A note on the stability of nonsurjective $\varepsilon$-isometries of Banach spaces
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- by Lixin Cheng and Yunbai Dong
- Proc. Amer. Math. Soc. 148 (2020), 4837-4844
- DOI: https://doi.org/10.1090/proc/15110
- Published electronically: July 30, 2020
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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
- Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR 1727673, DOI 10.1090/coll/048
- Lixin Cheng, Qingjin Cheng, Kun Tu, and Jichao Zhang, A universal theorem for stability of $\varepsilon$-isometries of Banach spaces, J. Funct. Anal. 269 (2015), no. 1, 199–214. MR 3345607, DOI 10.1016/j.jfa.2015.04.015
- Lixin Cheng, Duanxu Dai, Yunbai Dong, and Yu Zhou, Universal stability of Banach spaces for $\epsilon$-isometries, Studia Math. 221 (2014), no. 2, 141–149. MR 3200160, DOI 10.4064/sm221-2-3
- Lixin Cheng and Yunbai Dong, Corrigendum to “A universal theorem for stability of $\varepsilon$-isometries of Banach spaces” [J. Funct. Anal. 269 (2015) 199–214], J. Funct. Anal. 279 (2020), no. 1, 108518, 1. MR 4083780, DOI 10.1016/j.jfa.2020.108518
- Lixin Cheng, Yunbai Dong, and Wen Zhang, On stability of nonlinear non-surjective $\varepsilon$-isometries of Banach spaces, J. Funct. Anal. 264 (2013), no. 3, 713–734. MR 3003734, DOI 10.1016/j.jfa.2012.11.008
- Lixin Cheng and Yu Zhou, On perturbed metric-preserved mappings and their stability characterizations, J. Funct. Anal. 266 (2014), no. 8, 4995–5015. MR 3177328, DOI 10.1016/j.jfa.2014.02.019
- S. J. Dilworth, Approximate isometries on finite-dimensional normed spaces, Bull. London Math. Soc. 31 (1999), no. 4, 471–476. MR 1687544, DOI 10.1112/S0024609398005591
- Y. Dutrieux and G. Lancien, Isometric embeddings of compact spaces into Banach spaces, J. Funct. Anal. 255 (2008), no. 2, 494–501. MR 2419968, DOI 10.1016/j.jfa.2008.04.002
- T. Figiel, On nonlinear isometric embeddings of normed linear spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 185–188 (English, with Russian summary). MR 231179
- Julian Gevirtz, Stability of isometries on Banach spaces, Proc. Amer. Math. Soc. 89 (1983), no. 4, 633–636. MR 718987, DOI 10.1090/S0002-9939-1983-0718987-6
- G. Godefroy and N. J. Kalton, Lipschitz-free Banach spaces, Studia Math. 159 (2003), no. 1, 121–141. Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday. MR 2030906, DOI 10.4064/sm159-1-6
- D. H. Hyers and S. M. Ulam, On approximate isometries, Bull. Amer. Math. Soc. 51 (1945), 288–292. MR 13219, DOI 10.1090/S0002-9904-1945-08337-2
- D. H. Hyers and S. M. Ulam, Approximate isometries of the space of continuous functions, Ann. of Math. (2) 48 (1947), 285–289. MR 20717, DOI 10.2307/1969171
- S. Mazur and S. Ulam, Sur les transformations isométriques d’espaces vectoriels normés, C. R. Acad. Sci. Paris 194 (1932), 946–948.
- Matjaž Omladič and Peter emrl, On non linear perturbations of isometries, Math. Ann. 303 (1995), no. 1, 617–628. MR 1513292, DOI 10.1007/BF01461008
- Songwei Qian, $\epsilon$-isometric embeddings, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1797–1803. MR 1260178, DOI 10.1090/S0002-9939-1995-1260178-5
- Peter emrl and Jussi Väisälä, Nonsurjective nearisometries of Banach spaces, J. Funct. Anal. 198 (2003), no. 1, 268–278. MR 1962360, DOI 10.1016/S0022-1236(02)00049-6
- Igor A. Vestfrid, Stability of almost surjective $\varepsilon$-isometries of Banach spaces, J. Funct. Anal. 269 (2015), no. 7, 2165–2170. MR 3378871, DOI 10.1016/j.jfa.2015.04.009
- Igor A. Vestfrid, Hyers-Ulam stability of isometries and non-expansive maps between spaces of continuous functions, Proc. Amer. Math. Soc. 145 (2017), no. 6, 2481–2494. MR 3626505, DOI 10.1090/proc/13383
Bibliographic 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