Open Access

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

Journal of Inequalities and Applications20102010:482392

Received: 25 July 2010

Accepted: 17 August 2010

Published: 8 September 2010


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.

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, [810]).

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

Let be a nondecreasing sequence of positive numbers tending to such that and let be an admissible ideal of , let be an Orlicz function, and let be a 2-normed space. Further, let be a bounded sequence of positive real numbers. By we denote the space of all sequences defined over . Now, we define the following sequence spaces:


The following well-known inequality [16, page 190] will be used in the study.

for all and . Also for all .

Theorem 2.1.

are linear spaces.


We will prove the assertion for only and the others can be proved similarly. Assume that and , so
Since is a 2-norm, and is an Orlicz function the following inequality holds:
From the above inequality, we get

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.

For any fixed , is paranormed space with respect to the paranorm defined by


That and are easy to prove. So we omit them.
  1. (iii)
    Let us take and in . Let
Let and , then if , then, we have
Thus, and

 (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.

It should be noted that for a fixed the space

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) provided is such that .

(ii) .

  1. (i)
    For given first choose such that Now using the continuity of choose such that . Let . Now from the definition
Thus if then
that is,
that is,
that is,
Hence from above using the continuity of we must have
which consequently implies that
that is,
This shows that
and so belongs to . This proves the result.
  1. (ii)
    Let , then the fact

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.


We give the proof for only. Let and let be a sequence of scalars such that for all . Then we have

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

Authors’ Affiliations

Department of Mathematics, Istanbul Ticaret University


  1. Kostyrko P, Šalát T, Wilczyński W: -convergence. Real Analysis Exchange 2000, 26(2):669–686.MathSciNetMATHGoogle Scholar
  2. Lahiri BK, Das P: and -convergence in topological spaces. Mathematica Bohemica 2005, 130(2):153–160.MathSciNetMATHGoogle Scholar
  3. Kostyrko P, Mačaj M, Šalát T, Sleziak M: -convergence and extremal -limit points. Mathematica Slovaca 2005, 55(4):443–464.MathSciNetMATHGoogle Scholar
  4. Das P, Malik P: On the statistical and - variation of double sequences. Real Analysis Exchange 2008, 33(2):351–364.MathSciNetMATHGoogle Scholar
  5. Gähler S: 2-metrische Räume und ihre topologische Struktur. Mathematische Nachrichten 1963, 26: 115–148. 10.1002/mana.19630260109MathSciNetView ArticleMATHGoogle Scholar
  6. Gunawan H, Mashadi : On finite-dimensional 2-normed spaces. Soochow Journal of Mathematics 2001, 27(3):321–329.MathSciNetMATHGoogle Scholar
  7. Freese RW, Cho YJ: Geometry of Linear 2-Normed Spaces. Nova Science, Hauppauge, NY, USA; 2001:viii+301.MATHGoogle Scholar
  8. 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
  9. Gürdal M, Pehlivan S: Statistical convergence in 2-normed spaces. Southeast Asian Bulletin of Mathematics 2009, 33(2):257–264.MathSciNetMATHGoogle Scholar
  10. Gürdal M, Şahiner A, Açık I: Approximation theory in 2-Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(5–6):1654–1661. 10.1016/ ArticleMathSciNetMATHGoogle Scholar
  11. Krasnoselskii 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 1973, 25: 973–978. 10.4153/CJM-1973-102-9MathSciNetView 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. 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
  16. Maddox IJ: Elements of Functional Analysis. Cambridge University Press, London, UK; 1970:x+208.MATHGoogle Scholar


© E. Savaş. 2010

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.