A double sequence space

[i] **solid** or **normal** if

[ii] **symmetric** if

[iii] **sequence algebra** if

[iv] **convergence free** if

**Definition 1.3** Let

A canonical preimage of a sequence

A sequence space **monotone** if it contains the canonical preimages of all its step spaces.

The following subspaces

Let

## 2. Bounded variation sequence spaces defined by Orlicz function

In this section, we define and study the concepts of

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

(a)

(b)

*Proof.* (a) Let

Let

Write

Now, since

For

Since

This implies that,

Hence, we have

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

This implies that

(b) Let

and

Therefore

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

so we have

Hence,

**Corollary** *for* *and*

*Proof.* For this let

Now let

This implies that

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 2.2** The spaces

**Example 2.1** To show this let

Then we have

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

Hence

For this let us consider

Now consider,

Now taking the supremum on both sides, we get

Hence,

## 3. Paranorm bounded variation sequence spaces

In this section we study double sequence spaces by using the double sequences of strictly positive real numbers

We also denote

and

We can now state and proof the theorems based on these double sequence spaces which are as follows:

**Theorem 3.1** Let

where

*Proof.*

Let us take

which in terms give us

and

Therefore

in the sense that

Then, since the inequality

holds by subadditivity of g, the sequence

Therefore,

as

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

**Theorem 3.2** The spaces

**Example 3.1** By counter example, we prove the above result for the space

Let A be an infinite subset of increasing natural numbers, i.e.,

Let

Since A is infinite, so

Let

Since

We shall now introduce a theorem which improves our work.

**Theorem 3.3** Let

*Proof.* Let

Since

Let

Therefore,

**Remark 3.1** Let

**Theorem 3.4** The set

*Proof.* Let

We need to show that

(1)

(2) If

Since

For a given

Then

where

Then

(2) For the next step, let

then

where

such that

We have

Then we have a subset

Let

Therefore, for

Hence the result

Since the inclusions

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

and

We shall now consider important theorems of these double sequence spaces by using modulus function.

**Theorem 4.1** *For any modulus function* *, the classes of double sequence* *and* *are linear spaces.*

*Proof.* Suppose

Now, assume

be such that

This implies that

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 4.2** A sequence

*Proof.* Let

Fix

which holds for all

Hence

Conversely, suppose that

Then, being

Then, the set

Let

If we fix

Hence

That is

This shows that

where the

with the property that

Then there exists a

showing that

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 4.3** For any modulus function

*Proof.* We consider

Let

Let

Now, since

Therefore,

implies that

Thus we have

**Remark 4.1** The space

**Example 4.1** Here we give counter example for establishment of this result. Let

Let

Consider the sequence

Then,

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 4.4** For any modulus function

*Proof.* Let

Then, the sets

and

Therefore,

Thus, we have

**Remark 4.2** If

**Example 4.2** Let

Then, it is clearly seen that

Let

is a permutation on

But

## 5. Conclusion

In this chapter, we study different forms of

### Acknowledgements

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.