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Some normed binomial difference sequence spaces related to the [Formula: see text] spaces.

Meimei Song1, Jian Meng1

  • 1Department of Mathematics, Tianjin University of Technology, Tianjin, 300000 P.R. China.

Journal of Inequalities and Applications
|July 7, 2017
PubMed
Summary

This study introduces new normed binomial sequence spaces using binomial transformations and difference operators. These novel spaces are proven to be linearly isomorphic to existing sequence spaces, with their bases and duals fully characterized.

Keywords:
Schauder basismatrix domainsequence spaceα-, β- and γ-duals

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Area of Science:

  • Mathematics
  • Functional Analysis
  • Sequence Spaces

Background:

  • Sequence spaces are fundamental in functional analysis.
  • Binomial transformations and difference operators are key tools for constructing new sequence spaces.
  • Understanding the properties of these spaces is crucial for further mathematical research.

Purpose of the Study:

  • To introduce and define the normed binomial sequence spaces denoted as [Formula: see text].
  • To establish linear isomorphisms between the newly defined spaces and established sequence spaces ([Formula: see text] and [Formula: see text]).
  • To compute the Schauder bases and the alpha-, beta-, and gamma-duals of the introduced sequence spaces.

Main Methods:

  • Construction of the normed binomial sequence spaces [Formula: see text] by integrating binomial transformations and the difference operator.
  • Application of principles of functional analysis to demonstrate linear isomorphisms.
  • Utilizing established techniques in sequence space theory to determine Schauder bases and dual spaces.

Main Results:

  • Successful introduction of the normed binomial sequence spaces [Formula: see text].
  • Proof of linear isomorphism between [Formula: see text] and [Formula: see text], as well as [Formula: see text].
  • Complete computation of Schauder bases and the α-, β-, and γ-duals for the new sequence spaces.

Conclusions:

  • The newly defined normed binomial sequence spaces possess well-characterized structural properties.
  • The established isomorphisms provide connections to existing mathematical frameworks.
  • The computed bases and duals offer essential tools for further analysis and applications in functional analysis.