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Some notes on the greedy basis for Banach spaces under $\varepsilon$-isometry | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 147، دوره 14، شماره 1، فروردین 2023، صفحه 1881-1889 اصل مقاله (420.34 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2022.28677.3959 | ||
| نویسندگان | ||
| Minanur Rohman* 1، 2؛ Ilker Eryilmaz1 | ||
| 1Departement of Mathematics, Faculty of Science, Ondokuz Mayis University, Turkiye | ||
| 2Department of Primary School Teacher Education, School of Islamic Studies Ma'had Aly Al-Hikam Malang, Indonesia | ||
| تاریخ دریافت: 20 مهر 1401، تاریخ بازنگری: 02 آذر 1401، تاریخ پذیرش: 05 آذر 1401 | ||
| چکیده | ||
| 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 $\left\|{\psi }_n\right\|=a=\left\|x^*_n\right\|$. | ||
| کلیدواژهها | ||
| ε-isometry؛ greedy basis؛ Banach space؛ stability | ||
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