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|<3\varepsilon a\right\}\end{equation*} where $\left\|{\psi }_n\right\|=a=\left\|x^*_n\right\|$.