Open Access

On the Generalized -Riesz Difference Sequence Space and -Property

Journal of Inequalities and Applications20092009:385029

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

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 .

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

(1.1)

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

(1.2)

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

(1.3)

The difference sequence spaces , and were first defined and studied by Kızmaz in [3] and studied by several authors, [49]. 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,

(1.4)

where denotes the matrix defined by

(1.5)

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

(1.6)

where , and , where denotes the matrix defined by

(1.7)

for all and for any fixed .

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

(1.8)

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

(1.9)

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.

Let and be two sequence spaces and let be an infinite matrix of real 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
(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.

For any sequence space the matrix domain of an infinite matrix is defined by
(2.2)

Definition 2.3.

If a sequence space paranormed by contains a sequence with the property that for every there is a unique sequence of scalars such that
(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.

For the sequence spaces and , define the set by
(2.4)
With the notation of (2.2), the - - -duals of a sequence space which are, respectively, denoted by are defined by
(2.5)

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

Lemma 2.5 (see [13]).

( i) Let for every Then if and only if there exists an integer such that
(2.6)
( ii) Let for every Then if and only if
(2.7)

Lemma 2.6 (see [14]).

( i) Let for every Then if and only if there exists an integer such that
(2.8)
( ii) Let for every Then if and only if
(2.9)

Lemma 2.7 (see [14]).

Let for every Then if and only if (2.8), (2.9) hold, and
(2.10)

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,

(3.1)

After this, by we denote the matrix defined by

(3.2)

for all Then we define

(3.3)

If we take then we have

(3.4)

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

Theorem 3.1.

The sequence space is a complete linear metric space paranormed by
(3.5)

where and

Proof.

The linearity of with respect to the co-ordinatewise addition and scalar multiplication follows from the inequalites which are satisfied for [15]:
(3.6)
and for any [16], we have
(3.7)
It is obvious that and for all . Let :
(3.8)
(3.9)
Again the inequalities (3.7) and (3.9) yield the subadditivity of and
(3.10)
Let be any sequence of the elements of the space such that
(3.11)
and also be any sequence of scalars such that Then, since the inequality
(3.12)
holds by subadditivity of is bounded, and thus we have
(3.13)

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

(3.14)
for all If we use the definition of , we obtain for each fixed that
(3.15)
for which leads us to the fact that
(3.16)
is a Cauchy sequence of real numbers for every fixed Since is complete, it converges, so we write as Hence by using these infinitely many limits , we define the sequence . Since (3.14) holds for each and
(3.17)
Take any first let in (3.17) and then to obtain Finally, taking in (3.17) and letting we have Minkowski's inequality for each , that is,
(3.18)

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.

For the proof of the theorem, we should show the existence of a linear bijection between the spaces and for With the notation of
(3.19)

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

Let and define the sequence by
(3.20)
Then,
(3.21)
where
(3.22)

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.

Define the sequence of the elements of the space for every fixed by
(3.23)

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

(3.24)

where for all and

Proof.

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

Theorem 3.4.

( i) Let for every Define the set as follows:
(3.25)

Then;

( ii) Let for every Define the set by
(3.26)

Then;

Proof.

Let We easily derive with the notation
(3.27)
and the matrix which is defined by
(3.28)
for all , thus, by using the method in [1], [12] we deduce that whenever if and only if whenever From Lemma 2.5(i), we obtain the desired result that
(3.29)
This is easily obtained by proceeding as in the proof of (i), above by using the second part of Lemma 2.5. So we omit the detail.

Theorem 3.5.

( i) Let for every Define the set as follow:
(3.30)

Then;

( ii) Let for every Define the set by
(3.31)

Then;

Proof.

If we take the matrix by
(3.32)
for and if we carry out the method which is used in [1, 12], we get that whenever if and only if whenever Hence we deduce from Lemma 2.7 that
(3.33)
and exists which is shown that
(3.34)
This may be obtained in the similar way as in the proof of (i) above by using the second part of Lemmas 2.6 and 2.7. So we omit the detail.

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.

( i) Let for every Then if and only if there exists an integer such that
(3.35)

for each

( ii) Let for every Then if and only if
(3.36)

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.

If then for any and and for any there exists such that
(4.1)

whenever and

Proof.

Let and be given. Let and , there exists such that for all Let Thus for all . There exists such that
(4.2)
for all Set There exists such that
(4.3)
for all Set Assume that and We recall that and . With these notations, let and By using convexity of the function for all and the fact that for and where and , we have
(4.4)

Lemma 4.3.

If then for any there exists and such that
(4.5)

for all with

Proof.

Let be a real number such that Then there exists such that for all Let be a real number such that Then for each and we have
(4.6)

Theorem 4.4.

If then has property

Proof.

Let and with for Take There exists such that Let Since for each is bounded, by using the diagonal method, we have that for each , we can find a subsequence of such that converges for all with Since is Cauchy sequence for all there exists such that
(4.7)
for all Then we see that
(4.8)
Therefore, for each there exists such that
(4.9)
for all Hence, there is a sequence of positive integers with such that
(4.10)
for all By Lemma 4.3, there exists and such that
(4.11)
for all and Let be a real number corresponding to Lemma 4.2 with
(4.12)
and that is
(4.13)
whenever and Since we have that . Let be such that
(4.14)
Put and Then
(4.15)
Hence;
(4.16)
(4.17)
By using (4.17) and convexity of the function , we have
(4.18)
Hence So this implies that
(4.19)

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

(1)
Department of Mathematics, Sakarya University
(2)
Duzce MYO, Duzce University

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Copyright

© M. Başarir and M. Kayikçi. 2009

This article is published under license to BioMed Central Ltd. 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.