Abstract
Assume that X and Y are real Banach spaces with the same finite dimension. In this paper we show that if a standard coarse isometry f: X → Y satisfies an integral convergence condition or weak stability on a basis, then there exists a surjective linear isometry U: X → Y such that ‖f (x) − Ux‖ = o(‖x‖) as ‖x‖ → ∞. This is a generalization about the result of Lindenstrauss and Szankowski on the same finite dimensional Banach spaces without the assumption of surjectivity. As a consequence, we also obtain a stability result for ε-isometries which was established by Dilworth.
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Acknowledgements
The authors would like to thank the colleagues and graduate students in the Functional Analysis group of Xiamen University for their very helpful conversations and suggestions.
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Supported by National Natural Science Foundation of China (11731010 and 12071388).
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Sun, Y., Zhang, W. Coarse Isometries between Finite Dimensional Banach Spaces. Acta Math Sci 41, 1493–1502 (2021). https://doi.org/10.1007/s10473-021-0506-5
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DOI: https://doi.org/10.1007/s10473-021-0506-5