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On almost B-summable double sequence spaces.

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This study introduces new almost null and almost convergent double sequence spaces using a four-dimensional generalized difference matrix. The research establishes these spaces as Banach spaces and explores their duals and matrix classes.

Keywords:
[Formula: see text]-summablealmost convergent double sequence spacefour-dimensional matricesfour-dimensional matrix domainfour-dimensional matrix mappingα-, [Formula: see text]- and γ-dual

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Area of Science:

  • Mathematical Analysis
  • Sequence Spaces
  • Functional Analysis

Background:

  • Builds upon prior work on generalized difference matrices and their domains in double sequence spaces.
  • Addresses the need for new sequence spaces with specific convergence properties.

Purpose of the Study:

  • To introduce and define novel almost null and almost convergent double sequence spaces.
  • To investigate the topological and structural properties of these newly defined spaces.
  • To characterize matrix classes related to these sequence spaces.

Main Methods:

  • Utilizes the concept of a four-dimensional generalized difference matrix as a domain.
  • Employs techniques from functional analysis to prove Banach space properties.
  • Applies methods for determining dual spaces (α-dual, β-dual, γ-dual).
  • Characterizes matrix classes using established definitions and theorems.

Main Results:

  • Introduced the double sequence spaces [Formula: see text] and [Formula: see text].
  • Established that these spaces are Banach spaces under specific conditions.
  • Determined the inclusion relation between the new almost convergent double sequence spaces.
  • Identified the α-dual, β-dual, and γ-dual of the space [Formula: see text].
  • Characterized new matrix classes [Formula: see text] and [Formula: see text].

Conclusions:

  • The newly defined double sequence spaces possess the structure of Banach spaces.
  • The study provides a comprehensive analysis of the structural properties and duals of these spaces.
  • Significant results concerning matrix classes associated with these spaces were obtained.