## Abstract

In 1986, Atanassov introduced the concept of intuitionistic fuzzy set theory which is based on the extensions of definitions of fuzzy set theory given by Zadeh. This theory provides a variable model to elaborate uncertainty and vagueness involved in decision making problems. In this chapter, we concentrate our study on the ideal convergence of sequence spaces with respect to intuitionistic fuzzy norm and discussed their topological and algebraic properties.

### Keywords

- ideal
- intuitionistic fuzzy normed spaces
- Orlicz function
- compact operator
- I-convergence

## 1. Introduction

In recent years, the fuzzy theory has emerged as the most active area of research in many branches of mathematics, computer and engineering [1]. After the excellent work of Zadeh [2], a large number of research work have been done on fuzzy set theory and its applications as well as fuzzy analogues of the classical theories. It has a wide number of applications in various fields such as population dynamics [3], nonlinear dynamical system [4], chaos control [5], computer programming [6], etc. In 2006, Saadati and Park [7] introduced the concept of intuitionistic fuzzy normed spaces after that the concept of statistical convergence in intuitionistic fuzzy normed space was studied for single sequence in [8]. The study of intuitionistic fuzzy topological spaces [9], intuitionistic fuzzy 2-normed space [10] and intuitionistic fuzzy Zweier ideal convergent sequence spaces [11] are the latest developments in fuzzy topology.

First, let us recall some notions, basic definitions and concepts which are used in sequel.

**Definition 1.1.** (See Ref. [7]). The five-tuple

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

In this case

**Example 1.1.** *Let* *be a normed space. Denote* *and* *for all* *and let* *and* *be fuzzy sets on* *defined as follows:*

for all

**Definition 1.2.** Let

In 1951, the concept of statistical convergence was introduced by Steinhaus [12] and Fast [13] in their papers “Sur la convergence ordinaire et la convergence asymptotique” and “Sur la convergence statistique,” respectively. Later on, in 1959, Schoenberg [14] reintroduced this concept. It is a very useful functional tool for studying the convergence of numerical problems through the concept of density. The concept of ideal convergence, which is a generalization of statistical convergence, was introduced by Kostyrko et al. [15] and it is based on the ideal

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

A sequence **-convergent** [21, 22] to a number

## 2. IF-ideal convergent sequence spaces using compact operator

This section consists of some double sequence spaces with respect to intuitionistic fuzzy normed space and study the fuzzy topology on the said spaces. First we recall some basic definitions on compact operator.

**Definition 2.1.** (See [23]). Let

The set of all bounded linear operators

and

**Definition 2.2.** (See [23]). Let

(i)

(ii)

The set of all compact linear operators

In 2015, Khan et al. [11] introduced the following sequence spaces:

Motivated by this, we introduce the following sequence spaces with the help of compact operator in intuitionistic fuzzy normed spaces:

Here, we also define an open ball with center

Now, we are ready to state and prove our main results. This theorem is based on the linearity of new define sequence spaces which is stated as follows.

**Theorem 2.1.** The sequence spaces

*Proof.* Let

This implies

Now, we define the set

Let

and

Thus, we have

and

This implies that

Therefore, the sequence space

Similarly, we can proof for the other space. □

In the following theorems, we discussed the convergence problem in the said sequence spaces. For this, firstly we have to discuss about the topology of this space. Define

Then

**Theorem 2.2**. Let

*Proof*. Fix

such that

Conversely, if for each

There are some facts that arise in connection with the convergence of sequences in these spaces. Let us proceed to the next theorem on Ideal convergence of sequences in these new define spaces.

**Theorem 2.3.** A sequence

*Proof.* Suppose that

which implies that

Conversely, let us choose

Now, we want to show that there exists a number

For this, we define for each

So, we have to show that

In particular

which is not possible. On the other hand

In particular

which is not possible. Hence

## 3. IF-ideal convergent sequence spaces using Orlicz function

In this section, we have discussed the ideal convergence of sequences in Intuitionistic fuzzy

**Definition 3.1.** An Orlicz function is a function

**Remark 3.1.** If

In 2009, Mohiuddine and Lohani [18] introduced the concept of statistical convergence in intuitionistic fuzzy normed spaces in their paper published in Chaos, Solitons and Fractals. This motivated us to introduced some sequence spaces defined by compact operator and Orlicz function which are as follows:

We also define an open ball with center

We shall now consider some theorems of these sequence spaces and invite the reader to verify the linearity of these sequence spaces.

**Theorem 3.1.** Every open ball

*Proof.* Let

Let

Putting

Let

and

Thus

**Remark 3.2.**

Define

Then

In the above result we can easily verify that the open sets in these spaces are open ball in the same spaces. This theorem itself will have various applications in our future work.

**Theorem 3.2.** The topology

*Proof.*

**Theorem 3.3.**

*Proof.* Let

Putting

Putting

and

which is a contradiction. Hence,

## 4. Conclusion

The concept of defining intuitionistic fuzzy ideal convergent sequence spaces as it generalized the fuzzy set theory and give quite useful and interesting applications in many areas of mathematics and engineering. This chapter give brief introduction to intuitionistic fuzzy normed spaces with some basic definitions of convergence applicable on it. We have also summarized different types of sequence spaces with the help of ideal, Orlicz function and compact operator. At the end of this chapter some theorems and remarks based on these new defined sequence spaces are discussed for proper understanding.

## Conflict of interest

The authors declare that they have no competing interests.