Abstract
We study new sequence spaces associated to sequences in normed spaces and the band matrix F̂ defined by the Fibonacci sequence. We give some characterizations of continuous linear operators and weakly unconditionally Cauchy series by means of completeness of the new sequence spaces. Also, we characterize the barreledness of a normed space via weakly∗ unconditionally Cauchy series in X^{*}.
Highlights
We have ∞, c and c for the spaces of all bounded, convergent and null sequences x = (xk), respectively, normed by x ∞ = supk |xk|, where k ∈ N, the set of positive integers
By w, we denote the space of all real sequences x =
A sequence space λ with a linear topology is called a K-space provided each of the maps pi : λ → R defined by pi(x) = xi is continuous for all i ∈ N
Summary
We have ∞, c and c for the spaces of all bounded, convergent and null sequences x = (xk), respectively, normed by x ∞ = supk |xk|, where k ∈ N, the set of positive integers. We introduce the sets SF(x), SFw(x) and SFw∗(g) by means of sequences in normed spaces and the Fibonacci matrix F = (fnk). We will characterize wucs by means of completeness of the spaces SF(x) and SFw(x), and we will obtain necessary and sufficient conditions for the operator T : SF(x)( and SFw(x)) → X to be continuous.
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