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On Generalized Paranormed Statistically Convergent Sequence Spaces Defined by Orlicz Function
Journal of Inequalities and Applications volume 2009, Article number: 729045 (2009)
Abstract
We define generalized paranormed sequence spaces ,
,
, and
defined over a seminormed sequence space
. We establish some inclusion relations between these spaces under some conditions.
1. Introduction
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_IEq6_HTML.gif)
will represent the spaces of all, convergent, null, statistically convergent, statistically null, bounded, bounded statistically convergent, and bounded statistically null -valued sequence spaces throughout the paper, where
is a seminormed space, seminormed by
For
the space of complex numbers, these spaces represent the
which are the spaces of all, convergent, null, statistically convergent, statistically null, bounded, bounded statistically convergent, and bounded statistically null sequences, respectively. The zero sequence is denoted by
, where
is the zero element of
.
The idea of statistical convergence was introduced by Fast [1] and studied by various authors (see [2–4]). The notion depends on the density of subsets of the set of natural numbers. A subset
of
is said to have density
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ1_HTML.gif)
where is the characteristic function of
.
A sequence is said to be statistically convergent to the number
(i.e.,
) if for every
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ2_HTML.gif)
In this case, we write or stat
lim
Let be a mapping of the set of positive integers into itself. A continuous linear functional
on
, the space of real bounded sequences, is said to be an invariant mean or
-mean if and only if
(1) when the sequence
has
for all
,
(2), where
(3) for all
The mappings are one to one and such that
for all positive integers
and
where
denotes the
th iterate of the mapping
at
. Thus
extends the limit functional on
, the space of convergent sequences, in the sense that
for all
. In that case
is translation mapping
, a
-mean is often called a Banach limit, and
, the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequences [5].
If , set
It can be shown [6] that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ3_HTML.gif)
where
Several authors including Schaefer [7], Mursaleen [6], Sava [8], and others have studied invariant convergent sequences.
An Orlicz function is a function , which is continuous, nondecreasing, and convex with
for
and
as
. If the convexity of an Orlicz function
is replaced by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ4_HTML.gif)
then this function is called modulus function, introduced and investigated by Nakano [9] and followed by Ruckle [10], Maddox [11], and many others.
Lindenstrauss and Tzafriri [12] used the idea of Orlicz function to construct the sequence space
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ5_HTML.gif)
which is called an Orlicz sequence space.
The space becomes a Banach space with the norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ6_HTML.gif)
The space is closely related to the space
which is an Orlicz sequence space with
for
. Orlicz sequence spaces were introduced and studied by Parashar and Choudhary [13], Bhardwaj and Singh [14], and many others.
It is well known that since is a convex function and
then
for all
with
.
An Orlicz funtion is said to satisfy
-condition for all values of
, if there exists constant
, such that
. The
-condition is equivalent to the inequality
for all values of
and for
being satisfied [15].
The notion of paranormed space was introduced by Nakano [16] and Simons [17]. Later on it was investigated by Maddox [18], Lascarides [19], Rath and Tripathy [20], Tripathy and Sen [21], Tripathy [22], and many others.
The following inequality will be used throughout this paper. Let be a sequence of positive real numbers with
and let
. Then for
, the set of complex numbers for all
, we have [23]
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ7_HTML.gif)
2. Definitions and Notations
A sequence space is said to be solid (or normal) if
, whenever
and for all sequences
of scalars with
for all
.
A sequence space is said to be symmetric if
implies
, where
is a permutation of
.
A sequence space is said to be monotone if it contains the canonical preimages of its step spaces.
Throughout the paper will represent a sequence of positive real numbers and
a seminormed space over the field
of complex numbers with the seminorm
. We define the following sequence spaces:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ8_HTML.gif)
We write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ9_HTML.gif)
If ,
for each
and
then these spaces reduce to the spaces
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ10_HTML.gif)
defined by Tripathy and Sen [21].
Firstly, we give some results; those will help in establishing the results of this paper.
Lemma 2.1 ([21]).
For two sequences and
one has
if and only if
, where
such that
Lemma 2.2 ([21]).
Let and
, then the followings are equivalent:
(i) and
(ii)
Lemma 2.3 ([21]).
Let be an infinite subset of
such that
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ11_HTML.gif)
Then is uncountable.
Lemma 2.4 ([24]).
If a sequence space is solid then
is monotone.
3. Main Results
Theorem 3.1.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_IEq127_HTML.gif)
, ,
,
are linear spaces.
Proof.
Let . Then there exist
,
positive real numbers and
,
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ12_HTML.gif)
Let ,
be scalars and let
. Then by (1.7) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ13_HTML.gif)
Hence is a linear space.
The rest of the cases will follow similarly.
Theorem 3.2.
The spaces and
are paranormed spaces, paranormed by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ14_HTML.gif)
where .
Proof.
We prove the theorem for the space . The proof for the other space can be proved by the same way. Clearly
for all
and
. Let
Then we have
,
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ15_HTML.gif)
Let . Then by the convexity of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ16_HTML.gif)
Hence from above inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ17_HTML.gif)
For the continuity of scalar multiplication let be any complex number. Then by the definition of
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ18_HTML.gif)
where .
Since we have
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ19_HTML.gif)
and therefore converges to zero when
converges to zero or
converges to zero.
Hence the spaces and
are paranormed by
.
Theorem 3.3.
Let be complete seminormed space, then the spaces
and
are complete.
Proof.
We prove it for the case and the other case can be established similarly. Let
be a Cauchy sequence in
for all
. Then
, as
. For a given
, let
and
to be such that
. Then there exists a positive integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ20_HTML.gif)
Using definition of paranorm we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ21_HTML.gif)
Hence is a Cauchy sequence in
. Therefore for each
there exists a positive integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ22_HTML.gif)
Using continuity of , we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ23_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ24_HTML.gif)
Taking infimum of such s we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ25_HTML.gif)
for all and
. Since
and
is continuous, it follows that
. This completes the proof of the theorem.
Theorem 3.4.
Let and
be two Orlicz functions satisfying
-condition. Then
(i)
(ii)
where ,
,
and
.
Proof.
-
(i)
We prove this part for
and the rest of the cases will follow similarly. Let
. Then for a given
, there exists
such that there exists a subset
of
with
, where
(3.15)
If we take then
implies that
. Hence we have by convexity of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ27_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ28_HTML.gif)
Hence by (3.15) it follows that for a given , there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ29_HTML.gif)
Therefore
-
(ii)
We prove this part for the case
and the other cases will follow similarly.
Let . Then by using (1.7) it can be shown that
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ30_HTML.gif)
This completes the proof.
Theorem 3.5.
For any sequence of positive real numbers and for any two seminorms
and
on
one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ31_HTML.gif)
where ,
,
and
.
Proof.
The proof follows from the fact that the zero sequence belongs to each of the classes the sequence spaces involved in the intersection.
The proof of the following result is easy, so omitted.
Proposition 3.6.
Let be an Orlicz function which satisfies
condition, and let
and
be two seminorms on
. Then
(i)
(ii)
(iii) where
,
,
, and
,
(iv)if is stronger than
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ32_HTML.gif)
where ,
,
and
.
Theorem 3.7.
The spaces are not solid, where
and
.
Proof.
To show that the spaces are not solid in general, consider the following example. Let ,
for all
,
, where
and
for all
. Then we have
for all
. Consider the sequence
, where
is defined by
and
for each fixed
. Hence
for
and
. Let
if
is odd and
, otherwise. Then
for
and
. Thus
is not solid for
and
.
The proof of the following result is obvious in view of Lemma 2.4.
Proposition 3.8.
The space is solid as well as monotone for
and
.
Theorem 3.9.
The spaces are not symmetric, where
,
,
and
.
Proof.
To show that the spaces are not symmetric, consider the following examples. Let ,
for all
,
, where
and
for all
. Then we have
for all
. We consider the sequence
defined by
if
and
otherwise. Then
for
and
. Let
be a rearrangement of
, which is defined as
if
is odd and
, otherwise. Then
for
and
.
To show for and
, let
for all
odd and
for all
even. Let
and
, where
. Let
and
for all
. Then we have
for all
. We consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ33_HTML.gif)
Then for
and
. We consider the rearrengement
of
as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ34_HTML.gif)
Then for
and
. Thus the spaces
are not symmetric in general, where
,
,
and
.
Proposition 3.10.
For two sequences and
one has
if and only if
, where
such that
.
Proof.
The proof is obvious in view of Lemma 2.1.
The following result is a consequence of the above result.
Corollary 3.11.
For two sequences and
one has
if and only if
and
, where
such that
.
The following result is obvious in view of Lemma 2.2.
Proposition 3.12.
Let and
, then the followings are equivalent:
(i) and
(ii)
Theorem 3.13.
Let be a sequence of nonnegative bounded real numbers such that
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ35_HTML.gif)
Proof.
Let . Then for a given
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ36_HTML.gif)
where the vertical bar indicates the number of elements in the enclosed set.
From the above inequality it follows that .
Conversely let . Let
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ37_HTML.gif)
For a given , let
.
Let .
Since , so
, uniformly in
, as
. There exits a positive integer
such that
for all
. Then for all
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ38_HTML.gif)
Hence .
This completes the proof of the theorem.
The following result is a consequence of the above theorem.
Corollary 3.14.
Let and
be two bounded sequences of real numbers such that
and
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F729045/MediaObjects/13660_2009_Article_1998_Equ39_HTML.gif)
Since the inclusion relations and
are strict, we have the following result.
Corollary 3.15.
The spaces and
are nowhere dense subsets of
The following result is obvious in view of Lemma 2.3.
Proposition 3.16.
The spaces and
are not separable.
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The authors would like to express their gratitude to the reviewers for their careful reading and valuable suggestions which improved the presentation of the paper.
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Başarir, M., Altundağ, S. On Generalized Paranormed Statistically Convergent Sequence Spaces Defined by Orlicz Function. J Inequal Appl 2009, 729045 (2009). https://doi.org/10.1155/2009/729045
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DOI: https://doi.org/10.1155/2009/729045