The Scientific World Journal

Volume 2014 (2014), Article ID 398203, 7 pages

http://dx.doi.org/10.1155/2014/398203

Research Article

On Generalized Difference Hahn Sequence Spaces

Kuldip Raj¹ and Adem Kiliçman²

¹School of Mathematics, Shri Mata Vaishno Devi University, Katra 182320, India

²Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia

Received 17 March 2014; Accepted 2 May 2014; Published 13 May 2014

Academic Editor: S. A. Mohiuddine

Copyright © 2014 Kuldip Raj and Adem Kiliçman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We construct some generalized difference Hahn sequence spaces by mean of sequence of modulus functions. The topological properties and some inclusion relations of spaces are investigated. Also we compute the dual of these spaces, and some matrix transformations are characterized.

1. Introduction and Preliminaries

By a sequence space, we understand a linear subspace of the space of all real or complex-valued sequences, where denotes the complex field and . For , we write , , and for the classical spaces of all bounded, convergent, and null sequences, respectively. Also by , , and we denote the space of all bounded, convergent, and -absolutely convergent series, which are Banach spaces with the following norms: and , respectively. Additionally, the spaces and are defined by

A coordinate space (or a -space) is a vector space of numerical sequences, where addition and scalar multiplication are defined pointwise. That is, a sequence space with a linear topology is called a -space provided that each of the maps defined by is continuous for all . A -space is a -space, which is also a Banach space with continuous coordinate functionals , . A -space is called an -space provided that is a complete linear metric space. An -space whose topology is normable is called a -space. If a normed sequence space contains a sequence with the property that for every there is a unique sequence of scalars such that

then is called Schauder basis (or briefly basis) for . The series which has the sum is then called the expansion of with respect to and written as . An -space is said to have property, if and is a basis for , where is a sequence whose only nonzero term is in place for each and , the set of all finitely nonzero sequences. If is dense in , then is called an -space, and thus implies .

The notion of difference sequence spaces was introduced by Kizmaz [1], who defined the sequence spaces as follows:

where . The notion was further generalized by Et and Çolak [2] by introducing the spaces. Let be a nonnegative integer; then,

where and for all . The generalized difference sequence has the following binomial representation:

Later concept have been studied by Bektaş et al. [3] and Et and Esi [4]. Another type of generalization of the difference sequence spaces is due to Tripathy and Esi [5] who studied the spaces , , and . Recently, Esi et al. [6] and Tripathy et al. [7] have introduced a new type of generalized difference operators and unified those as follows.

Let , be nonnegative integers; then, for a given sequence space, we have

for , and , where and for all .

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 as and is a sequence of vectors with as , then as .

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 [8], Theorem 10.4.2, pp. 183). For more details about sequence spaces (see [9, 10]) and the references therein.

A modulus function is a function such that

(1) if and only if ,(2), for all ,(3) is increasing,(4) is continuous from the right at .

It follows that must be continuous everywhere on . The modulus function may be bounded or unbounded. For example, if we take , then is bounded. If , , then the modulus function is unbounded. Subsequentially, modulus function has been discussed in ([11–14]) and references therein.

Let and be two sequence spaces and be an infinite matrix of complex numbers, where , . Then, we say that defines a matrix mapping from into , and we denote it by writing for every sequence . The sequence , the -transform of , is in , where

For simplicity in notation, here and in what follows, the summation without limits runs from to . By , we denote the class of all matrices such that . Thus, if and only if the series on the right side of (7) converges for each and each and we have for all . A sequence is said to be -summable to if converges to which is called the -limit of .

The matrix domain of an infinite matrix in a sequence space is defined by

which is a sequence space (for several examples of matrix domains, see [15] p. 49–176). In [16], Başar and Altay have defined the sequence space which consists of all sequences such that -transforms of them are in , where denotes the matrix as follows:

for all , . The space has been studied by Başar et al. [17], where

Hahn [18] introduced the -space of all sequences such that

where , for all . The following norm:

was defined on the space by Hahn [18] (and also [19]). Rao ([20], Proposition 2.1) defined a new norm on as . G. Goes and S. Goes [19] proved that the space is a -space.

Hahn proved the following properties of the space .

Lemma 1. (i) is a Banach space.

(ii) .

(iii) .

In [19], G. Goes and S. Goes studied functional analytic properties of the -space . Additionally, G. Goes and S. Goes considered the arithmetic means of sequences in and and used an important fact which the sequence of arithmetic means of is a quasiconvex null sequence. And also G. Goes and S. Goes proved that .

Rao [20] studied some geometric properties of Hahn sequence space and gave the characterizations of some classes of matrix transformations. Balasubramanian and Pandiarani [21] defined the new sequence space called the Hahn sequence space of fuzzy numbers and proved that and duals of is the Cesàro space of the set of all fuzzy bounded sequences. Kirişci [22] compiled to studies on Hahn sequence space and defined a new Hahn sequence space by Cesàro mean in [23].

In [24], Kirişci introduce the sequence space by

where , for all . If we take , which are called Hahn sequence spaces. We denote the collection of all finite subsets of by .

Let be a sequence of modulus functions, be a bounded sequence of positive real numbers, and be a sequence of strictly positive real numbers. In the present paper we defined the following sequence space:

where and for all . Define the sequence , which will be frequently used, by -transform of a sequence ; that is,

where with

for all , .

If we take , and for all , then we get the sequence space defined by [24] Kirişci. By taking , , and for all , we obtained a Hahn sequence space defined by Hanh [18].

The following inequality will be used throughout the paper.

Let be a sequence of positive real numbers with , and let . Then, for the factorable sequences and in the complex plane, we have

The main purpose of this paper is to study some difference Hahn sequence spaces by mean of sequence of modulus functions. We will study some topological and algebraic properties of the sequence spaces in Section 2. In Section 3 we will determine the -, -, and -duals of the spaces . Finally, we also made an attempt to characterize some matrix transformations on the spaces .

2. Main Results

The purpose of this section is to study the properties like linearity, paranorm, and relevant inclusion relations in the spaces .

Theorem 2. The sequence space is a linear space over the complex field .

Proof. Let , and , . Then their exist integers and such that and . By using the inequality (17) and the properties of modulus function, we have

Thus, . This proves that is a linear space over the field of complex number .

Theorem 3. Let be a sequence of modulus functions and be any sequence of strictly positive real numbers. Then is a paranormed space with the paranorm defined by

where and .

Proof. Clearly for all . It is trivial that for . Since using Minkowski’s inequality, we have

Hence . Finally to check the continuity of scalar multiplication, let us take a complex number by definition, we have

where is a positive integer such that . Let for any fixed with . By definition for , we have

Also for , taking small enough, since is continuous for each , we have

Equations (22) and (23) imply that as . This completes the proof.

Theorem 4. Let be a sequence of modulus functions and . Then .

Proof. Let . By definition of , we have , for all . Since , we have for all . Let . Thus, we have

Which implies that . This completes the proof.

Theorem 5. .

Proof. We consider

Then, for ,

and from , , we obtain

For each positive integer , we get

and as ,

and . Then and

Let , and we consider

Then the series is convergent from the definition of . Also, , and therefore .

Then,

From (30) and (32), we have

Theorem 6. The sequence space is a BK-space with AK.

Proof. If is any sequence, we write . Let and . Then, their exists such that

for all . Now let be given. Then, we have for all by (34)

whence for all . This shows that .

Since is an AK-space and every AK-space is AD, we can give the following corollary.

Corollary 7. The sequence space is AD.

3. Duals of Hahn Sequence Space

In this section, we determining the , , and -duals of the sequence space . Let and be sequences, and be subsets of , and be an infinite matrix of complex numbers. We write , and = for the multiplier space of and . In the special cases of , we write , , and , , for the -dual, -dual, and -dual of . By , we denote the sequence in the -row of , and we write and , provided that for all .

Given an -space containing , its conjugate is denoted by and its -dual or sequential dual is denoted by and is given by . Let be a sequence space. Then is called perfect if , normal if whenever for some , and monotone if contains the canonical preimages of all its stepspace.

Lemma 8. (i) if and only if

(ii) if and only if

Lemma 9. (i) if and only if

(ii) if and only if (39) holds and

Lemma 10. (i) if and only if (38) holds.

(ii) if and only if (40) holds with .

Lemma 11. (i) if and only if (38) holds and

Lemma 12. (i) if and only if (41) holds and

Theorem 13. We define the set

Then, .

Proof. Let us take any . We define the matrix by

for all, , .

Consider the equation

It follows from (45) with Lemma 8(ii) that whenever if and only if , whenever . This means that , whenever if and only if . This gives the result that .

Theorem 14. Let . Then , where

Proof. Consider the equation

where are defined by

for all, . Thus, we deduce from Lemma 9(ii) with (47) that whenever if and only if whenever . Thus and by (39) and (40), respectively. Nevertheless, the inclusion holds, and thus, we have , whence .

Lemma 15. Let be FK-space with . Then,

(i); (ii)If has AK, ;(iii)If has AD, .

From Theorem 6, Corollary 7, and Lemma 15, we can write the following corollary.

Corollary 16. (i) ;

(ii) .

Lemma 17. Let be a sequence space. Then, the following assertions are true:

(i) is perfect is normal is monotone;(ii) is normal ;(iii) is monotone .

Combining Theorem 13, Theorem 14, and Lemma 17, we can give the following corollary.

Corollary 18. The space is not monotone and so it is neither normal nor perfect.

4. Matrix Transformations

In this section we characterize some matrix transformations on the space .

Lemma 19. Let , be any two sequence spaces, be an infinite matrix, and be a triangle matrix. Then, if and only if .

If we define , then we can give the following corollary from Lemma 19 with defined by (16).

Corollary 20. (i) if and only if

(ii) if and only if

Theorem 21. Suppose that the entries of the infinite matrices and are connected with the relation

for all , where and is any sequence space. Then if and only if , for all and .

Proof. Let be any given sequence spaces. Suppose that (51) holds between and , and take into account that the spaces and are norm isomorphic. Let and take any . Then, exists and , which yields that for each . Hence, exists, and thus,

for all . We have which leads us to the consequence . Conversely, let for all , and hold, and take any . Then, exists. Therefore, we obtain from the equality that

for all . Thus, and this shows that .

If we use the Corollary 20 and change the roles of the spaces with in Theorem 21, we can give the following theorem.

Theorem 22. Suppose that the entries of the infinite matrices and are connected with the relation for all and is any sequence space. Then if and only if .

Proof. Let and consider the following equality:

which yields that as for all . Therefore, one can observe from here that whenever if and only if whenever .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors express their sincere gratitude to the referees for the careful and detailed reading of the paper and for the very helpful suggestions that improved the paper substantially. The authors also gratefully acknowledge that this research was partially supported by the University Putra Malaysia under the GP-IBT Grant Scheme having Project no. GP-IBT/2013/9420100.

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