- Research Article
- Open access
- Published:
On the Generalized
-Riesz Difference Sequence Space and
-Property
Journal of Inequalities and Applications volume 2009, Article number: 385029 (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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ1_HTML.gif)
Then the matrix of the Riesz mean
, which is triangle limitation matrix, is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ3_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ4_HTML.gif)
where denotes the matrix
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ5_HTML.gif)
The idea of difference sequences is generalized by Çolak and Et [11]. They defined the sequence spaces:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ6_HTML.gif)
where , and
, where
denotes the matrix
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ7_HTML.gif)
for all and for any fixed
.
Recently, Başarir and Öztürk [12] introduced the Riesz difference sequence space as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ8_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ9_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ10_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ11_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ12_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ13_HTML.gif)
With the notation of (2.2), the -
-
-duals of a sequence space
which are, respectively, denoted by
are defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ14_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ15_HTML.gif)
( ii) Let for every
Then
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ16_HTML.gif)
Lemma 2.6 (see [14]).
( i) Let for every
Then
if and only if there exists an integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ17_HTML.gif)
( ii) Let for every
Then
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ18_HTML.gif)
Lemma 2.7 (see [14]).
Let for every
Then
if and only if (2.8), (2.9) hold, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ19_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ20_HTML.gif)
After this, by we denote the matrix
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ21_HTML.gif)
for all Then we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ22_HTML.gif)
If we take then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ23_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ24_HTML.gif)
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]:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ25_HTML.gif)
and for any [16], we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ26_HTML.gif)
It is obvious that and
for all
. Let
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ27_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ28_HTML.gif)
Again the inequalities (3.7) and (3.9) yield the subadditivity of and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ29_HTML.gif)
Let be any sequence of the elements of the space
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ30_HTML.gif)
and also be any sequence of scalars such that
Then, since the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ31_HTML.gif)
holds by subadditivity of is bounded, and thus we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ32_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ33_HTML.gif)
for all If we use the definition of
, we obtain for each fixed
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ34_HTML.gif)
for which leads us to the fact that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ35_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ36_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ37_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ38_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ39_HTML.gif)
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ40_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ41_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ42_HTML.gif)
Then; the sequence is a basis for the space
and any
has a unique representation of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ43_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ44_HTML.gif)
Then;
( ii) Let for every
Define the set
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ45_HTML.gif)
Then;
Proof.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_IEq227_HTML.gif)
Let We easily derive with the notation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ46_HTML.gif)
and the matrix which is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ47_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ48_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_IEq235_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ49_HTML.gif)
Then;
( ii) Let for every
Define the set
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ50_HTML.gif)
Then;
Proof.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_IEq244_HTML.gif)
If we take the matrix by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ51_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ52_HTML.gif)
and exists
which is shown that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ53_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_IEq253_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ54_HTML.gif)
for each
( ii) Let for every
Then
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ55_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ56_HTML.gif)
whenever and
Proof.
Let and
be given. Let
and
, there exists
such that
for all
Let
Thus
for all
. There exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ57_HTML.gif)
for all Set
There exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ58_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ59_HTML.gif)
Lemma 4.3.
If then for any
there exists
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ60_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ61_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ62_HTML.gif)
for all Then we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ63_HTML.gif)
Therefore, for each there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ64_HTML.gif)
for all Hence, there is a sequence of positive integers
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ65_HTML.gif)
for all By Lemma 4.3, there exists
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ66_HTML.gif)
for all and
Let
be a real number corresponding to Lemma 4.2 with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ67_HTML.gif)
and that is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ68_HTML.gif)
whenever and
Since
we have that
. Let
be such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ69_HTML.gif)
Put and
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ70_HTML.gif)
Hence;
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ71_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ72_HTML.gif)
By using (4.17) and convexity of the function , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ73_HTML.gif)
Hence So this implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F385029/MediaObjects/13660_2009_Article_1946_Equ74_HTML.gif)
for some Finally; we can say that the sequence space
has property
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Başarir, M., Kayikçi, M. On the Generalized -Riesz Difference Sequence Space and
-Property.
J Inequal Appl 2009, 385029 (2009). https://doi.org/10.1155/2009/385029
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DOI: https://doi.org/10.1155/2009/385029