- Research Article
- Open Access

# On the Generalized -Riesz Difference Sequence Space and -Property

- Metin Başarir
^{1}Email author and - Mustafa Kayikçi
^{2}

**2009**:385029

https://doi.org/10.1155/2009/385029

© M. Başarir and M. Kayikçi. 2009

**Received:**1 May 2009**Accepted:**17 July 2009**Published:**23 August 2009

## Abstract

We introduce the generalized Riesz difference sequence space which is defined by where is the Riesz sequence space defined by Altay and Başar. We give some topological properties, compute the duals, and determine the Schauder basis of this space. Finally; we study the characterization of some matrix mappings on this sequence space. At the end of the paper, we investigate some geometric properties of and we have proved that this sequence space has property for .

## Keywords

- Positive Integer
- Real Number
- Basic Fact
- Matrix Mapping
- Sequence Space

## 1. Introduction

Let be the space of all real valued sequences. We write for the sequence spaces of all bounded, convergent, and null sequences, respectively. Also by , , and , we denote the sequence spaces of all convergent, absolutely and -absolutely, convergent series, respectively; where

Let be a sequence of positive numbers and

Then the matrix of the Riesz mean , which is triangle limitation matrix, is given by

It is well known that the matrix is regular if and only if as

Altay and Başar [1, 2] introduced the Riesz sequence space , and of nonabsolute type which is the set of all sequences whose -transforms are in the space , and respectively. Here and afterwards, will be used as a bounded sequence of strictly positive real numbers with and and denotes the collection of all finite subsets of where . The Riesz sequence space introduced in [1] by Altay and Başar is

The difference sequence spaces , and were first defined and studied by Kızmaz in [3] and studied by several authors, [4–9]. Başar and Altay [10] have studied the sequence space as the set of all sequences such that their -transforms are in the space ; that is,

where denotes the matrix defined by

The idea of difference sequences is generalized by Çolak and Et [11]. They defined the sequence spaces:

where , and , where denotes the matrix defined by

for all and for any fixed .

Recently, Başarir and Öztürk [12] introduced the Riesz difference sequence space as follows:

Başar and Altay defined the matrix which generalizes the matrix . Now we define the matrix and if we take , then it corresponds to the matrix . We define

The results related to the matrix domain of the matrix are more general and more comprehensive than the corresponding consequences of matrix domain of

Our main subject in the present paper is to introduce the generalized Riesz difference sequence space which consists of all the sequences such that their -transforms are in the space and to investigate some topological and geometric properties with respect to paranorm on this space.

## 2. Basic Facts and Definitions

In this section we give some definitions and lemmas which will be frequently used.

Definition 2.1.

By we denote the class of all matrices such that Thus, if and only if the series on the right side of (2.1) 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

Definition 2.2.

Definition 2.3.

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 is written as

Definition 2.4.

Now we give some lemmas which we need to prove our theorems.

Lemma 2.5 (see [13]).

Lemma 2.6 (see [14]).

Lemma 2.7 (see [14]).

also holds.

## 3. Some Topological Properties of Generalized -RieszDifference Sequence Space

Let us define the sequence which will be used for the -transform of a sequence , that is,

After this, by we denote the matrix defined by

for all Then we define

If we take then we have

Here are some topological properties of the generalized Riesz difference sequence space.

Theorem 3.1.

where and

Proof.

which tends to zero as Hence the continuity of the scalar multiplication has shown. Finally; it is clear to say that is a paranorm on the space

Moreover; we will prove the completeness of the space Let be any Cauchy sequence in the space where Then, for a given there exists a positive integer such that

which implies that . Since for all it follows that as so is complete.

Theorem 3.2.

Let for each Then the difference sequence space is linearly isomorphic to the space where

Proof.

define the transformation from to by . However, is a linear transformation, moreover; it is obviuos that whenever and hence is injective.

and is a paranorm on . Thus, we have that Consequently; is surjective and is paranorm preserving. Hence, is a linear bijection and this explains that the spaces and are linearly isomorphic.

Now, the Schauder basis for the space will be given in the following theorem.

Theorem 3.3.

Then; the sequence is a basis for the space and any has a unique representation of the form

where for all and

Proof.

This can be easily obtained by [12, Theorem 5] so we omit the proof.

Theorem 3.4.

Then;

Then;

Proof.

Theorem 3.5.

Then;

Then;

Proof.

Now we will characterize the matrix mappings from the space to the space . It can be proved by applying the method in [1, 12]. So we omit the proof.

Theorem 3.6.

for each

for each

## 4. -Property of Generalized Riesz Difference Sequence Space

In the previous section; we show that the sequence space which is the space of all real sequences such that is a complete paranormed space. It is paranormed by for all where We recall that a paranormed space is total if implies Every total paranormed space becomes a linear metric space with the metric given by It is clear that is a total paranormed space.

In this section, we investigate some geometric properties of . First we give the definition of the property in a paranormed space and we will use the method in [17] to prove the property Consequently, we obtain that has property for

From here, for a sequence and for , we use the notation and .

Now we give the definition of the property in a linear metric space.

Definition 4.1.

A linear metric space is said to have the property if for each and there exists such that for each element and each sequence in with for all there is an index for which

Lemma 4.2.

whenever and

Proof.

Lemma 4.3.

for all with

Proof.

Theorem 4.4.

If then has property

Proof.

for some Finally; we can say that the sequence space has property

## Declarations

### Acknowledgment

We would like to express our gratitude to the reviewer for his/her careful reading and valuable suggestions which improved the presentation of the paper.

## Authors’ Affiliations

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## Copyright

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