The Scientific World Journal

Volume 2014 (2014), Article ID 134534, 6 pages

http://dx.doi.org/10.1155/2014/134534

Research Article

Spaces of Ideal Convergent Sequences

M. Mursaleen¹ and Sunil K. Sharma²

¹Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

²Department of Mathematics, Model Institute of Engineering & Technology, Kot Bhalwal, J&K 181122, India

Received 20 August 2013; Accepted 26 November 2013; Published 28 January 2014

Academic Editors: F. Başar and J. Xu

Copyright © 2014 M. Mursaleen and Sunil K. Sharma. 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.

Abstract.

In the present paper, we introduce some sequence spaces using ideal convergence and Musielak-Orlicz function . We also examine some topological properties of the resulting sequence spaces.

1. Introduction and Preliminaries

The notion of ideal convergence was first introduced by Kostyrko et al. [1] as a generalization of statistical convergence which was further studied in topological spaces by Das et al. see [2]. More applications of ideals can be seen in [2, 3]. We continue in this direction and introduce -convergence of generalized sequences with respect to Musielak-Orlicz function.

A family of subsets of a nonempty set is said to be an ideal in if

(1),(2) imply ,(3), imply ,

while an admissible ideal of further satisfies for each ; see [1]. A sequence in is said to be -convergent to . If for each , the set belongs to ; see [1]. For more details about ideal convergent sequence spaces, see [4–10] and references therein.

Mursaleen and Noman [11] introduced the notion of -convergent and -bounded sequences as follows.

Let be a strictly increasing sequence of positive real numbers tending to infinity; that is,

The sequence is -convergent to the number , called the -limit of , if , as , where

The sequence is -bounded if . It is well known [11] that if in the ordinary sense of convergence, then

This implies that

which yields that and hence is -convergent to .

Let be a linear metric space. A function is called paranorm if

(1), for all ,(2), for all ,(3), for all ,(4)if is a sequence of scalars with as and is a sequence of vectors with as , then as .

A paranorm for which implies that is called total paranorm and the pair is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [12, Theorem 10.4.2, P-183]). For more details about sequence spaces, see [13–15] and references therein.

An Orlicz function is a function which is continuous, nondecreasing, and convex with , for and as .

Lindenstrauss and Tzafriri [16] used the idea of Orlicz function to define the following sequence space. Let be the space of all real or complex sequences . Then,

which is called an Orlicz sequence space. The space is a Banach space with the norm

It is shown in [16] that every Orlicz sequence space contains a subspace isomorphic to . The -condition is equivalent to for all values of and for .

A sequence of Orlicz function is called a Musielak-Orlicz function see; [17, 18]. A sequence defined by

is called the complementary function of a Musielak-Orlicz function . For a given Musielak-Orlicz function , the Musielak-Orlicz sequence space and its subspace are defined as follows:

where is a convex modular defined by

We consider equipped with the Luxemburg norm

or equipped with the Orlicz norm

Let be a Musielak-Orlicz function and let be a bounded sequence of positive real numbers. We define the following sequence spaces:

We can write

If we take , for all , we have

The following inequality will be used throughout the paper. If , , then

for all , and . Also for all .

The main aim of this paper is to study some ideal convergent sequence spaces defined by a Musielak-Orlicz function . We also make an effort to study some topological properties and prove some inclusion relations between these spaces.

2. Main Results

Theorem 1. Let be a Musielak-Orlicz function and let be a bounded sequence of positive real numbers. Then, the spaces , , , and are linear.

Proof. Let and let , be scalars. Then, there exist positive numbers and such that

For a given , we have

Let . Since is nondecreasing convex function, so by using inequality (15), we have

Now, by (17), we have

Therefore, . Hence is a linear space. Similarly, we can prove that , , and are linear spaces.

Theorem 2. Let be a Musielak-Orlicz function. Then,

Proof. Let . Then, there exist and such that

We have

Taking supremum over on both sides, we get . The inclusion is obvious. Thus,

This completes the proof of the theorem.

Theorem 3. Let be a Musielak-Orlicz function and let be a bounded sequence of positive real numbers. Then, is a paranormed space with paranorm defined by

Proof. It is clear that . Since , we get . Let us take . Let

Let and . If , then we have

Thus, and

Let , where and let as . We have to show that as . Let

If and , then we observe that

From the above inequality, it follows that

and, consequently,

This completes the proof.

Theorem 4. Let and be Musielak-Orlicz functions that satisfy the -condition. Then,

(i),

(ii) for .

Proof. (i) Let . Then, there exists such that

Let and choose with such that for . Write and consider

Since satisfies -condition, we have

For , we have

Since is nondecreasing and convex, it follows that

Since satisfies -condition, we have

Hence,

From (32), (34), and (38), we have . Thus, . Similarly, we can prove the other cases.

(ii) Let . Then, there exists such that

The rest of the proof follows from the following equality:

Corollary 5. Let be a Musielak-Orlicz function which satisfies -condition. Then, holds for , and .

Proof. The proof follows from Theorem 3 by putting and .

Theorem 6. The spaces and are solid.

Proof. We will prove for the space . Let . Then, there exists such that

Let be a sequence of scalars with . Then, the result follows from the following inequality:

and this completes the proof. Similarly, we can prove for the space .

Corollary 7. The spaces and are monotone.

Proof. It is easy to prove, so we omit the details.

Theorem 8. The spaces and are sequence algebra.

Proof. Let . Then,

Let . Then, we can show that

Thus, . Hence, is a sequence algebra. Similarly, we can prove that is a sequence algebra.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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