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Algebraic duals of some sets of difference sequences defined by Orlicz functions
Journal of Inequalities and Applications volume 2014, Article number: 300 (2014)
In this paper, we first give a description of some sets of sequences generated by difference operators and defined by Orlicz functions. Then their algebraic duals such as the α-, β-, γ- and null-duals are computed.
MSC:46A45, 47N40, 65J99, 46A20.
Let w denote the space of all scalar sequences, and any subspace of w is called a sequence space. Let , c and be the linear spaces of bounded, convergent and null sequences with complex terms, respectively, normed by , where , the set of positive integers.
Lindenstrauss and Tzafriri  used the Orlicz function and introduced the sequence space as follows:
They proved that is a Banach space normed by
Throughout this section X will denote one of the sequence spaces , c and .
The notion of difference sequence spaces was introduced by Kizmaz . For some other works on difference sequences, Orlicz functions and related literature, we refer to [3–8]. Let be any fixed sequence of non-zero complex numbers. Et and Esi  generalized the above sequence spaces to the following sequence spaces:
In this paper, for an Orlicz function M, we can have the following spaces in the line of the spaces studied by Mursaleen et al. :
In fact, we get the following spaces:
where , for all .
Bektaş et al.  introduced the difference operator and defined it as follows:
Using this difference operator, we can construct the following sequence space:
is defined by
Now, for subsequent use, we slightly generalize the above definition as follows.
Now, for , let us define
It can be shown that is a BK-space.
Again for , let us define
It can be shown that is a BK-space.
It is trivial that if and only if . Also the norms and are equivalent.
Let us define the operator
by , where . It is trivial that D is a bounded linear operator on , , c and . Furthermore, the set
is a subspace of and normed by
and are equivalent as topological spaces since
is a linear homeomorphism.
are isometric isomorphisms for .
Hence , and are isometrically isomorphic to , and , respectively.
Moreover, for , which follows from the following inequality and convexity of M:
Investigation of spaces is often combined with that of duals. The algebraic dual space is defined for all vector spaces. When defined for a topological vector space, there is a subspace of this dual space, corresponding to continuous linear functionals, which constitutes a continuous dual space. For any finite-dimensional normed vector space or topological vector space, such as Euclidean n-space, the continuous dual and the algebraic dual coincide. This is, however, false for any infinite-dimensional normed space. For some related literature on duality relevant to this paper, we refer to [9, 11, 13, 14]. Our results of this paper will generalize few existing results as well as generate some new results in the literature of algebraic duality within the field of functional analysis.
2 Computation of algebraic duals
In this section we compute the α-, β-, γ- and N-duals of the spaces , and .
Definition 2.1 
Let X be a sequence space and define
then , , and are called the α-, β-, γ- and N-(or null) duals of X, respectively. It is known that if , then for .
Lemma 2.1 
Let m be a positive integer. Then there exist positive constants and such that
Lemma 2.2 implies for some .
Proof Let , then
Then there exists such that
Taking , for an arbitrary fixed positive integer k, by the subadditivity of modulus, the monotonicity and convexity of M:
Then the above inequality, the inequality
and the convexity of M imply
Hence we have the desired result. □
Hence we have the following lemma.
implies for some ,
Theorem 2.4 Let M be an Orlicz function. Then
Proof (i) Let , then . Now, for any , we have . Then we have
Conversely, suppose that for . Then for each . So we can take
This implies that .
Again suppose that and . Then there exists a strictly increasing sequence of positive integers with such that
Then we have
This contradicts . Hence . This completes the proof of (i).
The proof is similar to that of part (i). □
If we take for all in Theorem 2.4, then we obtain the following corollary.
Corollary 2.5 Let M be an Orlicz function. Then
Theorem 2.6 Let M be an Orlicz function. Then , where
Proof The proof is immediate using Lemma 2.3(ii). □
The following lemma will be used in the next theorem.
Lemma 2.7 
Let be a sequence of positive numbers increasing monotonically to infinity.
If , then .
If is convergent, then .
Theorem 2.8 Let M be an Orlicz function and denote the set of all positive null sequences. Then
Proof We give the proof for part (i) for , and the proof of other parts follows similarly using Lemma 2.7. For details, one may refer to .
For each , there exists one and only one such that
for sufficiently large k by (1.1). Let . Then, using the same technique as applied in [, p.429], we can show that .
Let . Again, using the same technique as applied in [, pp.429-430], we can show that .
This completes the proof. □
Although we conclude this paper here, the following further suggestion remains open:
What is the N-dual of the space ?
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The authors declare that they have no competing interests.
HD carried out the initial description of the spaces. IHJ proposed the structure of the paper. HD and IHJ both formulated and proved the results. All authors read and approved the final manuscript.
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Dutta, H., Jebril, I.H. Algebraic duals of some sets of difference sequences defined by Orlicz functions. J Inequal Appl 2014, 300 (2014). https://doi.org/10.1186/1029-242X-2014-300
- difference sequences
- Orlicz function
- algebraic duals