- 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.

## 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,

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

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

Definition 1.2 (see [7]).

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:

## 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.

Theorem 2.2.

sequences space is a linear space.

Proof.

and this completes the proof.

Theorem 2.3.

Theorem 2.4.

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.

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.

Proof.

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

## Declarations

### Acknowledgments

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

## Authors’ Affiliations

## References

- Gähler S:
**Lineare 2-normierte Räume.***Mathematische Nachrichten*1965,**28:**1–43.View ArticleGoogle Scholar - Gähler S:
**Über die Uniformisierbarkeit 2-metrischer Räume.***Mathematische Nachrichten*1965,**28:**235–244.MathSciNetView ArticleMATHGoogle Scholar - Gunawan H:
**The space of -summable sequences and its natural -norm.***Bulletin of the Australian Mathematical Society*2001,**64**(1):137–147. 10.1017/S0004972700019754MathSciNetView ArticleMATHGoogle Scholar - Freese RW, Cho YJ:
*Geometry of Linear 2-Normed Spaces*. Nova Science Publishers, Hauppauge, NY, USA; 2001:viii+301.MATHGoogle Scholar - Şahiner A, Gürdal M, Saltan S, Gunawan H:
**Ideal convergence in 2-normed spaces.***Taiwanese Journal of Mathematics*2007,**11**(5):1477–1484.MathSciNetMATHGoogle Scholar - Gürdal M, Pehlivan S:
**Statistical convergence in 2-normed spaces.***Southeast Asian Bulletin of Mathematics*2009,**33**(2):257–264.MathSciNetMATHGoogle Scholar - Gunawan H, Mashadi M:
**On -normed spaces.***International Journal of Mathematics and Mathematical Sciences*2001,**27**(10):631–639. 10.1155/S0161171201010675MathSciNetView ArticleMATHGoogle Scholar - Sahiner A, Gurdal M:
**New sequence spaces in -normed spaces with respect to an Orlicz function.***The Aligarh Bulletin of Mathematics*2008,**27**(1):53–58.MathSciNetGoogle Scholar - Savaş E:
**On some new sequence spaces in 2-normed spaces using ideal convergence and an Orlicz function.***journal of Inequalities and Applications*2010,**2010:**-8. Article ID 482392Google Scholar - Savaş E:
**-strongly summable sequences spaces in 2-normed spaces defined by ideal convergence and an Orlicz function.***Applied Mathematics and Computation*2010,**217**(1):271–276. 10.1016/j.amc.2010.05.057MathSciNetView ArticleMATHGoogle Scholar - Krasnoselski MA, Rutisky YB:
*Convex Function and Orlicz Spaces*. Noordhoff, Groningen, The Netherlands; 1961.Google Scholar - Parashar SD, Choudhary B:
**Sequence spaces defined by Orlicz functions.***Indian Journal of Pure and Applied Mathematics*1994,**25**(4):419–428.MathSciNetMATHGoogle Scholar - Ruckle WH:
**spaces in which the sequence of coordinate vectors is bounded.***Canadian Journal of Mathematics. Journal Canadien de Mathématiques*1973,**25:**973–978.MathSciNetView ArticleMATHGoogle Scholar - Maddox IJ:
**Sequence spaces defined by a modulus.***Mathematical Proceedings of the Cambridge Philosophical Society*1986,**100**(1):161–166. 10.1017/S0305004100065968MathSciNetView ArticleMATHGoogle Scholar - Gunawan H:
**On -inner products, -norms, and the Cauchy-Schwarz inequality.***Scientiae Mathematicae Japonicae*2002,**55**(1):53–60.MathSciNetGoogle Scholar - Maddox IJ:
*Elements of Functional Analysis*. Cambridge University Press, London, UK; 1970:x+208.MATHGoogle Scholar - Kampthan PK, Gupta M:
*Sequence Spaces and Series, Lecture Notes in Pure and Applied Mmathematics*.*Volume 65*. Marcel Dekker, New York, NY, USA; 1981.Google Scholar - Savas E, Patterson RF:
**On some double sequence spaces defined by a modulus.***Math. Slovaca*2011,**61**(2):1–12.MathSciNetView ArticleGoogle Scholar

## Copyright

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