- Research Article
- Open Access

# On Some New Sequence Spaces in 2-Normed Spaces Using Ideal Convergence and an Orlicz Function

- E Savaş
^{1}Email author

**2010**:482392

https://doi.org/10.1155/2010/482392

© E. Savaş. 2010

**Received: **25 July 2010

**Accepted: **17 August 2010

**Published: **8 September 2010

## Abstract

The purpose of this paper is to introduce certain new sequence spaces using ideal convergence and an Orlicz function in 2-normed spaces and examine some of their properties.

## Keywords

- Topological Space
- Sequence Space
- Positive Real Number
- Scalar Multiplication
- Statistical Convergence

## 1. Introduction

The notion of ideal convergence was introduced first by Kostyrko et al. [1] as a generalization of statistical convergence which was further studied in topological spaces [2]. More applications of ideals can be seen in [3, 4].

The concept of 2-normed space was initially introduced by Gähler [5] as an interesting nonlinear generalization of a normed linear space which was subsequently studied by many authors (see, [6, 7]). Recently, a lot of activities have started to study summability, sequence spaces and related topics in these nonlinear spaces (see, [8–10]).

Recall in [11] that an Orlicz function is continuous, convex, nondecreasing function such that and for , and as .

Subsequently Orlicz function was used to define sequence spaces by Parashar and Choudhary [12] and others.

If convexity of Orlicz function, is replaced by , then this function is called Modulus function, which was presented and discussed by Ruckle [13] and Maddox [14].

Note that if is an Orlicz function then for all with

Let be a normed space. Recall that a sequence of elements of is called to be statistically convergent to if the set has natural density zero for each .

A family of subsets a nonempty set is said to be an ideal in if (i) (ii) imply (iii) imply , while an admissible ideal of further satisfies for each , [9, 10].

Given is a nontrivial ideal in . The sequence in is said to be -convergent to if for each the set belongs to , [1, 3].

Let be a real vector space of dimension where A 2-norm on is a function which satisfies (i) if and only if and are linearly dependent, (ii) , (iii) , and (iv) . The pair is then called a -normed space [6].

Recall that is a 2-Banach space if every Cauchy sequence in is convergent to some in

Quite recently Savaş [15] defined some sequence spaces by using Orlicz function and ideals in 2-normed spaces.

In this paper, we continue to study certain new sequence spaces by using Orlicz function and ideals in 2-normed spaces. In this context it should be noted that though sequence spaces have been studied before they have not been studied in nonlinear structures like -normed spaces and their ideals were not used.

## 2. Main Results

Theorem 2.1.

Proof.

Two sets on the right hand side belong to and this completes the proof.

It is also easy to see that the space is also a linear space and we now have the following.

Theorem 2.2.

Proof.

(iv) Finally using the same technique of Theorem of Savaş [15] it can be easily seen that scalar multiplication is continuous. This completes the proof.

Corollary 2.3.

which is a subspace of the space is a paranormed space with the paranorms for and .

Theorem 2.4.

Let , be Orlicz functions. Then we have

- (i)

gives us the result.

Definition 2.5.

Let be a sequence space. Then is called solid if whenever for all sequences of scalars with for all .

Theorem 2.6.

The sequence spaces are solid.

Proof.

where . Hence for all sequences of scalars with for all whenever .

## Authors’ Affiliations

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## Copyright

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