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

In this paper, we discuss some conditions of a greedy basis for Banach space X under a standard \varepsilon-isometry mapping. We show that if X and Y are Banach spaces, \left(x_n\right) is a greedy basis for X, and f:X\to Y is a standard \varepsilon-isometry, then \left(f\left(x_n\right)\right) is a greedy basis for a subspace of Y. As a result, if f is a surjective standard \varepsilon-isometry, then \left(f\left(x_n\right)\right) is a greedy basis for Y. We also show that {span\left\{\left(f\left(x_n\right)\right)\right\}}^* is isomorphic with \mathrm{\Psi }\subset Y^* where \mathrm{\Psi } is defined as\begin{equation*}    \mathrm{\Psi }\mathrm{:=}\overline{span}\left\{{\psi }_n:\ {\psi }_n\in Y^*\ and\ \left|\left\langle x^*_n,x\right\rangle -\left\langle {\psi }_n,f\left(x\right)\right\rangle \right|<۳\varepsilon a\right\}\end{equation*} where \left\|{\psi }_n\right\|=a=\left\|x^*_n\right\|.

Keywords:

ε-isometry , greedy basis , Banach space , stability