- Research Article
- Open Access
© M. Başarir and M. Kayikçi. 2009
Received: 1 May 2009
Accepted: 17 July 2009
Published: 23 August 2009
We introduce the generalized Riesz difference sequence space which is defined by where is the Riesz sequence space defined by Altay and Başar. We give some topological properties, compute the duals, and determine the Schauder basis of this space. Finally; we study the characterization of some matrix mappings on this sequence space. At the end of the paper, we investigate some geometric properties of and we have proved that this sequence space has property for .
Let be the space of all real valued sequences. We write for the sequence spaces of all bounded, convergent, and null sequences, respectively. Also by , , and , we denote the sequence spaces of all convergent, absolutely and -absolutely, convergent series, respectively; where
Altay and Başar [1, 2] introduced the Riesz sequence space , and of nonabsolute type which is the set of all sequences whose -transforms are in the space , and respectively. Here and afterwards, will be used as a bounded sequence of strictly positive real numbers with and and denotes the collection of all finite subsets of where . The Riesz sequence space introduced in  by Altay and Başar is
The difference sequence spaces , and were first defined and studied by Kızmaz in  and studied by several authors, [4–9]. Başar and Altay  have studied the sequence space as the set of all sequences such that their -transforms are in the space ; that is,
The idea of difference sequences is generalized by Çolak and Et . They defined the sequence spaces:
Recently, Başarir and Öztürk  introduced the Riesz difference sequence space as follows:
Our main subject in the present paper is to introduce the generalized Riesz difference sequence space which consists of all the sequences such that their -transforms are in the space and to investigate some topological and geometric properties with respect to paranorm on this space.
2. Basic Facts and Definitions
In this section we give some definitions and lemmas which will be frequently used.
By we denote the class of all matrices such that Thus, if and only if the series on the right side of (2.1) converges for each and every and we have for all A sequence is said to be -summable to if converges to which is called as the -limit of
Now we give some lemmas which we need to prove our theorems.
Lemma 2.5 (see ).
Lemma 2.6 (see ).
Lemma 2.7 (see ).
Here are some topological properties of the generalized Riesz difference sequence space.
This can be easily obtained by [12, Theorem 5] so we omit the proof.
In the previous section; we show that the sequence space which is the space of all real sequences such that is a complete paranormed space. It is paranormed by for all where We recall that a paranormed space is total if implies Every total paranormed space becomes a linear metric space with the metric given by It is clear that is a total paranormed space.
In this section, we investigate some geometric properties of . First we give the definition of the property in a paranormed space and we will use the method in  to prove the property Consequently, we obtain that has property for
We would like to express our gratitude to the reviewer for his/her careful reading and valuable suggestions which improved the presentation of the paper.
- Altay B, Başar F: On the paranormed Riesz sequence spaces of non-absolute type. Southeast Asian Bulletin of Mathematics 2003,26(5):701–715.MathSciNetMATHGoogle Scholar
- Altay B, Başar F: Some Euler sequence spaces of nonabsolute type. Ukrainian Mathematical Journal 2005,57(1):1–17. 10.1007/s11253-005-0168-9MathSciNetView ArticleMATHGoogle Scholar
- Kızmaz H: On certain sequence spaces. Canadian Mathematical Bulletin 1981,24(2):169–176. 10.4153/CMB-1981-027-5MathSciNetView ArticleMATHGoogle Scholar
- Altay B, Polat H: On some new Euler difference sequence spaces. Southeast Asian Bulletin of Mathematics 2006,30(2):209–220.MathSciNetMATHGoogle Scholar
- Polat H, Başar F: Some Euler spaces of difference sequences of order . Acta Mathematica Scientia. Series B 2007,27(2):254–266. 10.1016/S0252-9602(07)60024-1MathSciNetView ArticleMATHGoogle Scholar
- Malkowsky E, Parashar SD: Matrix transformations in spaces of bounded and convergent difference sequences of order . Analysis 1997,17(1):87–97.MathSciNetView ArticleMATHGoogle Scholar
- Et M, Başarir M: On some new generalized difference sequence spaces. Periodica Mathematica Hungarica 1997,35(3):169–175. 10.1023/A:1004597132128MathSciNetView ArticleMATHGoogle Scholar
- Başarir M, Nuray F: Paranormed difference sequence spaces generated by infinite matrices. Pure and Applied Mathematika Sciences 1991,34(1–2):87–90.MathSciNetMATHGoogle Scholar
- Polat H, Başarir M: New Taylor difference sequence spaces of order . International Mathematical Journal 2004,5(3):211–223.MathSciNetMATHGoogle Scholar
- Başar F, Altay B: On the space of sequences of -bounded variation and related matrix mappings. Ukrainian Mathematical Journal 2003,55(1):136–147. 10.1023/A:1025080820961MathSciNetView ArticleMATHGoogle Scholar
- Çolak R, Et M: On some generalized difference sequence spaces and related matrix transformations. Hokkaido Mathematical Journal 1997,26(3):483–492.MathSciNetView ArticleMATHGoogle Scholar
- Başarir M, Öztürk M: On the Riesz difference sequence space. Rendiconti del Circolo Matematico di Palermo 2008,57(3):377–389. 10.1007/s12215-008-0027-2MathSciNetView ArticleMATHGoogle Scholar
- Grosse-Erdmann K-G: Matrix transformations between the sequence spaces of Maddox. Journal of Mathematical Analysis and Applications 1993,180(1):223–238. 10.1006/jmaa.1993.1398MathSciNetView ArticleMATHGoogle Scholar
- Lascarides CG, Maddox IJ: Matrix transformations between some classes of sequences. Proceedings of the Cambridge Philosophical Society 1970, 68: 99–104. 10.1017/S0305004100001109MathSciNetView ArticleMATHGoogle Scholar
- Maddox IJ: Elements of Functional Analysis. Cambridge University Press, Cambridge, UK; 1970:x+208.MATHGoogle Scholar
- Maddox IJ: Paranormed sequence spaces generated by infinite matrices. Proceedings of the Cambridge Philosophical Society 1968, 64: 335–340. 10.1017/S0305004100042894MathSciNetView ArticleMATHGoogle Scholar
- Khompurngson K: Geometric properties of some paranormed sequence spaces, M.S. thesis. Chiang Mai University, Chiang Mai, Thailand; 2004.Google Scholar
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