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Sequence Spaces Defined by Musielak-Orlicz Function over -Normed Spaces
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  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  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 , 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.  as
Strongly almost convergent sequence was introduced and studied by Maddox  and Freedman et al. . Parashar and Choudhary  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  and recently in .
Mursaleen and Noman  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  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  in the mid 1960s, while for that of -normed spaces one can see Misiak . Since then, many others have studied this concept and obtained various results; see Gunawan ([21, 22]) and Gunawan and Mashadi . 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
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.
i If for all , then
ii If , for all