Some notes on the greedy basis for Banach spaces under ε-isometry

Document Type : Research Paper

Authors

1 Departement of Mathematics, Faculty of Science, Ondokuz Mayis University, Turkiye

2 Department of Primary School Teacher Education, School of Islamic Studies Ma'had Aly Al-Hikam Malang, Indonesia

Abstract

In this paper, we discuss some conditions of a greedy basis for Banach space X under a standard ε-isometry mapping. We show that if X and Y are Banach spaces, (xn) is a greedy basis for X, and f:XY is a standard ε-isometry, then (f(xn)) is a greedy basis for a subspace of Y. As a result, if f is a surjective standard ε-isometry, then (f(xn)) is a greedy basis for Y. We also show that span{(f(xn))} is isomorphic with ΨY where Ψ is defined as
Ψ:=span{ψn: ψnY and |xn,xψn,f(x)|<3εa}
 where ψn=a=xn.

Keywords

Volume 14, Issue 1
January 2023
Pages 1881-1889
  • Receive Date: 12 October 2022
  • Revise Date: 23 November 2022
  • Accept Date: 26 November 2022