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.
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.
2 Computation of algebraic duals
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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