Abstract
On paranorm I-convergent double sequence spaces defined by a compact operator
Highlights
Definition 1.1: Let X and Y be two normed linear spaces
The notions of paranorm I– convergence of double sequence spaces defined by compact operator have been defined here and some elementary properties of these notions are obtained
These definitions and results provide new tools to deal with the convergence problems of sequences in the advance settings, occurring in many branches of science and engineering
Summary
Definition 1.1: Let X and Y be two normed linear spaces. An operator T: X → Y is said to be a compact linear operator (or completely continuous linear operator), if: 1. T is linear 2. Definition 1.4: A double sequence x = (xij) ∈ 2ω is said to be I-convergent to a number L if for every ε > 0, we have. A function g: X → R is called paranorm, if for all x, y ∈ X, Definition 1.5: A double sequence x = (xij) ∈ 2ω is said to be I - null if L=0 Definition 1.8: A double sequence space E is said to be solid or normal if (xij) ∈ E implies that (αijxij) ∈ E for all sequence of scalars (αij) with |αij| < 1 for all (i, j) ∈ N × N. Definition 1.11: A double sequence space E is said to be convergence free if (yij) ∈ E whenever (xij) ∈ E and xij = 0 implies yij = 0, for all (i, j) ∈ N × N.
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