Abstract
We introduced the weak ideal convergence of new sequence spaces combining an infinite matrix of complex numbers and Musielak-Orlicz function over normed spaces. We also study some topological properties and inclusion relation between these spaces.
1. Introduction
Throughout the paper , , , , and denote the classes of all, bounded, convergent, null, and -absolutely summable sequences of complex numbers. The sets of natural numbers and real numbers will be denoted by , , respectively, and will denote an admissible ideal in ; , will denote a normed linear space and its continuous dual, respectively. Many authors studied various sequence spaces using normed or seminormed linear spaces. In this paper, using an infinite matrix of complex numbers and the notion of weak ideal, we aimed to introduce some new sequence spaces with Musielak-Orlicz function in normed spaces. By an ideal we mean a family of subsets of a nonempty set satisfying the following: (i) ; (ii) imply ; (iii) imply , while an admissible ideal of further satisfies for each . The notion of ideal convergence was introduced first by . Kostyrko et al. [1] as a generalization of statistical convergence. Given that is a nontrivial ideal in , the sequence in a normed space is said to be -convergent to if, for each , A sequence in a normed space is said to be -bounded if there exists such that A sequence in a normed space is said to be -Cauchy if, for each , there exists a positive integer such that Recently different classes of sequences have been introduced using ideal convergence; see [2, 3]. Following [4, 5], Pehlivan et al. [6] have introduced the concepts of weak -convergence and weak -Cauchy sequence in a normed space and investigated their basic properties. A sequence in a normed space is said to be weak -convergent to if, for each and for each , the set A sequence in a normed space is said to be weak -bounded for each if there exists such that A sequence in a normed space is said to be weak -Cauchy if, for each and for each , there exists a positive integer such that An Orlicz function is a function which is continuous, nondecreasing, and convex with , for , and , as . If convexity of is replaced by , then it is called a modulus function, introduced by Nakano [7]. Ruckle [8] and Maddox [9] used the idea of a modulus function to construct some spaces of complex sequences. An Orlicz function is said to satisfy -condition for all values of if there exists a constant , such that . The -condition is equivalent to for all values of and for . Lindenstrauss and Tzafriri [10] used the idea of an Orlicz function to define the following sequence spaces: which is a Banach space with the Luxemburg norm defined by The space is closely related to the space , which is an Orlicz sequence space with for . Recently different classes of sequences have been introduced using Orlicz functions. See [11–14]. A sequence of Orlicz functions for all is called a Musielak-Orlicz function.
2. Definitions and Preliminaries
Let be a sequence; then denotes the set of all permutations of the elements of ; that is,
Definition 1. A sequence space is said to be symmetric if for all .
Definition 2. A sequence space is said to be normal (or solid) if , whenever and for all sequence of scalars with for all .
Let and let be a sequence space. A -step space of is a sequence space . A canonical preimage of a sequence is a sequence defined as
A canonical preimage of a step space is a set of canonical preimages of all elements in ; that is, is in canonical preimage of if and only if is canonical preimage of some .
Definition 3. A sequence space is said to be monotone if contains the canonical preimages of all its step spaces.
Lemma 4. Every normal space is monotone.
For any bounded sequence of positive numbers, we have the following well known inequality.
If and , then for all and .
3. Main Results
In this section, we define some new weak ideal convergent sequence spaces and investigate their linear topological structures. We find out some relations related to these sequence spaces. Let be a weak admissible ideal of , let be a Musielak-Orlicz function, and let and be two nonempty subsets of the space of complex sequences. Let , be an infinite matrix of complex numbers. We write if converges for each . Further, let be any bounded sequence of positive real numbers:
Let us consider a few special cases of the above sets.(1)If for all , then the above classes of sequences are denoted by , , , and , respectively.(2)If for all , then the above classes of sequences are denoted by , , , and , respectively.(3)If for all and , then the above classes of sequences are denoted by , , , and , respectively.(4)If we take for all and as then we denote the above classes of sequences by , , , and , respectively.(5)If we take and as where is a nondecreasing sequence of positive numbers tending to , , and , then we denote the above classes of sequences by , , , and .(6)By a lacunary , , where , we will mean an increasing sequence of nonnegative integers with as . The interval determined by will be denoted by and and let as Then we denote the above classes of sequences by , , , and , respectively.(7)If , for all and , then the above classes of sequences are denoted by , , and , respectively.
Theorem 5. The spaces , , and are linear spaces.
Proof. We will prove the assertion for ; the others can be proved similarly. Assume that , , and . Then, there exist and such that the sets Since is linear and the Orlicz function is convex for all , the following inequality holds: where . On the other hand from the above inequality we get Since the two sets on the right hand side belong to , this completes the proof.
Theorem 6. The spaces , , and are paranormed spaces with respect to the paranorm defined by where .
Proof. Clearly and , where is the zero element of . Let and . Then, for , we set
Let , , and ; then we have
Let where , and let as . We have to show that as . We set
If and , by using nondecreasing and convexity of the Orlicz function for all , we obtain that
From the above inequality, it follows that
and consequently
Note that for all . Hence, by our assumption, the right hand of (24) tends to 0 as , and the result follows. This completes the proof of the theorem.
Theorem 7. Let , , and be Musielak-Orlicz functions. Then, the following hold:(a), provided be such that ,(b).
Proof. (a) Let be given. Choose such that . Using the continuity of the Orlicz function , choose such that implies that .
Let be any element in ; put
Then, by definition of ideal convergent, we have the set . If , then we have
Using the continuity of the Orlicz function for all and the relation (26), we have
Consequently, we get
This shows that
This proves the assertion.
(b) Let be any element in . Then, by the following inequality, the results follow:
Theorem 8. Let for all ; then .
Proof. Let ; then there exists some such that This implies that for sufficiently large value of . Since for all is nondecreasing, we get Thus, . This completes the proof of the theorem.
Theorem 9. (i) If , then .
(ii) If , then .
Proof. (i) Let ; since , then we have
and hence .
(ii) Let and . Then for each there exists a positive integer such that
for all . This implies that
Thus and this completes the proof.
Theorem 10. For any sequence of Orlicz functions which satisfies -condition, one has .
Proof. Let , and let be given. Then, there exists such that the set By taking and let and choose with such that for all and for . Consider Since is continuous for all , we have For , we use the fact that . Since is nondecreasing and convex, it follows that Since satisfies -condition, Hence By putting (39) and (42) in (38), we get This proves that .
Theorem 11. Let and let be bounded; then
Proof. Let ; we put and for all . Then for all . Let be such that for all . Define the sequences and as follows: for , let and ; for let and . Then clearly, for all we have , , , and . Therefore, we have Hence .
Theorem 12. For any two sequences and of positive real numbers and for any two norms and on , the following holds: where , , and .
Proof. Proof of the theorem is obvious, because the zero element belongs to each of the sequence spaces involved in the intersection.
Theorem 13. The sequence spaces are normal as well as monotone, where .
Proof. We will give the proof for only. Let and let be a sequence of scalars such that for all . Then, we have where ; hence, . By Lemma 4, we have that the space is monotone.
Note. It is clear from the definitions that
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
Many thanks are due to the editor and anonymous referee for careful reading of the paper and valuable suggestions which helped in improving an earlier version of it.