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

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.

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

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:

(2.1)

where

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

(2.2)

then

(2.3)

for all and . Also for all .

Theorem 2.1.

are linear spaces.

Proof.

We will prove the assertion for only and the others can be proved similarly. Assume that and , so

(2.4)

Since is a 2-norm, and is an Orlicz function the following inequality holds:

(2.5)

where

(2.6)

From the above inequality, we get

(2.7)

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

(2.8)

Proof.

That and are easy to prove. So we omit them.

• (iii) Let us take and in . Let

  

  (2.9)

Let and , then if , then, we have

(2.10)

Thus, and

(2.11)

 (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

(2.12)

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)⊆.

Proof.

• (i) For given first choose such that Now using the continuity of choose such that . Let . Now from the definition

  

  (2.13)

Thus if then

(2.14)

that is,

(2.15)

that is,

(2.16)

that is,

(2.17)

Hence from above using the continuity of we must have

(2.18)

which consequently implies that

(2.19)

that is,

(2.20)

This shows that

(2.21)

and so belongs to . This proves the result.

• (ii) Let , then the fact

  

  (2.22)

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.

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

(2.23)

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