- Research Article
- Open Access
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.
- Positive Integer
- Real Number
- Complex Number
- Similar Argument
- Standard Technique
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
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.
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