 Research Article
 Open access
 Published:
Generalized VectorValued Sequence Spaces Defined by Modulus Functions
Journal of Inequalities and Applications volume 2010, Article number: 457892 (2010)
Abstract
We introduce the vectorvalued sequence spaces , , and , and , using a sequence of modulus functions and the multiplier sequence of nonzero complex numbers. We give some relations related to these sequence spaces. It is also shown that if a sequence is strongly Cesàro summable with respect to the modulus function then it is statistically convergent.
1. Introduction
Let be the set of all sequences real or complex numbers and , , and be, respectively, the Banach spaces of bounded, convergent, and null sequences with the usual norm , where , the set of positive integers.
The studies on vectorvalued sequence spaces are done by Das and Choudhury [1], Et [2], Et et al. [3], Leonard [4], Rath and Srivastava [5], J. K. Srivastava and B. K. Srivastava [6], Tripathy et al. [7, 8], and many others.
Let be a sequence of seminormed spaces such that for each . We define
It is easy to verify that is a linear space under usual coordinatewise operations defined by and , where .
Let be a sequences of nonzero scalar. Then for a sequence space , the multiplier sequence space , associated with the multiplier sequence , is defined as .
The notion of a modulus was introduced by Nakano [9]. We recall that a modulus is a function from to such that
(i) if and only if ,
(ii) for ,
(iii) is increasing,
(iv) is continuous from the right at 0.
It follows that must be continuous everywhere on . Maddox [10] and Ruckle [11] used a modulus function to construct some sequence spaces.
After then some sequence spaces defined by a modulus function were introduced and studied by Bilgin [12], Pehlivan and Fisher [13], Waszak [14], Bhardwaj [15], Altın [16], and many others.
The notion of difference sequence spaces was introduced by Kızmaz [17] and it was generalized by Et and Çolak [18]. Let be a fixed positive integer. Then we write
for , and , where , , and so we have
2. Main Results
In this section, we prove some results involving the sequence spaces , , and .
Definition 2.1.
Let be a sequence of seminormed spaces such that for each , a sequence of strictly positive real numbers, a sequence of seminorms, a sequence of modulus functions, and any fixed sequence of nonzero complex numbers . We define the following sequence spaces:
Throughout the paper will denote any one of the notation 0,1 or .
If and for all , we will write instead of .
If and for all , we will write instead of .
If , we say that is strongly Cesàro summable with respect to the modulus function and we will write and will be called limit of with respect to the modulus .
The proofs of the following theorems are obtained by using the known standard techniques; therefore, we give them without proofs.
Theorem 2.2.
Let the sequence be bounded. Then the spaces are linear spaces.
Theorem 2.3.
Let be a modulus function and the sequence be bounded; then
and the inclusions are strict.
Theorem 2.4.
is a paranormed (need not total paranorm) space with
where .
Theorem 2.5.
Let and be any two sequences of modulus functions. For any bounded sequences and of strictly positive real numbers and for any two sequences of seminorms and , we have
(i);
(ii);
(iii);
(iv)If is stronger than for each , then ;
(v)If equivalent to for each , then ;
(vi).
Proof.

(i)
We will only prove (i) for and the other cases can be proved by using similar arguments. Let and choose with such that for and for all . Write and consider
(2.4)
where the first summation is over and second summation is over . Since is continuous, we have
By the definition of , we have for ,
Hence
From (2.5) and (2.7), we obtain .
The following result is a consequence of Theorem 2.5(i).
Corollary 2.6.
Let be a modulus function. Then .
Theorem 2.7.
Let and be bounded; then .
Proof.
If we take for all and using the same technique of Theorem 5 of Maddox [19], it is easy to prove the theorem.
Theorem 2.8.
Let be a modulus function; if , then .
Proof.
Omitted.
3. Statistical Convergence
The notion of statistical convergence were introduced by Fast [20] and Schoenberg [21], independently. Over the years and under different names, statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory, and number theory. Later on it was further investigated from the sequence space point of view and linked with summability theory by Šalát [22], Fridy [23], Connor [24], Mursaleen [25], Işik [26], Malkowsky and Sava [27], and many others. In recent years, generalizations of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces. Statistical convergence and its generalizations are also connected with subsets of the StoneČech compactification of the natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability. The notion depends on the density of subsets of the set of natural numbers.
A subset of is said to have density positive integers which is defined by if
where is the characteristic function of . It is clear that any finite subset of have zero natural density and .
In this section, we introduce statistically convergent sequences and give some inclusion relations between statistically convergent sequences and summable sequences.
Definition 3.1.
A sequence is said to be statistically convergent to if for every ,
In this case, we write . The set of all statistically convergent sequences is denoted by . In the case , we will write instead of .
Theorem 3.2.
Let be a modulus function; then
(i)If , then ;
(ii)If and , then ;
(iii),
where .
Proof.
Omitted.
In the following theorems, we will assume that the sequence is bounded and .
Theorem 3.3.
Let be a modulus function. Then .
Proof.
Let and let be given. Let and denote the sums over with and , respectively. Then
Hence .
Theorem 3.4.
Let be bounded; then .
Proof.
Suppose that is bounded. Let and and be denoted in previous theorem. Since is bounded, there exists an integer such that , for all . Then
Hence .
Theorem 3.5.
if and only if is bounded.
Proof.
Let be bounded. By Theorems 3.3 and 3.4, we have .
Conversely suppose that is unbounded. Then there exists a sequence of positive numbers with , for . If we choose
then we have
for all and so , but for , and for all . This contradicts to .
References
Das NR, Choudhury A: Matrix transformation of vector valued sequence spaces. Bulletin of the Calcutta Mathematical Society 1992, 84(1):47–54.
Et M: Spaces of Cesàro difference sequences of order r defined by a modulus function in a locally convex space. Taiwanese Journal of Mathematics 2006, 10(4):865–879.
Et M, Gökhan A, Altinok H: On statistical convergence of vectorvalued sequences associated with multiplier sequences. Ukraïns'kiĭ Matematichniĭ Zhurnal 2006, 58(1):125–131. translation in Ukrainian Mathematical Journal, vol. 58, no. 1, pp. 139–146, 2006
Leonard IE: Banach sequence spaces. Journal of Mathematical Analysis and Applications 1976, 54(1):245–265. 10.1016/0022247X(76)902481
Ratha A, Srivastava PD: On some vector valued sequence spaces . Ganita 1996, 47(1):1–12.
Srivastava JK, Srivastava BK: Generalized sequence space . Indian Journal of Pure and Applied Mathematics 1996, 27(1):73–84.
Tripathy BC, Sen M: Vector valued paranormed bounded and null sequence spaces associated with multiplier sequences. Soochow Journal of Mathematics 2003, 29(3):313–326.
Tripathy BC, Mahanta S: On a class of vectorvalued sequences associated with multiplier sequences. Acta Mathematicae Applicatae Sinica. English Series 2004, 20(3):487–494. 10.1007/s1025500401867
Nakano H: Concave modulars. Journal of the Mathematical Society of Japan 1953, 5: 29–49. 10.2969/jmsj/00510029
Maddox IJ: Sequence spaces defined by a modulus. Mathematical Proceedings of the Cambridge Philosophical Society 1986, 100(1):161–166. 10.1017/S0305004100065968
Ruckle WH: FK spaces in which the sequence of coordinate vectors is bounded. Canadian Journal of Mathematics 1973, 25: 973–978. 10.4153/CJM19731029
Bilgin T: The sequence space ℓ(p,f,q,s) on seminormed spaces. Bulletin of the Calcutta Mathematical Society 1994, 86(4):295–304.
Pehlivan S, Fisher B: On some sequence spaces. Indian Journal of Pure and Applied Mathematics 1994, 25(10):1067–1071.
Waszak A: On the strong convergence in some sequence spaces. Fasciculi Mathematici 2002, (33):125–137.
Bhardwaj VK: A generalization of a sequence space of Ruckle. Bulletin of the Calcutta Mathematical Society 2003, 95(5):411–420.
Altın Y: Properties of some sets of sequences defined by a modulus function. Acta Mathematica Scientia B. English Edition 2009, 29(2):427–434.
Kızmaz H: On certain sequence spaces. Canadian Mathematical Bulletin 1981, 24(2):169–176. 10.4153/CMB19810275
Et M, Çolak R: On some generalized difference sequence spaces. Soochow Journal of Mathematics 1995, 21(4):377–386.
Maddox IJ: On strong almost convergence. Mathematical Proceedings of the Cambridge Philosophical Society 1979, 85(2):345–350. 10.1017/S0305004100055766
Fast H: Sur la convergence statistique. Colloqium Mathematicum 1951, 2: 241–244.
Schoenberg IJ: The integrability of certain functions and related summability methods. The American Mathematical Monthly 1959, 66: 361–375. 10.2307/2308747
Šalát T: On statistically convergent sequences of real numbers. Mathematica Slovaca 1980, 30(2):139–150.
Fridy JA: On statistical convergence. Analysis 1985, 5(4):301–313.
Connor JS: The statistical and strong p Cesàro convergence of sequences. Analysis 1988, 8(1–2):47–63.
Mursaleen M: λ statistical convergence. Mathematica Slovaca 2000, 50(1):111–115.
Işik M: On statistical convergence of generalized difference sequences. Soochow Journal of Mathematics 2004, 30(2):197–205.
Malkowsky E, Savas E: Some λ sequence spaces defined by a modulus. Archivum Mathematicum 2000, 36(3):219–228.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Işik, M. Generalized VectorValued Sequence Spaces Defined by Modulus Functions. J Inequal Appl 2010, 457892 (2010). https://doi.org/10.1155/2010/457892
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/457892