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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 817659, 6 pages
The Hahn Sequence Space Defined by the Cesáro Mean
Department of Mathematical Education, Hasan Ali Yücel Education Faculty, Istanbul University, Vefa, Fatih, 34470 Istanbul, Turkey
Received 11 April 2013; Revised 4 July 2013; Accepted 1 August 2013
Academic Editor: Feyzi Başar
Copyright © 2013 Murat Kirişci. 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.
The -space of all sequences is given as such that converges and is a null sequence which is called the Hahn sequence space and is denoted by . Hahn (1922) defined the space and gave some general properties. G. Goes and S. Goes (1970) studied the functional analytic properties of this space. The study of Hahn sequence space was initiated by Chandrasekhara Rao (1990) with certain specific purpose in the Banach space theory. In this paper, the matrix domain of the Hahn sequence space determined by the Cesáro mean first order, denoted by , is obtained, and some inclusion relations and some topological properties of this space are investigated. Also dual spaces of this space are computed and, matrix transformations are characterized.
By a sequence space, we understand a linear subspace of the space of all complex sequences which contains , the set of all finitely nonzero sequences, where denotes the complex field and . 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, absolutely, and -absolutely convergent series, 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 , for all . 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 the Schauder basis (or briefly basis) for . The series which has the sum is then called the expansion of with respect to , and it is 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 the place for each and , the set of all finitely nonzero sequences. If is dense in , then is called an -space; thus, implies .
Let and be two sequence spaces, and let be an infinite matrix of the complex numbers , where . Then, we say that defines a matrix mapping from into , and we denote it by writing if exists and belongs to for every sequence , where , the -transform of with
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 (3) 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. If is triangle, that is, and for all , then one can easily observe that the sequence spaces and are linearly isomorphic; that is, . There are several examples of the matrix domain of an infinite matrix in a sequence space in Chapter in . By , we will denote the collection of all finite subsets of .
Hahn  introduced the space and proved that the following statements hold:(i) is a Banach space with the norm ,(ii),(iii).
2. The New Hahn Sequence Space
Hahn  proved that . Now, we give some inclusion relations.
Theorem 1. The following inclusions are strict: (a),(b),(c).
Proof. (a) It is clear that from . Now, we show that this inclusion is strict. Let us consider the sequence , , and . Then, . Since the sequence is in but not in , then .
(b) Since  and , then .
(c) We choose the sequence . Since , then is not in , but it is in . Thus, we see that is strict.
Theorem 2. The sequence space is a -space with the norm
Lemma 3 (see ). The -space has an property.
Since , then one has the following.
Theorem 4. The -space does not have an property.
Theorem 5. One has the following:
Proof. It is similar to the proof of [3, Theorem 3.2].
Theorem 6. The sequence space is norm isomorphic to the space ; that is, .
Proof. To prove this, we will show the existence of a linear bijection between the spaces and . Consider the transformation defined, with the notation of (7), from to by . The linearity of is clear. Furthermore, it is trivial that whenever , and, hence, is injective.
Let , and define the sequence by . Then, we have Consequently, we see from here that is surjective and is norm preserving. Hence, is a linear bijection which, therefore, shows that the spaces and are norm isomorphic, as desired.
Theorem 7. Define a sequence of elements of the space for every fixed by Then, the sequence is a basis for the space , and any has a unique representation of the form where for all .
Proof. It is clear that , since
Let be given. For every nonnegative integer , we put Then, we obtain by applying to (14) with (13) that Given that , then there is an integer such that for all . Hence, for all , which proves that is represented as in (12).
To show the uniqueness of this representation, we assume that . Since the linear transformation , from to , used in Theorem 6 is continuous, then we have at this stage that which contradicts the fact that for all . Hence, the representation (12) of is unique.
3. Duals of the Sequence Space
In this section, we state and prove the theorems determining the -, -, and -duals of the sequence space .
The set defined by is called the multiplier space of the sequence spaces and . One can easily observe for a sequence space with that the inclusions hold. With the notation of (19), the alpha-, beta-, and gamma-duals of a sequence space , which are, respectively, denoted by , , and are defined by
The alpha-, beta-, and gamma-duals of a sequence space are also referred to as the Köthe-Toeplitz dual, the generalized Köthe-Toeplitz dual, and the Garling dual of a sequence space, respectively.
Given an -space containing , its conjugate is denoted by , and its -dual or sequential dual is denoted by and is given by all sequences .
We need the following lemmas.
Lemma 8 (see ). Let be defined via a sequence , and let the inverse of the triangle matrix be defined by for all . Then,
Theorem 10. The -dual of the space is the set
Proof. Let . We define the matrix via the sequence by Bearing in mind the relation (7), we immediately derive that We, therefore, observe by (30) that whenever if and only if whenever . Then, we derive by Lemma 9 (i) that which yields the result that .
Hahn  proved that .
Theorem 11. Consider the following:
Theorem 12. One has the following:
Proof. This is obtained in the similar way used in the proof of Theorem 11.
4. Matrix Transformations
Let us suppose throughout that the sequences and are connected with (7), and let the -transform of the sequence be , and let the -transform of the sequence be ; that is, It is clear here that the method is applied to the -transform of the sequence , while the method is directly applied to the terms of the sequence . So the methods and are essentially different.
Following Şengönül and Başar , we give some knowledge about the dual summability methods of the new type. Let us assume the existence of the matrix product . We will say in this situation that the methods and in (36) are the dual of the new type if is reduced to (or becomes ) under the application of the formal summation by parts. This leads us to the fact that exists and is equal to and formally holds, if one side exists. This statement is equivalent to the relation
Now, we may give the following theorem.
Theorem 13. Let and be the dual matrices of the new type, and be any given sequence space. Then, let if and only if and for every fixed .
Proof. Suppose that and are dual matrices of the new type; that is to say that (37) holds; let be any given sequence space, and take account that the spaces and are linearly isomorphic.
Let , and take any . Then, exists, and , which yields that for each . Hence, exists, and, thus, letting in the equality we have by (37) that , which leads us to the consequence that .
Conversely, let , and (38) hold, and take any . Then, we have , which gives together with (38) that for each . Hence, exists. Therefore, we obtain from the equality as that , and this shows that . This completes the proof.
By the changing roles of the spaces and in Theorem 13, we have the following.
Theorem 14. Suppose that the elements of the infinite matrices and are connected with the relation for all , and let be any given sequence space. Then, if and only if .
Corollary 15. (i) if and only if (23) hold with instead of .
(ii) if and only if (24) and (25) hold with instead of .
(iii) if and only if and (24) hold with instead of .
(iv) if and only if (24) hold with instead of .
(v) if and only if (26) and (27) hold with instead of .
Hahn  defined the space and gave some of its general properties. G. Goes and S. Goes  studied the functional analytic properties of the space . The study of the Hahn sequence space was initiated by Chandrasekhara Rao  with a certain specific purpose in the Banach space theory. Also Chandrasekhara Rao  computed some matrix transformations. Chandrasekhara Rao and Subramanian  introduced a new class of sequence spaces called semi-replete spaces. Chandrasekhara Rao and Subramanian  defined the semi-Hahn space and proved that the intersection of all the semi-Hahn spaces is the Hahn space. Balasubramanian and Pandiarani  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.
The sequence space was defined by Hahn , and G. Goes and S. Goes  and Chandrasekhara Rao et al. [5, 8, 10] investigated some properties of the space . In exception of these works, there has not been any work related to the Hahn sequence space. In this paper, the Hahn sequence space defined by the Cesáro mean worked as follows. In Section 2, the new Hahn sequence space is determined by the Cesáro mean, and some properties of this space are investigated. In Section 3, -, -, and -duals of the new Hahn sequence space are computed. In Section 4, the matrix classes and are characterized, where is an arbitrary sequence space, and some results of these characterizations are given.
We can define the matrix domain of an arbitrary triangle , compute its -, -, and -duals, and characterize the matrix transformations on them into the classical sequence spaces, and almost the convergent sequence space is a new result.
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