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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 364743, 10 pages

http://dx.doi.org/10.1155/2013/364743

## Sequence Spaces Defined by Musielak-Orlicz Function over -Normed Spaces

^{1}Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India^{2}Department of Mathematics, Model Institute of Engineering & Technology, Kot Bhalwal 181122, Jammu and Kashmir, India^{3}Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

Received 21 July 2013; Accepted 16 September 2013

Academic Editor: Abdullah Alotaibi

Copyright © 2013 M. Mursaleen et al. 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 over *n*-normed spaces defined
by a Musielak-Orlicz function . We also study some topological properties and prove
some inclusion relations between these spaces.

#### 1. Introduction and Preliminaries

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

Lindenstrauss and Tzafriri [1] 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 as an Orlicz sequence space. The space is a Banach space with the norm

It is shown in [1] 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 functions is called a Musielak-Orlicz function (see [2, 3]). 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 linear metric space. A function : is called paranorm if(1) for all ,(2) for all , (3) for all , (4) is a sequence of scalars with and is a sequence of vectors with ; then .

A paranorm for which implies 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 [4], Theorem , pp. 183). For more details about sequence spaces, see [5–12] and references therein.

A sequence of positive integers is called lacunary if , and as . The intervals determined by will be denoted by and . The space of lacunary strongly convergent sequences was defined by Freedman et al. [13] as

Strongly almost convergent sequence was introduced and studied by Maddox [14] and Freedman et al. [13]. Parashar and Choudhary [15] have introduced and examined some properties of four sequence spaces defined by using an Orlicz function , which generalized the well-known Orlicz sequence spaces , , and . It may be noted here that the space of strongly summable sequences was discussed by Maddox [16] and recently in [17].

Mursaleen and Noman [18] 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, and it is said that a sequence is -convergent to the number , called the -limit of if as , where

The sequence is -bounded if . It is well known [18] that if in the ordinary sense of convergence, then This implies that which yields that and hence is -convergent to .

The concept of 2-normed spaces was initially developed by Gähler [19] in the mid 1960s, while for that of -normed spaces one can see Misiak [20]. Since then, many others have studied this concept and obtained various results; see Gunawan ([21, 22]) and Gunawan and Mashadi [23]. Let and let be a linear space over the field , where is the field of real or complex numbers of dimension , where . A real valued function on satisfying the following four conditions (1) if and only if are linearly dependent in ;(2) is invariant under permutation; (3) for any ; (4)is called an -norm on , and the pair is called an -normed space over the field .

For example, if we may take being equipped with the -norm = the volume of the -dimensional parallelepiped spanned by the vectors which may be given explicitly by the formula where for each , leting be an -normed space of dimension and be linearly independent set in , then the following function on defined by defines an -norm on with respect to .

A sequence in an -normed space is said to converge to some if

A sequence in an -normed space is said to be Cauchy if

If every Cauchy sequence in converges to some , then is said to be complete with respect to the -norm. Any complete -normed space is said to be -Banach space.

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

If we take , we get

If we take for all , we have

The following inequality will be used throughout the paper. If , , then for all and . Also for all .

In this paper, we introduce sequence spaces defined by a Musielak-Orlicz function over -normed spaces. We 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 spaces over the field of complex number .*

*Proof. *Let , and let . In order to prove the result, we need to find some such that

Since , there exist positive numbers such that
Define . Since is nondecreasing, convex function and by using inequality (20), we have
Thus, we have . Hence, is a linear space. Similarly, we can prove that and are linear spaces.

Theorem 2. *Let be a Musielak-Orlicz function, and let be a bounded sequence of positive real numbers. Then is a topological linear space paranormed by
**
where .*

*Proof. *Clearly for . Since , we get . Again if , then
This implies that for a given , there exist some such that
Thus,
Suppose that for each . This implies that for each . Let , then
It follows that
which is a contradiction. Therefore, for each , and thus for each . Let and be the case such that
Let ; then, by using Minkowski's inequality, we have
Since , and are nonnegative, we have
Therefore, . Finally we prove that the scalar multiplication is continuous. Let be any complex number. By definition,
Thus,
where . Since , we have
So the fact that scalar multiplication is continuous follows from the above inequality. This completes the proof of the theorem.

Theorem 3. *Let be a Musielak-Orlicz function. If for all fixed , then .*

*Proof. *Let ; then there exists positive number such that
Define . Since is nondecreasing and convex and by using inequality (20), we have
Hence, .

Theorem 4. *Let and let be Musielak-Orlicz functions satisfying -condition, then one has*(i)*;
*(ii)*;
*(iii)*. *

*Proof. *Let , then we have

Let and choose with such that for . Let for all . We can write
So, we have
For . Since are nondecreasing and convex, it follows that
Since satisfies -condition, we can write
Hence,
From (40) and (43), we have . This completes the proof of . Similarly we can prove that

Theorem 5. *Let . Then for a Musielak-Orlicz function which satisfies -condition, one has*(i)*;
*(ii)*;
*(iii)*. *

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

Theorem 6. *Let be a Musielak-Orlicz function and let . Then if and only if
**
for some .*

*Proof. *Let . Suppose that (45) does not hold. Therefore, there are subinterval of the set of interval and a number , where
such that
Let us define as follows:
Thus, by (47),. But . Hence, (45) must hold.

Conversely, suppose that (45) holds and let . Then for each ,
Suppose that . Then for some number , there is a number such that for a subinterval , of the set of interval ,
From properties of sequence of Orlicz functions, we obtain
which contradicts (45), by using (49). Hence, we get

This completes the proof.

Theorem 7. *Let be a Musielak-Orlicz function. Then the following statements are equivalent:*(i)*;*(ii)*;*(iii)*. *

*Proof. *(i) (ii). Let (i) hold. To verify (ii), it is enough to prove
Let . Then for , there exists , such that
Hence, there exists such that
So, we get .

(ii) (iii). Let (ii) hold. Suppose (iii) does not hold. Then for some
and therefore we can find a subinterval , of the set of interval , such that
Let us define as follows:

Then . But by (57), , which contradicts (ii). Hence, (iii) must holds.

(iii) (i). Let (iii) hold and suppose that . Suppose that ; then
Let for each ; then by (59),
which contradicts (iii). Hence, (i) must hold.

Theorem 8. *Let be a Musielak-Orlicz function. Then the following statements are equivalent:*(i)*;*(ii)*;*(iii)*. *

*Proof. *(i) (ii). It is obvious.

(ii) (iii). Let (ii) hold. Suppose that (iii) does not hold. Then
and we can find a subinterval , of the set of interval , such that
Let us define as follows:

Thus, by (62), , but , which contradicts (ii). Hence, (iii) must hold.

(iii) (i). Let (iii) hold. Suppose that . Then
Again suppose that ; for some number and a subinterval , of the set of interval , we have
Then from properties of the Orlicz function, we can write
Consequently, by (64), we have
which contradicts (iii). Hence, (i) must hold.

Theorem 9. *Let for all and let be bounded. Then
*

*Proof. *Let ; write
and for all . Then for all . Take for . Define sequences and as follows.

For , let and , and for , let and . Then clearly for all , we have
Now it follows that and . Therefore,
Now for each ,
and so
Hence, . This completes the proof of the theorem.

Theorem 10. *
i If for all , then
**
ii If , for all , then
*

*Proof. *(i) Let ; then
Since , this implies that
therefore,
Hence,
(ii) Let for each and . Let ; then for each , we have
Since , we have
Therefore, , for each . Hence,

This completes the proof of the theorem.

Theorem 11. *If , for all , then
*

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

#### Acknowledgment

The authors are very grateful to the referees for their valuable suggestions and comments. The third author also acknowledges the partial support by University Putra Malaysia under the project ERGS 1-2013/5527179.

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