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An Orlicz extension of difference sequences on real linear n-normed spaces
Journal of Inequalities and Applications volume 2013, Article number: 232 (2013)
In this paper, we present an extension of some classes of difference sequences by considering them in a base space X, a real linear n-normed space via a sequence of Orlicz functions. We investigate the spaces for linearity, existence of norms and completeness under different conditions. We also show that they are convex spaces and compute their topologically equivalent spaces. Further some results on equivalence of various norms on such extended spaces are presented.
MSC:40A05, 46B20, 46B99, 46A19, 46A99.
1 Introduction and preliminaries
Let w denote the space of all real or complex sequences. By c, and , we denote the Banach spaces of convergent, null and bounded sequences , respectively, normed by
An Orlicz function is a function which is continuous, non-decreasing and convex with , for and as .
Lindenstrauss and Tzafriri  used the Orlicz function and introduced the sequence space as follows:
They proved that is a Banach space normed by
The study of Orlicz sequence spaces was initiated with a certain specific purpose in Banach space theory. Indeed, Lindberg got interested in Orlicz spaces in connection with finding Banach spaces with symmetric Schauder bases having complementary subspaces isomorphic to or (). Subsequently, Lindenstrauss and Tzafriri  studied these Orlicz sequence spaces in more detail, and solved many important and interesting structural problems in Banach spaces. Later on, different classes of sequence spaces defined by an Orlicz function were studied by different authors. For details, one may refer to Kamthan and Gupta .
The notion of a difference sequence space was introduced by Kizmaz  who studied the difference sequence spaces , and . The notion was further generalized by Et and Colak  by introducing the spaces , and . Another type of generalization of the difference sequence spaces is due to Tripathy and Esi  who studied the spaces , and . Tripathy et al.  generalized the above notions and unified these as follows.
Let m, s be non-negative integers. Then, for a given sequence space Z, we have
where and for all , which is equivalent to the following binomial representation:
Let m, s be non-negative integers. Then, for a given sequence space Z, Dutta  introduced the following spaces:
where and for all , which is equivalent to the following binomial representation:
The concept of 2-normed spaces was introduced and studied by Gähler, a German Mathematician who worked at German Academy of Science, Berlin, in a series of papers in the German language published in Mathematische Nachrichten; see, for example, references [8–13]. This notion, which is nothing but a two-dimensional analogue of a normed space, got the attention of a wider audience after the publication of a paper by White  in 1969 entitled 2-Banach spaces. In the same year Gähler published another paper on this theme in the same journal. Siddiqi delivered a series of lectures on this theme in various conferences in India and Iran. His joint paper with Gähler and Gupta  of 1975 also provided valuable results related to the theme of this paper. The notion of n-normed spaces can be found in Misiak . Since then, many others have studied this concept and obtained various results; see, for instance, Gunawan [17, 18], Gunawan and Mashadi [8, 19], Dutta [20–22] and Gürdal et al. . For some related and recent works in this area, one may refer to Chu and Ku  and Tanaka and Saito .
Let and X be a real vector space of dimension d, where . A real-valued function on satisfying the following four conditions:
(N1) if and only if are linearly dependent,
(N2) is invariant under permutation,
(N3) for any ,
(N4) is called an n-norm on X, and the pair is called an n-normed space.
A trivial example of an n-normed space is equipped with the following Euclidean n-norm:
where for each .
If is an n-normed space of dimension and is a linearly independent set in X, then the following function on defined by
defines an -norm on X with respect to , and this is known as derived -norm on X.
The standard n-norm on X, which is a real inner product space of dimension , is given as follows:
where denotes the inner product on X. If , then this n-norm is exactly the same as the Euclidean n-norm mentioned earlier. For , this n-norm is the usual norm .
A sequence in an n-normed space is said to converge to some in the n-norm if
A sequence in an n-normed space is said to be Cauchy with respect to the n-norm if
If every Cauchy sequence in X converges to some , then X is said to be complete with respect to the n-norm. Any complete n-normed space is said to be an n-Banach space.
Now we state the following useful results on the n-norm as lemmas which were given in .
Lemma 1.1 Every n-normed space is an -normed space for all . In particular, every n-normed space is a normed space.
Lemma 1.2 A standard n-normed space is complete if and only if it is complete with respect to the usual norm .
Lemma 1.3 On a standard n-normed space X, the derived -norm , defined with respect to an orthonormal set , is equivalent to the standard -norm . Precisely, we have
for all , where .
Let be a real linear n-normed space and let denote an X-valued sequence space. Then, for a sequence of Orlicz functions , we define the following difference sequence spaces:
Similarly, we can define , and .
In the above definition of spaces, the n-norm on X is either a standard n-norm or a non-standard n-norm. In general, we write , and for a standard case, we write . For a derived norm, we use .
It is obvious that . Again, follows from the following inequality:
Similarly, we have . Also, it is obvious that for , , and , .
When , , and for all and , the above spaces deduce to the famous and very useful spaces c, and .
2 Main results
In this section we investigate some results on the n-norm as well as the main results of this article involving the sequence spaces , , , , and .
The proofs of the following two propositions are easy and so they are omitted.
Proposition 2.1 Let and X be a real vector space of dimension d, where . Let be the collection of linearly independent sets B with elements. For , let us define
Then is a seminorm on X and the family of seminorms generates a locally convex topology on X.
Proposition 2.2 The seminorms ’s have the following properties:
for , and , we have
Hence we have the following proposition.
Proposition 2.3 The seminorms defined by Proposition 2.1 satisfy the axiom of an n-norm.
Example 2.1 Consider the linear space of real polynomials of degree ≤m on the interval . Let be arbitrary but distinct fixed points in . For in , let us define
Then is an n-norm on .
Proof The proof is a routine verification and so it is omitted. □
Theorem 2.1 The spaces , , , , and are linear.
Proof The proof is a routine verification and thus it is omitted. □
Theorem 2.2 (i) , and are normed linear spaces by
, and are normed linear spaces by(2.2)
where is the derived 1-norm (norm) on X.
Proof (i) If , then clearly . Conversely, assume . Then using (2.1), we have
This implies that for a given , there exists some () such that
Hence, for every ,
Suppose for some i. Let , then .
It follows that as for some . This is a contradiction.
So, we must have for all . Let , then and so , by taking , for . Thus, taking , we can easily conclude that for all .
Let and be any two elements. Then there exist such that
Let . Then, by the convexity of M, we have
Finally, let α be any scalar. Then
This completes the proof.
The proof follows by applying similar arguments as above. □
Remark It is obvious that if and only if for . Moreover, it is clear that the norms and are equivalent.
Theorem 2.3 (i) The spaces , and are topologically isomorphic with the spaces , and , respectively.
The spaces , and are topologically isomorphic with the spaces , and , respectively, where is a subspace of , .
, , , , and are convex sets.
Proof (i) For , let us consider the mapping , defined by
Clearly T is linear homeomorphism.
In this case we consider the mapping , defined by
Clearly is a linear homeomorphism.
The proof follows by using the convexity of Orlicz functions. □
Remark Let be a linearly independent set in X. Then , is a derived -norm on X for each and for each .
Hence we have the following theorem.
Theorem 2.4 Let be any linearly independent set in X. Then , and are normed linear spaces by
We call these norms derived norms.
Proof Proof is similar to that of Theorem 2.2. □
Theorem 2.5 Let X be an n-Banach space. Then , and are Banach spaces under the norm (2.1).
Proof Let Y be any one of the spaces , and . Let be any Cauchy sequence Y. Let be fixed and be such that for a , and . Then there exists a positive integer such that
Using (2.1), we get
Hence we have
It follows that for every ,
For with , for all , we have
This implies that
Hence is a Cauchy sequence in X for all . Since X is an n-Banach space, is convergent in X for all . For simplicity, let for each . By taking , we can conclude that
Now we can find that
Then, using the continuity of Orlicz functions, we have
Hence we have
It follows that . Since and Y is a linear space, so we have .
This completes the proof of the theorem. □
The following corollary is due to Lemma 1.2.
Corollary 2.6 If X is a Banach space under the standard n-norm, then , and are Banach spaces under the norm
For the following results, let us assume Y to be any one of the spaces , and .
Theorem 2.7 If converges to an x in Y in the norm defined by (2.1), then also converges to x in the derived norm defined by (2.3) for .
Proof Let converge to x in Y in the norm . Then
Using (2.1), we get
So, for any linearly independent set , we have
Hence by (1.1), we get
Hence converges to x in the norm . □
If X is equipped with the standard n-norm and the derived norm is with respect to an orthonormal set, then the converse of the above theorem is also true.
Theorem 2.8 Let X be a standard n-normed space and the derived -norm on X is with respect to an orthonormal set. Then is convergent in Y in the norm defined by (2.1) if and only if is convergent in Y in the derived norm defined by (2.3) for .
Proof In view of the above theorem, it is enough to prove that is convergent in the norm implies is convergent in the norm .
Let converge to x in Y in the norm . Then
Using (2.3) for , we get
Now one may observe that
where and on the right-hand side denote the standard -norm and the usual norm on X, respectively. Since the derived -norm on X is with respect to an orthonormal set, using Lemma 1.3, we have
and in this case on the right-hand side is the derived -norm which we used to define the norm .
It follows that
Hence as .
That is, converges to x in Y in the norm . □
Using Lemma 1.3, we get the following corollary.
Corollary 2.9 Let X be a standard n-normed space and let the derived -norms on X be with respect to an orthonormal set. Then a sequence in Y is convergent in the norm defined by (2.1) if and only if it is convergent in the derived norm and, by induction, in the derived norm defined by (2.3) for all . In particular, a sequence in Y is convergent in the norm if and only if it is convergent in the derived norm , defined by
Theorem 2.10 Let X be a standard n-normed space and let the derived -norms on X for all be with respect to an orthonormal set. Then Y is complete with respect to the norm defined by (2.1) if and only if it is complete with respect to the derived norm defined by (2.3). By induction, Y is complete with respect to the norm if and only if it is complete with respect to the derived norm defined by (2.4).
Proof By replacing the phrases ‘ converges to x’ with ‘ is Cauchy’ and ‘’ with ‘’, we see that the analogues of Theorem 2.7, Theorem 2.8 and Corollary 2.9 hold for Cauchy sequences. This completes the proof. □
Remark Analogues of Theorem 2.4, Theorem 2.5, Corollary 2.6, Theorem 2.7, Theorem 2.8, Corollary 2.9 and Theorem 2.10 hold for the spaces , and .
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The author is grateful to three anonymous referees for careful reading of the paper and for helpful comments that improved the presentation of the paper. The author is also grateful to the University Grant Commission, New Delhi-110002, India for sponsoring a MRP under F. No. 39-935/2010 (SR) in the area of this research paper.
The author declares that they have no competing interests.
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Dutta, H. An Orlicz extension of difference sequences on real linear n-normed spaces. J Inequal Appl 2013, 232 (2013). https://doi.org/10.1186/1029-242X-2013-232
- Orlicz function
- difference operator
- convex space
- topological equivalence