- Open Access
Algebraic duals of some sets of difference sequences defined by Orlicz functions
© Dutta and Jebril; licensee Springer. 2014
- Received: 29 May 2014
- Accepted: 8 July 2014
- Published: 19 August 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.
- difference sequences
- Orlicz function
- algebraic duals
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.
Throughout this section X will denote one of the sequence spaces , c and .
where , for all .
Now, for subsequent use, we slightly generalize the above definition as follows.
It can be shown that is a BK-space.
It can be shown that is a BK-space.
It is trivial that if and only if . Also the norms and are equivalent.
is a linear homeomorphism.
are isometric isomorphisms for .
Hence , and are isometrically isomorphic to , and , respectively.
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.
In this section we compute the α-, β-, γ- and N-duals of the spaces , and .
Definition 2.1 
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 
Lemma 2.2 implies for some .
Hence we have the desired result. □
Hence we have the following lemma.
implies for some ,
This implies that .
The proof is similar to that of part (i). □
If we take for all in Theorem 2.4, then we obtain the following corollary.
Proof The proof is immediate using Lemma 2.3(ii). □
The following lemma will be used in the next theorem.
Lemma 2.7 
If , then .
If is convergent, 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 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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