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On Some New Sequence Spaces in 2Normed Spaces Using Ideal Convergence and an Orlicz Function
Journal of Inequalities and Applications volume 2010, Article number: 482392 (2010)
Abstract
The purpose of this paper is to introduce certain new sequence spaces using ideal convergence and an Orlicz function in 2normed 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 2normed 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 2norm 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 2Banach 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 2normed spaces.
In this paper, we continue to study certain new sequence spaces by using Orlicz function and ideals in 2normed 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 2normed 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:
where
The following wellknown inequality[16, page 190] will be used in the study.
then
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
Since is a 2norm, and is an Orlicz function the following inequality holds:
where
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
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
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)⊆.
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
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.

(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
where . Hence for all sequences of scalars with for all whenever .
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Savaş, E. On Some New Sequence Spaces in 2Normed Spaces Using Ideal Convergence and an Orlicz Function. J Inequal Appl 2010, 482392 (2010). https://doi.org/10.1155/2010/482392
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DOI: https://doi.org/10.1155/2010/482392