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Coarse Isometries between Finite Dimensional Banach Spaces

Authors
  • Sun, Yuqi1
  • Zhang, Wen1
  • 1 Xiamen University, Xiamen, 361005, China , Xiamen (China)
Type
Published Article
Journal
Acta Mathematica Scientia
Publisher
Springer-Verlag
Publication Date
Jun 29, 2021
Volume
41
Issue
5
Pages
1493–1502
Identifiers
DOI: 10.1007/s10473-021-0506-5
Source
Springer Nature
Keywords
Disciplines
  • Article
License
Yellow

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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