Abstract
In this manuscript we characterize the completeness of a normed space through the strong lacunary ( N θ ) and lacunary statistical convergence ( S θ ) of series. A new characterization of weakly unconditionally Cauchy series through N θ and S θ is obtained. We also relate the summability spaces associated with these summabilities with the strong p-Cesàro convergence summability space.
Highlights
Let X be a normed space, a sequence ⊂ X is said to be strongly 1-Cesàro summable to L ∈ X if 1 n lim ∑ k xk − Lk = 0. n→∞ n k =1Hardy-Littlewood [1] and Fekete [2] introduced this type of summability, which is related to the convergence of Fourier series
The strong lacunary summability Nθ was presented by Freedman et al [9] by introducing lacunary sequences and showed that Nθ is a larger class of BK-spaces which had many of the characteristics of
Fridy [10,11] showed the concept of statistical lacunary summability and they related it with the statistical convergence and the Nθ summability
Summary
Let X be a normed space, a sequence ( xk ) ⊂ X is said to be strongly 1-Cesàro summable In [14] the authors introduced the space of convergence S(∑ xi ) associated with the series ∑ xi , which is defined as the space of sequences ( a j ) in∞ such that ∑ ai xi converges. In [16,17] a Banach space is characterized by means of the strong p-Cesàro summability (w p ) and ideal-convergence. In this manuscript, the Nθ and Sθ summabilities are used along with the concept of weakly unconditionally series to characterize a Banach space.
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