- Research Article
- Open Access

# Some New Double Sequence Spaces Defined by Orlicz Function in -Normed Space

- Ekrem Savaş
^{1}Email author

**2011**:592840

https://doi.org/10.1155/2011/592840

© Ekrem Savaş. 2011

**Received:**1 January 2011**Accepted:**17 February 2011**Published:**9 March 2011

## Abstract

The aim of this paper is to introduce and study some new double sequence spaces with respect to an Orlicz function, and also some properties of the resulting sequence spaces were examined.

## Keywords

- Linear Space
- Normed Space
- Sequence Space
- Positive Real Number
- Modulus Function

## 1. Introduction

We recall that the concept of a 2-normed space was first given in the works of Gähler ([1, 2]) as an interesting nonlinear generalization of a normed linear space which was subsequently studied by many authors (see, [3, 4]). Recently, a lot of activities have started to study summability, sequence spaces, and related topics in these nonlinear spaces (see, e.g., [5–9]). In particular, Savaş [10] combined Orlicz function and ideal convergence to define some sequence spaces using 2-norm.

In this paper, we introduce and study some new double-sequence spaces, whose elements are form -normed spaces, using an Orlicz function, which may be considered as an extension of various sequence spaces to -normed spaces. We begin with recalling some notations and backgrounds.

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

Subsequently, Orlicz function was used to define sequence spaces by Parashar and Choudhary [12] and others. An Orlicz function can always be represented in the following integral form: , where is the known kernel of , right differential for , , for , is nondecreasing, and as .

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].

Remark 1.1.

If is a convex function and , then for all with .

Let and be real vector space of dimension , where . An -norm on is a function which satisfies the following four conditions:

(i) if and only if are linearly dependent,

(ii) are invariant under permutation,

(iii) , ,

(iv) .

The pair is then called an -normed space [3].

is defines an norm on with respect to (see, [15]).

Definition 1.2 (see [7]).

for every .

Definition 1.3 (see [16]).

Let be a linear space. Then, a map is called a paranorm (on ) if it is satisfies the following conditions for all and scalar:

(i) ,

(ii) ,

(iii) ,

(iv) and imply .

## 2. Main Results

Let be any -normed space, and let denote -valued sequence spaces. Clearly is a linear space under addition and scalar multiplication.

Definition 2.1.

for each .

Theorem 2.2.

sequences space is a linear space.

Proof.

and this completes the proof.

Theorem 2.3.

where , .

This gives us .

Theorem 2.4.

If for each and , then .

Proof.

Thus, . This completes the proof.

The following result is a consequence of the above theorem.

Theorem 2.6.

, where is the double space of bounded sequences and .

Proof.

and this completes the proof.

Theorem 2.7.

Proof.

Let ; when adding the above inequality from to we get and this completes the proof.

Definition 2.8 (see [10]).

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

Definition 2.9.

Let be a sequence space. Then, is called monotone if it contains the canonical preimages of all its step spaces (see, [17]).

Theorem 2.10.

The sequence space is solid.

Proof.

and this completes the proof.

We have the following result in view of Remark 1.1 and Theorem 2.10.

Corollary 2.11.

The sequence space is monotone.

Definition 2.12 (see [18]).

Such transformation is said to be nonnegative if is nonnegative for all , and .

Definition 2.13.

for each . If , then we say is summable to , where .

Theorem 2.14.

is linear spaces.

Proof.

This can be proved by using the techniques similar to those used in Theorem 2.2.

- (2)
Let for each and . Let .

Thus, , and this completes the proof.

## Declarations

### Acknowledgments

The author wishes to thank the referees for their careful reading of the paper and for their helpful suggestions.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.