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  • Research Article
  • Open Access

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

Journal of Inequalities and Applications20112011:592840

  • Received: 1 January 2011
  • Accepted: 17 February 2011
  • Published:


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.


  • 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., [59]). 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].

Let be equipped with the -norm, then the volume of the -dimensional parallelepiped spanned by the vectors, which may be given explicitly by the formula
where denotes inner product. Let be an -normed space of dimension and a linearly independent set in . Then, the function on is defined by

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

Definition 1.2 (see [7]).

A sequence in -normed space is aid to be convergent to an in (in the -norm) if

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.

Let be an Orlicz function and any -normed space. Further, let be a bounded sequence of positive real numbers. Now, we define the following new double sequence space as follows:

for each .

The following inequalities will be used throughout the paper. Let be a double sequence of positive real numbers with , and let . Then, for the factorable sequences and in the complex plane, we have as in Maddox [16]

Theorem 2.2.

sequences space is a linear space.


Now, assume that and . Then,
Since is a -norm on , and is an Orlicz function, we get

and this completes the proof.

Theorem 2.3.

space is a paranormed space with the paranorm defined by

where , .

  1. (i)
    Clearly, and (ii) . (iii) Let , then there exists such that
So, we have
and thus
  1. (iv)
    Now, let and . Since

This gives us .

Theorem 2.4.

If for each and , then .


If , then there exists some such that
This implies that
for sufficiently large values of and . Since is nondecreasing, we are granted

Thus, . This completes the proof.

The following result is a consequence of the above theorem.

Corollary 2.5.
  1. (i)
    If for each and , then
  1. (ii)
    If for each and , then

Theorem 2.6.

, where is the double space of bounded sequences and .


. Then, there exists an such that for each , . We want to show . But

and this completes the proof.

Theorem 2.7.

Let and be Orlicz function. Then, we have


We have

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.


Let ; that is,
Let ( ) be double sequence of scalars such that for all . Then, the result follows from the following inequality:

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]).

Let denote a four-dimensional summability method that maps the complex double sequences into the double-sequence , where the th term to is as follows:

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

Definition 2.13.

Let be a nonnegative matrix. Let be an Orlicz function and a factorable double sequence of strictly positive real numbers. Then, we define the following sequence spaces:

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

If we take and for all , then we have

Theorem 2.14.

is linear spaces.


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

Theorem 2.15.
  1. (1)
    If , then
  1. (2)
    If , then
  1. (1)
    Let ; since , we have
and hence .
  1. (2)

    Let for each and . Let .

Then, for each , there exists a positive integer such that
for all . This implies that

Thus, , and this completes the proof.



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

Authors’ Affiliations

Department of Mathematics, Istanbul Commerce University, Uskudar, 34672, Istanbul, Turkey


  1. Gähler S: Lineare 2-normierte Räume. Mathematische Nachrichten 1965, 28: 1–43.View ArticleGoogle Scholar
  2. Gähler S: Über die Uniformisierbarkeit 2-metrischer Räume. Mathematische Nachrichten 1965, 28: 235–244.MathSciNetView ArticleMATHGoogle Scholar
  3. 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
  4. Freese RW, Cho YJ: Geometry of Linear 2-Normed Spaces. Nova Science Publishers, Hauppauge, NY, USA; 2001:viii+301.MATHGoogle Scholar
  5. 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
  6. Gürdal M, Pehlivan S: Statistical convergence in 2-normed spaces. Southeast Asian Bulletin of Mathematics 2009,33(2):257–264.MathSciNetMATHGoogle Scholar
  7. Gunawan H, Mashadi M: On -normed spaces. International Journal of Mathematics and Mathematical Sciences 2001,27(10):631–639. 10.1155/S0161171201010675MathSciNetView ArticleMATHGoogle Scholar
  8. 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
  9. 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
  10. 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
  11. Krasnoselski MA, Rutisky YB: Convex Function and Orlicz Spaces. Noordhoff, Groningen, The Netherlands; 1961.Google Scholar
  12. Parashar SD, Choudhary B: Sequence spaces defined by Orlicz functions. Indian Journal of Pure and Applied Mathematics 1994,25(4):419–428.MathSciNetMATHGoogle Scholar
  13. 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
  14. 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
  15. Gunawan H: On -inner products, -norms, and the Cauchy-Schwarz inequality. Scientiae Mathematicae Japonicae 2002,55(1):53–60.MathSciNetGoogle Scholar
  16. Maddox IJ: Elements of Functional Analysis. Cambridge University Press, London, UK; 1970:x+208.MATHGoogle Scholar
  17. 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
  18. Savas E, Patterson RF: On some double sequence spaces defined by a modulus. Math. Slovaca 2011,61(2):1–12.MathSciNetView ArticleGoogle Scholar


© Ekrem Savaş. 2011

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