#### 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