![MathJax Logo](/templates/jsp/_style2/_tandf/pb2/images/math-jax.gif)
Abstract
In this article, we introduce the Zweier Paranorm I-convergent double sequence spaces and
for
, a sequence of positive real numbers. We study some algebraic and topological properties on these spaces.
Public Interest Statement
The term sequence has a great role in analysis. Sequence spaces play an important role in various fields of real analysis, complex analysis, functional analysis and Topology. They are very useful tools in demonstrating abstract concepts through constructing examples and counter examples. Convergence of sequences has always remained a subject of interest to the researchers. Later on, the idea of statistical convergence came into existence which is the generalization of usual convergence. Statistical convergence has several applications in different fields of Mathematics like Number Theory, Trigonometric Series, Summability Theory, Probability Theory, Measure Theory, Optimization and Approximation Theory. The notion of Ideal convergence (I-convergence) is a generalization of the statistical convergence and equally considered by the researchers for their research purposes since its inception.
1. Introduction
Let ,
and
be the sets of all natural, real and complex numbers, respectively. We write
the space of all real or complex sequences.
Let and
denote the Banach spaces of bounded, convergent and null sequences, respectively, normed by
.
The following subspaces of were first introduced and discussed by Maddox (Citation1969).
where is a sequence of strictly positive real numbers.
After then Lascarides (Citation1971,Citation1983) defined the following sequence spaces
where , for all
.
A double sequence of complex numbers is defined as a function . We denote a double sequence as
where the two subscripts run through the sequence of natural numbers independent of each other. A number
is called a double limit of a double sequence
if for every
there exists some
such that (Khan & Sabiha, Citation2011)
Therefore we have,
the space of all real or complex double sequences.
Each linear subspace of , for example,
is called a sequence space.
The notion of I-convergence is a generalization of the statistical convergence. At the initial stage it was studied by Kostyrko, Šalát, and Wilczynski (Citation2000). Later on it was studied by Šalát, Tripathy, and Ziman (Citation2004), Tripathy and Hazarika (Citation2009) and Demirci (Citation2001).
2. Preliminaries and definitions
Here, we give some preliminaries about the notion of I-convergence and Zweier sequence spaces. For more details one refer to Das, Kostyrko, Malik, and Wilczyński (Citation2008), Gurdal and Ahmet (Citation2008), Khan and Khan (Citation2014a,Citation2014b), Mursaleen and Mohiuddine (Citation2010,Citation2012).
Definition 2.1
If is a metric space, a set
is said to be nowhere dense if its closure
contains no sphere, or equivalently if
has no interior points.
Definition 2.2
Let X be a non-empty set. Then a family of sets I(
denoting the power set of X) is said to be an ideal in X if
(i) |
| ||||
(ii) | I is finitely additive i.e. A, B | ||||
(iii) | I is hereditary i.e. A |
A non-trivial ideal I is maximal if there cannot exist any non-trivial ideal JI containing I as a subset.
Definition 2.3
A double sequence is said to be
(i) | I-convergent to a number L if for every | ||||
(ii) | A double sequence | ||||
(iii) | A double sequence | ||||
(iv) | A double sequence |
Definition 2.4
A double sequence space E is said to be solid or normal if E implies
E for all sequence of scalars
with
for all i,j
.
Definition 2.5
Let X be a linear space. A function is called a paranorm, if for all
(i) | |||||
(ii) | |||||
(iii) | |||||
(iv) | If |
A sequence space with linear topology is called a K-space provided each of maps
defined by
is continuous for all
.
A K-space is called an FK-space provided
is a complete linear metric space.
An FK-space whose topology is normable is called a BK-space.
Let and
be two sequence spaces and
is an infinite matrix of real or complex numbers
, where
. Then we say that A defines a matrix mapping from
to
, and we denote it by writing
.
If for every sequence the sequence
, the A transform of x is in
, where
(1)
(1)
By , we denote the class of matrices A such that
Thus, if and only if series on the right side of (1) converges for each
and every
The approach of constructing new sequence spaces by means of the matrix domain of a particular limitation method have been recently employed by Altay, Başar, and Mursaleen (Citation2006), Başar and Altay (Citation2003), Malkowsky(Citation1997), Ng and Lee (Citation1978) and Wang (Citation1978).
Şengönül (Citation2007) defined the sequence which is frequently used as the
transform of the sequence
i.e.
where and
denotes the matrix
defined by
Following Basar and Altay (Citation2003), Şengönül (Citation2007), introduced the Zweier sequence spaces and
as follows
Here, we quote below some of the results due to Şengönül (Citation2007) which we will need in order to establish the results of this article.
Theorem 2.1
The sets and
are the linear spaces with the co-ordinate wise addition and scalar multiplication which are the BK-spaces with the norm
Theorem 2.2
The sequence spaces and
are linearly isomorphic to the spaces c and
, respectively, i.e.
and
.
Theorem 2.3
The inclusions strictly hold for
.
The following Lemma and the inequality has been used for establishing some results of this article.
Lemma 2.4
If I and M
N. If M
I then M
N
(Şengönül, Citation2007).
Let be the bounded sequence of positive reals numbers. For any complex
, whenever
we have
Also, whenever
we have
where
. (Maddox, Citation1969) cf. (Khan Ebadullah, Ayhan Esi, Khan, & Shafiq, Citation2013a; Khan & Khan, Citation2014b; Khan & Sabiha, Citation2011; Malkowsky, Citation1997; Ng & Lee, Citation1978).
Recently Khan Ebadullah, Ayhan Esi, Khan, and Shafiq (Citation2013a) introduced various Zweier sequence spaces the following sequence spaces.
We also denote byIn this article, we introduce the following sequence spaces. For any
, we have
We also denote by
and
where is a double sequence of positive real numbers.
Throughout the article, for the sake of convenience now we will denote by for all
.
3. Main results
Theorem 3.1
The sequence spaces are linear spaces.
Proof
We shall prove the result for the space .
The proof for the other spaces will follow similarly.
Let and let
be scalars. Then for a given
we have
where
and
Let
be such that . Then
Thus Hence
Therefore
is a linear space. Proof of
follows since it is a special case of
.
Remark
The sequence spaces , are linear spaces since each is an intersection of two of the linear spaces in Theorem 3.1.
Theorem 3.2
Let . Then
and
are paranormed spaces, paranormed by
where
Proof
Let
(1) | Clearly, | ||||
(2) |
| ||||
(3) | Since | ||||
(4) | Let |
Theorem 3.3
is a closed subspace of
Proof
Let be a Cauchy sequence in
such that
. We show that
. Since
, then there exists
such that
We need to show that
(1) |
| ||||
(2) | If |
For a given , we have
Then . Let
, where
Then .
We choose , then for each
, we have
Then is a Cauchy sequence of scalars in
, so there exists a scalar
such that
, as
.
For the next part let be given. Then we show that if
, then
.
Since , then there exists
such that
which implies that . The number
can be so chosen that together with (1), we have
such that
Since . Then we have a subset S of
such that
, where
Let , where
Therefore for each , we have
Then the result follows.
Theorem 3.4
The spaces and
are nowhere dense subsets of
Proof
Since the inclusions and
are strict so in view of Theorem 3.3 we have the following result.
Theorem 3.5
The spaces and
are not separable.
Proof
We shall prove the result for the space . The proof for the other spaces will follow similarly.
Let M be an infinite subset of of such that
. Let
Let
Clearly is uncountable.
Consider the class of open balls
Let be an open cover of
containing
.
Since is uncountable, so
cannot be reduced to a countable subcover for
.
Thus is not separable.
Theorem 3.6
Let and
Then the following results are equivalent.
(a) |
| ||||
(b) |
Proof
Suppose that and
, then the inequalities
hold for any
and for all
Therefore, the equivalence of (a) and (b) is obvious.
Theorem 3.7
Let and
be two sequences of positive real numbers. Then
if and only if
and
where
such that
Proof
Let and
Then there exists
such that
for all sufficiently large
Since
for a given
we have
Let Then
Then for all sufficiently large
Therefore
The converse part of the result follows obviously.
The other inclusion follows by symmetry of the two inequalities.
4. Conclusion
The notion of Ideal convergence (I-convergence) is a generalization of the statical convergence and equally considered by the researchers for their research purposes since its inception. Along with this the very new concept of double sequences has also found its place in the field of analysis. It is also being further discovered by mathematicians all over the world. In this article, we introduce paranorm ideal convergent double sequence spaces using Zweier transform. We study some topological and algebraic properties. Further we prove some inclusion relations related to these new spaces.
Acknowledgements
The authors would like to record their gratitude to the reviewer for his careful reading and making some useful corrections which improved the presentation of the paper.
Additional information
Funding
Notes on contributors
Vakeel A. Khan
Vakeel A. Khan received the MPhil and PhD degrees in Mathematics from Aligarh Muslim University, Aligarh, India. Currently he is a senior assistant professor at Aligarh Muslim University, Aligarh, India. A vigorous researcher in the area of Sequence Spaces , he has published a number of research papers in reputed national and international journals, including Numerical Functional Analysis and Optimization (Taylor’s and Francis), Information Sciences (Elsevier), Applied Mathematics Letters Applied Mathematics (Elsevier), A Journal of Chinese Universities (Springer- Verlag, China).
Nazneen Khan
Nazneen Khan received the MPhil and PhD degrees in Mathematics from Aligarh Muslim University, Aligarh, India. Currently she is an assistant professor at Taibah University, Kingdom of Saudi Arabia, Madina. Her research interests are Functional Analysis, sequence spaces and double sequences.
Yasmeen Khan
Yasmeen Khan received MSc and MPhil from Aligarh Muslim University, and is currently a PhD scholar at Aligarh Muslim University. Her research interests are Functional Analysis, sequence spaces and double sequences.
References
- Altay, B., Ba\c{s}ar, F., & Mursaleen, M. (2006). On the Euler sequence space which include the spaces l\textsubscript{p} and .I. Information Sciences, 176, 1450–1462.
- Ba\c{s}ar, F., & Altay, B. (2003). On the spaces of sequences of p-bounded variation and related matrix mappings. Ukrainian Mathematical Journal, 55, 136–147
- Das, P., Kostyrko, P., Malik, P., & Wilczy\’{n}ski, W. (2008). I and -convergence of double sequences. Mathematica Slovaca, 58, 605–620.
- Demirci, K. (2001). I-limit superior and limit inferior. Mathematical Communications, 6, 165–172.
- Esi, A., & Sapsizo\v{g}lu, A. (2012). On some lacunary -strong Zweier convergent sequence spaces. ROMAI Journal, 8, 61–70.
- Fadile Karababa, Y., & Esi, A. (2012). On some strong Zweier convergent sequence spaces. Acta University Apulennsis, 29, 9–15.
- Gurdal, M., & Ahmet, S. (2008). Extremal I-limit points of double sequences. Applied Mathematics E-Notes, 8, 131–137.
- Khan, V. A., Ebadullah, K., Esi, A., Khan, N., & Shafiq, M. (2013a). On Paranorm Zweier I-convergent sequence spaces. Journal of Mathematics, 2013, 613501, 1–6.
- Khan, V. A., Ebadullah, K., Esi, A., & Shafiq, M. (2013b). On some Zweier I-convergent sequence spaces defined by a modulus function. Afrika Matematika. doi:10.1007/s13370-013-0186-y
- Khan, V. A., & Khan, N. (2014a). On some Zweier I-convergent double sequence spaces defined by a modulus function. Analysis. doi:10.1515/anly-2014-1242
- Khan, V. A., & Khan, N. (2014b). On some Zweier I-convergent double sequence spaces defined by Orlicz function. Journal of Applied Mathematics & Informatics, 32, 687–695.
- Khan, V. A., & Sabiha, T. (2011). On some new double sequence spaces of invariant means defined by Orlicz function. Communications Faculty of Sciences, 60, 11–21.
- Kostyrko, P., .\v{S}al\’{a}t, T., & Wilczy\’{n}ski, W. (2000). I-convergence. Real Analysis Exchange, 26, 669–686.
- Lascarides, C. G. (1971). A study of certain sequence spaces of Maddox and generalization of a theorem of Iyer. Pacific Journal of Mathematics, 38, 487–500.
- Lascarides, C. G. (1983). On the equivalence of certain sets of sequences. Indian Journal of Mathematics, 25, 41–52.
- Maddox, I. J. (1969). Some properties of paranormed sequence spaces. Journal London Mathematical Society, 1, 316–322.
- Malkowsky, E. (1997). Recent results in the theory of matrix transformation in sequence spaces. Matematicki Vesnik, 49, 187–196.
- Mursaleen, M., & Mohiuddine, S. A. (2010). On ideal convergence of double sequences in probabilistic normed spaces. Mathematical Reports, 12, 359–371.
- Mursaleen, M., & Mohiuddine, S. A. (2012). On ideal convergence in probabilistic normed spaces. Mathematica Slovaca, 62, 49–62.
- Ng, P. N., & Lee, P. Y. (1978). Cesaro sequence spaces of non-absolute type. Commentationes Mathematicae (Prace Matematyczne), 20, 429–433.
- \v{S}al\’{a}t, T., Tripathy, B. C., & Ziman, M. (2004). On some properties of I-convergence. Tatra Mountains Mathematical Publications, 28, 279–286.
- \c{S}eng\"{o}n\"{u}l, M. (2007). On the Zweier sequence space. Demonstratio Mathematica, XL, 181–196.
- Tripathy, B. C., & Hazarika, B. (2009). Paranorm I-convergent sequence spaces. Mathematica Slovaca, 59, 485–494.
- Wang, C. S. (1978). On N\"{o}rlund sequence spaces. Tamkang Journal of Mathematics, 9, 269–274.