Abstract
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 -Cesaro summable with respect to the modulus function then it is -statistically convergent.
Highlights
Let w be the set of all sequences real or complex numbers and ∞, c, and c0 be, respectively, the Banach spaces of bounded, convergent, and null sequences x xk with the usual norm x sup |xk|, where k ∈ Æ {1, 2, . . .}, the set of positive integers.The studies on vector-valued sequence spaces are done by Das and Choudhury 1, Et 2, Et et al 3, Leonard 4, Rath and Srivastava 5, J
Let Ek, qk be a sequence of seminormed spaces such that Ek 1 ⊂ Ek for each k ∈ Æ, p pk a sequence of strictly positive real numbers, Q qk a sequence of seminorms, F fk a sequence of modulus functions, and u uk any fixed sequence of nonzero complex numbers uk
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 Salat, Fridy, Connor, Mursaleen, Isik, Malkowsky and Savas, and many others
Summary
Let w be the set of all sequences real or complex numbers and ∞, c, and c0 be, respectively, the Banach spaces of bounded, convergent, and null sequences x xk with the usual norm x sup |xk|, where k ∈ Æ {1, 2, . . .}, the set of positive integers. Let w be the set of all sequences real or complex numbers and ∞, c, and c0 be, respectively, the Banach spaces of bounded, convergent, and null sequences x xk with the usual norm x sup |xk|, where k ∈ Æ {1, 2, . The studies on vector-valued sequence spaces are done by Das and Choudhury 1 , Et 2 , Et et al 3 , Leonard 4 , Rath and Srivastava 5 , J. Let Ek, qk be a sequence of seminormed spaces such that Ek 1 ⊂ Ek for each k ∈ Æ. It is easy to verify that w E is a linear space under usual coordinatewise operations defined by x y xk yk and αx αxk , where α ∈. Let u uk be a sequences of nonzero scalar. For a sequence space E, the multiplier sequence space E u , associated with the multiplier sequence u, is defined as E u { xk ∈ w : ukxk ∈ E}
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