Spaces of Ideal Convergent Sequences

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.