Abstract and Applied Analysis
Volume 2011 (2011), Article ID 213878, 14 pages
doi:10.1155/2011/213878

Research Article

Some Topological and Geometrical Properties of a New Difference Sequence Space

Serkan Demiriz¹ and Celal Çakan²

¹Department of Mathematics, Faculty of Arts and Science, Gaziosmanpaşa University, 60250 Tokat, Turkey
²Faculty of Education, Inönü University, 44280 Malatya, Turkey

Received 9 August 2010; Accepted 25 January 2011

Academic Editor: Narcisa C. Apreutesei

Copyright © 2011 Serkan Demiriz and Celal Çakan. Serkan Demiriz and . 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 introduce the new difference sequence space . Further, it is proved that the space is the BK-space including the space , which is the space of sequences of pbounded variation. We also show that the spaces , and are linearly isomorphic for . Furthermore, the basis and the , and duals of the space are determined. We devote the final section of the paper to examine some geometric properties of the space .

1. Preliminaries, Background, and Notation

By , we will denote the space of all real valued sequences. Any vector subspace of is called as a sequence space. We will write , and for the spaces of all bounded, convergent and null sequences, respectively. Also, by and ; we denote the spaces of all bounded, convergent, absolutely, and -absolutely convergent series, respectively, where .

A sequence space with a linear topology is called a -space, provided each of the maps defined by is continuous for all , where denotes the complex field and . A -space is called an FK-space, provided is a complete linear metric space. An -space whose topology is normable is called a BK-space (see [1, pages 272-273]).

Let be two sequence spaces and an infinite matrix of real or complex numbers , where . Then, we say that defines a matrix mapping from into , and we denote it by writing ; if 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 0 to . By we denote the class of all matrices such that . Thus, if and only if the series on the right side of (1.1) converges for each and every , and we have for all . A sequence is said to be -summable to if converges to which is called as the -limit of .

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 a Schauder basis (or briefly basis) for . The series which has the sum is then called the expansion of with respect to and written as

For a sequence space , the matrix domain of an infinite matrix is defined by which is a sequence space. The new sequence space generated by the limitation matrix from the space either includes the space or is included by the space , in general; that is, the space is the expansion or the contraction of the original space .

We will define by for all and denote the collection of all finite subsets of by . We will also use the convention that any term with negative subscript is equal to naught.

The approach constructing a new sequence space by means of the matrix domain of a particular limitation method has been recently employed by Wang [2], Ng and Lee [3], Malkowsky [4], and Altay et al. [5]. They introduced the sequence spaces in [2], in [3], and in [4] and in [5]; where and denote the Nörlund, arithmetic, Riesz and Euler means, respectively, and .

Recently, there has been a lot of interest in investigating geometric properties of sequence spaces besides topological and some other usual properties. In literature, there are many papers concerning the geometric properties of different sequence spaces. For example, in [6], Mursaleen et al. studied some geometric properties of normed Euler sequence space. Şimşek and Karakaya [7] investigated the geometric properties of sequence space equipped with Luxemburg norm. Further information on geometric properties of sequence space can be found in [8, 9].

The main purpose of the present paper is to introduce the difference sequence space together with matrix domain and is to derive some inclusion relations concerning with . Also, we investigate some topological properties of this new space and furthermore characterize geometric properties concerning Banach-Saks type .

2. Difference Sequence Space

In the present section, we introduce the difference sequence space and emphasize its some properties. Although the difference sequence space corresponding to the space was defined by Kızmaz [10] as follows: the difference sequence space corresponding to the space was not examined, where denotes the anyone of the spaces or . So, Başar and Altay have recently studied the sequence space , the space of -bounded variation, in [11] defined by which fills up the gap in the existing literature. Recently, Aydín and Başar [12] studied the sequence spaces and , defined by Aydín and Başar [13] introduced the difference sequence spaces and , defined by Aydín [14] introduced sequence space, defined by Define the matrix by As was made by Başar and Altay in [11], we treat slightly more different than Kızmaz and the other authors following him and employ the technique obtaining a new sequence space by the matrix domain of a triangle limitation method. We will introduce the sequence space which is a natural continuation of Aydín and Başar [13], as follows: With the notation of (1.3), we may redefine the space by

Define the sequence which will be frequently used as the -transform of a sequence , that is, Now, we may begin with the following theorem which is essential in the text.

Theorem 2.1. The set becomes the linear space with the coordinatewise addition and scalar multiplication which is the BK-space with the norm where .

Proof. Since the proof is routine, we omit the details of the proof.

Theorem 2.2. The space is linearly isomorphic to the space ; that is, , where .

Proof. It is enough to show the existence of a linear bijection between the spaces and for . Consider the transformation defined, with the notation of (2.9), from to by . The linearity of is clear. Furthermore, it is trivial that whenever , and hence, is injective.
We assume that for and define the sequence by Then, since we get that Thus, we have that . In addition, one can derive that which means that is surjective and is norm preserving. Hence, is a linear bijection.

We wish to exhibit some inclusion relations concerning with the space .

Theorem 2.3. The inclusion strictly holds for .

Proof. To prove the validity of the inclusion for , it suffices to show the existence of a number such that for every .
Let and . Then, we obtain as expected, for .
Furthermore, let us consider the sequence defined by Then, the sequence is in , as asserted.

Lemma 2.4 (see [11, Theorem  2.4]). The inclusion strictly holds for .

Combining Lemma 2.4 and Theorem 2.3, we get the following corollary.

Corollary 2.5. The inclusion strictly holds for .

3. The Basis for the Space

In the present section, we will give a sequence of the points of the space which forms a basis for the space , where .

Theorem 3.1. Define the matrix of elements of the space for every fixed by for every fixed . Then, the sequence is a basis for the space , and any has a unique representation of the form where for all and .

Proof. It is clear that , since for ; here, is the sequence whose only nonzero term is 1 in the th place for each . Let be given. For every nonnegative integer , we set Then, by applying to (3.4), we obtain with (3.3) that where . For a given , there is an integer such that for all . Hence, for all , which proves that is represented as in (3.2).
Let us show the uniqueness of representation for given by (3.2). Assume, on the contrary, that there exists a representation Since the linear transformation , from to , used in Theorem 2.2 is continuous, at this stage, we have which contradicts the fact that for all . Hence, the representation (3.2) of is unique. This step concludes the proof.

4. The -, -, and -Duals of the Space

In this section, we state and prove theorems determining the -, -, and -duals of the space . Since the case can be proved by the same analogy and can be found in the literature, we omit the proof of that case and consider only the case in the proof of Theorems 4.4–4.6.

For the sequence spaces and , define the set by With the notation of (4.1), -, - and -duals of a sequence space , which are, respectively, denoted by and , are defined by It is well-known for the sequence spaces and that and whenever , where .

We begin with to quoting the lemmas due to Stieglitz and Tietz [15], which are needed in the proof of the following theorems.

Lemma 4.1. if and only if

Lemma 4.2. if and only if

Lemma 4.3. if and only if (4.5) holds.

Theorem 4.4. Define the set by where is defined via the sequence by for all . Then, , where .

Proof. Bearing in mind the relation (2.9), we immediately derive that It follows from (4.8) that whenever if and only if whenever . This means that if and only if . Then, we derive by Lemma 4.1 with instead of that This yields the desired consequence that .

Theorem 4.5. Define the sets , and by where is defined by for all . Then, , where .

Proof. Consider the equation Thus, we deduce from Lemma 4.2 with (4.12) that whenever if and only if whenever . That is to say that if and only if . Therefore, we derive from (4.4) and (4.5) that .

Theorem 4.6. , where .

Proof. It is natural that the present theorem may be proved by the same technique used in the proof of Theorems 4.4 and 4.5, above. But, we prefer here the following classical way.
Let and . Then, we obtain by applying Hölder’s inequality that which gives us by taking supremum over that This means that . Hence, Conversely, let and . Then, one can easily see that whenever . This shows that the triangle matrix , defined by (4.11), is in the class . Hence, the condition (4.5) holds with instead of which yields that . That is to say that Therefore, by combining the inclusions (4.15) and (4.16), we deduce that the -dual of the space is the set , and this step completes the proof.

5. Some Geometric Properties of the Space

In this section, we study some geometric properties of the space .

A Banach space is said to have the Banach-Saks property if every bounded sequence in admits a subsequence such that the sequence is convergent in the norm in [16], where A Banach space is said to have the weak Banach-Saks property whenever given any weakly null sequence and there exists a subsequence of such that the sequence strongly convergent to zero.

In [17], García-Falset introduce the following coefficient: where denotes the unit ball of .

Remark 5.1. A Banach space with has the weak fixed point property, [18].

Let . A Banach space is said to have the Banach-Saks type or property , if every weakly null sequence has a subsequence such that for some , for all (see [19]).

Now, we may give the following results related to the some geometric properties, mentioned above, of the space .

Theorem 5.2. The space has the Banach-Saks type .

Proof. Let be a sequence of positive numbers for which , and also let be a weakly null sequence in . Set and . Then, there exists such that Since is a weakly null sequence implies coordinatewise, there is an such that where . Set . Then, there exists an such that By using the fact that coordinatewise, there exists an such that where .
If we continue this process, we can find two increasing subsequences and such that for each and where . Hence, On the other hand, it can be seen that . Therefore, . We have Hence, we obtain By using the fact for all , we have Hence, has the Banach-Saks type . This completes the proof of the theorem.

Remark 5.3. Note that , since is linearly isomorphic to .

Hence, by the Remarks 5.1 and 5.3, we have the following.

Theorem 5.4. The space has the weak fixed point property, where .

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