Abstract
In this paper, we introduce the paranormed sequence space $\mathcal{M}_{u}(t)$ corresponding to the normed space $\mathcal{M}_{u}$ of bounded double sequences. We examine general topological properties of this space and determine its alpha-, beta- and gamma-duals. Furthermore, we characterize some classes of four-dimensional matrix transformations concerning this space and its dual spaces.
Highlights
The main drawback of the Pringsheim’s convergence is that a p−convergent double sequence need not be bounded
∞, for example k,l means that ∞ k,l=0, and we assume that θ denotes any of the convergence rule symbols p, bp or r
Let λ be a double sequence space converging with respect to some linear convergence rule θ − lim : λ → C
Summary
The main drawback of the Pringsheim’s convergence is that a p−convergent double sequence need not be bounded. In the present study; we define the paranormed double sequence space Mu(t) of all bounded double sequences, as follows: Mu(t) := x = (xkl) ∈ Ω : sup |xkl|tkl < ∞ . One can observe by similar approach used in single sequences that the set Mu(t) is complete paranormed space with the paranorm g(x) = sup |xkl|tkl/M
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