Generalized Vector-Valued Sequence Spaces Defined by Modulus Functions
© Mahmut Işik. 2010
Received: 17 June 2010
Accepted: 16 December 2010
Published: 21 December 2010
We introduce the vector-valued 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.
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 vector-valued sequence spaces are done by Das and Choudhury , Et , Et et al. , Leonard , Rath and Srivastava , J. K. Srivastava and B. K. Srivastava , Tripathy et al. [7, 8], and many others.
The notion of a modulus was introduced by Nakano . We recall that a modulus is a function from to such that
2. Main Results
The proofs of the following theorems are obtained by using the known standard techniques; therefore, we give them without proofs.
and the inclusions are strict.
The following result is a consequence of Theorem 2.5(i).
If we take for all and using the same technique of Theorem 5 of Maddox , it is easy to prove the theorem.
The notion of statistical convergence were introduced by Fast  and Schoenberg , 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 , Fridy , Connor , Mursaleen , Işik , Malkowsky and Sava , 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 Stone-Čech 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.
- Das NR, Choudhury A: Matrix transformation of vector valued sequence spaces. Bulletin of the Calcutta Mathematical Society 1992, 84(1):47–54.MathSciNetMATHGoogle Scholar
- 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.MathSciNetMATHGoogle Scholar
- Et M, Gökhan A, Altinok H: On statistical convergence of vector-valued 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, 2006MATHMathSciNetGoogle Scholar
- Leonard IE: Banach sequence spaces. Journal of Mathematical Analysis and Applications 1976, 54(1):245–265. 10.1016/0022-247X(76)90248-1MathSciNetView ArticleMATHGoogle Scholar
- Ratha A, Srivastava PD: On some vector valued sequence spaces . Ganita 1996, 47(1):1–12.MathSciNetMATHGoogle Scholar
- Srivastava JK, Srivastava BK: Generalized sequence space . Indian Journal of Pure and Applied Mathematics 1996, 27(1):73–84.MathSciNetMATHGoogle Scholar
- 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.MathSciNetMATHGoogle Scholar
- Tripathy BC, Mahanta S: On a class of vector-valued sequences associated with multiplier sequences. Acta Mathematicae Applicatae Sinica. English Series 2004, 20(3):487–494. 10.1007/s10255-004-0186-7MathSciNetView ArticleMATHGoogle Scholar
- Nakano H: Concave modulars. Journal of the Mathematical Society of Japan 1953, 5: 29–49. 10.2969/jmsj/00510029MathSciNetView ArticleMATHGoogle Scholar
- Maddox IJ: Sequence spaces defined by a modulus. Mathematical Proceedings of the Cambridge Philosophical Society 1986, 100(1):161–166. 10.1017/S0305004100065968MathSciNetView ArticleMATHGoogle Scholar
- Ruckle WH: FK spaces in which the sequence of coordinate vectors is bounded. Canadian Journal of Mathematics 1973, 25: 973–978. 10.4153/CJM-1973-102-9MathSciNetView ArticleMATHGoogle Scholar
- Bilgin T: The sequence space ℓ(p,f,q,s) on seminormed spaces. Bulletin of the Calcutta Mathematical Society 1994, 86(4):295–304.MathSciNetMATHGoogle Scholar
- Pehlivan S, Fisher B: On some sequence spaces. Indian Journal of Pure and Applied Mathematics 1994, 25(10):1067–1071.MathSciNetMATHGoogle Scholar
- Waszak A: On the strong convergence in some sequence spaces. Fasciculi Mathematici 2002, (33):125–137.Google Scholar
- Bhardwaj VK: A generalization of a sequence space of Ruckle. Bulletin of the Calcutta Mathematical Society 2003, 95(5):411–420.MathSciNetMATHGoogle Scholar
- Altın Y: Properties of some sets of sequences defined by a modulus function. Acta Mathematica Scientia B. English Edition 2009, 29(2):427–434.View ArticleMATHMathSciNetGoogle Scholar
- Kızmaz H: On certain sequence spaces. Canadian Mathematical Bulletin 1981, 24(2):169–176. 10.4153/CMB-1981-027-5MathSciNetView ArticleMATHGoogle Scholar
- Et M, Çolak R: On some generalized difference sequence spaces. Soochow Journal of Mathematics 1995, 21(4):377–386.MathSciNetMATHGoogle Scholar
- Maddox IJ: On strong almost convergence. Mathematical Proceedings of the Cambridge Philosophical Society 1979, 85(2):345–350. 10.1017/S0305004100055766MathSciNetView ArticleMATHGoogle Scholar
- Fast H: Sur la convergence statistique. Colloqium Mathematicum 1951, 2: 241–244.MathSciNetMATHGoogle Scholar
- Schoenberg IJ: The integrability of certain functions and related summability methods. The American Mathematical Monthly 1959, 66: 361–375. 10.2307/2308747MathSciNetView ArticleMATHGoogle Scholar
- Šalát T: On statistically convergent sequences of real numbers. Mathematica Slovaca 1980, 30(2):139–150.MathSciNetMATHGoogle Scholar
- Fridy JA: On statistical convergence. Analysis 1985, 5(4):301–313.MathSciNetView ArticleMATHGoogle Scholar
- Connor JS: The statistical and strong p -Cesàro convergence of sequences. Analysis 1988, 8(1–2):47–63.MathSciNetView ArticleMATHGoogle Scholar
- Mursaleen M: λ -statistical convergence. Mathematica Slovaca 2000, 50(1):111–115.MathSciNetMATHGoogle Scholar
- Işik M: On statistical convergence of generalized difference sequences. Soochow Journal of Mathematics 2004, 30(2):197–205.MathSciNetMATHGoogle Scholar
- Malkowsky E, Savas E: Some λ -sequence spaces defined by a modulus. Archivum Mathematicum 2000, 36(3):219–228.MathSciNetMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.