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On Some New Sequence Spaces in 2-Normed Spaces Using Ideal Convergence and an Orlicz Function
Journal of Inequalities and Applications volume 2010, Article number: 482392 (2010)
Abstract
The purpose of this paper is to introduce certain new sequence spaces using ideal convergence and an Orlicz function in 2-normed spaces and examine some of their properties.
1. Introduction
The notion of ideal convergence was introduced first by Kostyrko et al. [1] as a generalization of statistical convergence which was further studied in topological spaces [2]. More applications of ideals can be seen in [3, 4].
The concept of 2-normed space was initially introduced by Gähler [5] as an interesting nonlinear generalization of a normed linear space which was subsequently studied by many authors (see, [6, 7]). Recently, a lot of activities have started to study summability, sequence spaces and related topics in these nonlinear spaces (see, [8–10]).
Recall in [11] that an Orlicz function is continuous, convex, nondecreasing function such that
and
for
, and
as
.
Subsequently Orlicz function was used to define sequence spaces by Parashar and Choudhary [12] and others.
If convexity of Orlicz function, is replaced by
, then this function is called Modulus function, which was presented and discussed by Ruckle [13] and Maddox [14].
Note that if is an Orlicz function then
for all
with
Let be a normed space. Recall that a sequence
of elements of
is called to be statistically convergent to
if the set
has natural density zero for each
.
A family of subsets a nonempty set
is said to be an ideal in
if (i)
(ii)
imply
(iii)
imply
, while an admissible ideal
of
further satisfies
for each
, [9, 10].
Given is a nontrivial ideal in
. The sequence
in
is said to be
-convergent to
if for each
the set
belongs to
, [1, 3].
Let be a real vector space of dimension
where
A 2-norm on
is a function
which satisfies (i)
if and only if
and
are linearly dependent, (ii)
, (iii)
, and (iv)
. The pair
is then called a
-normed space [6].
Recall that is a 2-Banach space if every Cauchy sequence in
is convergent to some
in
Quite recently Savaş [15] defined some sequence spaces by using Orlicz function and ideals in 2-normed spaces.
In this paper, we continue to study certain new sequence spaces by using Orlicz function and ideals in 2-normed spaces. In this context it should be noted that though sequence spaces have been studied before they have not been studied in nonlinear structures like -normed spaces and their ideals were not used.
2. Main Results
Let be a nondecreasing sequence of positive numbers tending to
such that
and let
be an admissible ideal of
, let
be an Orlicz function, and let
be a 2-normed space. Further, let
be a bounded sequence of positive real numbers. By
we denote the space of all sequences defined over
. Now, we define the following sequence spaces:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482392/MediaObjects/13660_2010_Article_2163_Equ1_HTML.gif)
where
The following well-known inequality[16, page 190] will be used in the study.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482392/MediaObjects/13660_2010_Article_2163_Equ2_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482392/MediaObjects/13660_2010_Article_2163_Equ3_HTML.gif)
for all and
. Also
for all
.
Theorem 2.1.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482392/MediaObjects/13660_2010_Article_2163_IEq76_HTML.gif)
are linear spaces.
Proof.
We will prove the assertion for only and the others can be proved similarly. Assume that
and
, so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482392/MediaObjects/13660_2010_Article_2163_Equ4_HTML.gif)
Since is a 2-norm, and
is an Orlicz function the following inequality holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482392/MediaObjects/13660_2010_Article_2163_Equ5_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482392/MediaObjects/13660_2010_Article_2163_Equ6_HTML.gif)
From the above inequality, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482392/MediaObjects/13660_2010_Article_2163_Equ7_HTML.gif)
Two sets on the right hand side belong to and this completes the proof.
It is also easy to see that the space is also a linear space and we now have the following.
Theorem 2.2.
For any fixed ,
is paranormed space with respect to the paranorm defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482392/MediaObjects/13660_2010_Article_2163_Equ8_HTML.gif)
Proof.
That and
are easy to prove. So we omit them.
-
(iii)
Let us take
and
in
. Let
(2.9)
Let and
, then if
, then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482392/MediaObjects/13660_2010_Article_2163_Equ10_HTML.gif)
Thus, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482392/MediaObjects/13660_2010_Article_2163_Equ11_HTML.gif)
(iv) Finally using the same technique of Theorem of Savaş [15] it can be easily seen that scalar multiplication is continuous. This completes the proof.
Corollary 2.3.
It should be noted that for a fixed the space
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482392/MediaObjects/13660_2010_Article_2163_Equ12_HTML.gif)
which is a subspace of the space is a paranormed space with the paranorms
for
and
.
Theorem 2.4.
Let ,
be Orlicz functions. Then we have
(i) provided
is such that
.
(ii)⊆
.
Proof.
-
(i)
For given
first choose
such that
Now using the continuity of
choose
such that
. Let
. Now from the definition
(2.13)
Thus if then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482392/MediaObjects/13660_2010_Article_2163_Equ14_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482392/MediaObjects/13660_2010_Article_2163_Equ15_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482392/MediaObjects/13660_2010_Article_2163_Equ16_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482392/MediaObjects/13660_2010_Article_2163_Equ17_HTML.gif)
Hence from above using the continuity of we must have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482392/MediaObjects/13660_2010_Article_2163_Equ18_HTML.gif)
which consequently implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482392/MediaObjects/13660_2010_Article_2163_Equ19_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482392/MediaObjects/13660_2010_Article_2163_Equ20_HTML.gif)
This shows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482392/MediaObjects/13660_2010_Article_2163_Equ21_HTML.gif)
and so belongs to . This proves the result.
-
(ii)
Let
, then the fact
(2.22)
gives us the result.
Definition 2.5.
Let be a sequence space. Then
is called solid if
whenever
for all sequences
of scalars with
for all
.
Theorem 2.6.
The sequence spaces are solid.
Proof.
We give the proof for only. Let
and let
be a sequence of scalars such that
for all
. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F482392/MediaObjects/13660_2010_Article_2163_Equ23_HTML.gif)
where . Hence
for all sequences of scalars
with
for all
whenever
.
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Savaş, E. On Some New Sequence Spaces in 2-Normed Spaces Using Ideal Convergence and an Orlicz Function. J Inequal Appl 2010, 482392 (2010). https://doi.org/10.1155/2010/482392
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DOI: https://doi.org/10.1155/2010/482392