Abstract
The purpose of this paper is to introduce the notion of weighted almost convergence of a sequence and prove that this sequence endowed with the sup-norm {Vert cdot Vert } _{infty} is a BK-space. We also define the notions of weighted almost conservative and regular matrices and obtain necessary and sufficient conditions for these matrix classes. Moreover, we define a weighted almost A-summable sequence and prove the related interesting result.
Highlights
1 Introduction and preliminaries Let ω denote the space of all complex sequences s =∞ j=0 (or write s =)
By N we denote the set of natural numbers, and by R the set of real numbers
We use the standard notation ∞, c and c0 to denote the sets of all bounded, convergent and null sequences of real numbers, respectively, where each of the sets is a Banach space with the sup-norm . ∞ defined by s ∞ = supj∈N |sj|
Summary
We use the standard notation ∞, c and c0 to denote the sets of all bounded, convergent and null sequences of real numbers, respectively, where each of the sets is a Banach space with the sup-norm . K k=0 converges for each n ∈ N and the sequence As = (Ans) belongs to Y , we say that matrix A maps X into Y . The sequence s = (sk) of ∞ is said to be almost convergent, denoted by f , if all of its Banach limits [1] are equal.
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