A note on the stability of nonsurjective $\varepsilon$-isometries of Banach spaces
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- by Lixin Cheng and Yunbai Dong PDF
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Abstract:
Let $X, Y$ be Banach spaces, and let $f: X\rightarrow Y$ be an $\varepsilon$-isometry with $f(0)=0$ for some $\varepsilon \geq 0$. In this paper, we show that for every $x^*\in X^*$, there exists $\varphi \in Y^*$ with $\|\varphi \|=\|x^*\|\equiv r$ such that \begin{equation} \big |\langle x^*,x\rangle -\langle \varphi ,f(x)\rangle \big |\leq 3r\varepsilon ,\;\;\forall \;x\in X.\nonumber \end{equation} Moreover, the constant “3” is the best one in general.References
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Additional Information
- Lixin Cheng
- Affiliation: School of Mathematical Sciences, Xiamen University, Xiamen, 361005, People’s Republic of China
- Email: lxcheng@xmu.edu.cn
- Yunbai Dong
- Affiliation: Research Center of Nonlinear Science and School of Mathematics and Computer, Wuhan Textile University, Wuhan, 430073, People’s Republic of China
- Email: baiyunmu301@126.com
- Received by editor(s): February 7, 2020
- Received by editor(s) in revised form: March 17, 2020, and March 24, 2020
- Published electronically: July 30, 2020
- Additional Notes: The first author was supported by NSFC, grant 11731010.
The second author is the corresponding author
The second author was supported by NSFC, grant 11671314. - Communicated by: Stephen Dilworth
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 4837-4844
- MSC (2010): Primary 46B04; Secondary 46B20, 47A58
- DOI: https://doi.org/10.1090/proc/15110
- MathSciNet review: 4143398