Acta Mathematica Scientia

, Volume 39, Issue 4, pp 1163–1172 | Cite as

Stability of a Pair of Banach Spaces for ε-Isometries

  • Duanxu Dai (戴端旭)Email author
  • Bentuo Zheng (郑本拓)


A pair of Banach spaces (X, Y) is said to be stable if for every ε-isometry f : XY, there exist γ > 0 and a bounded linear operator T : L(f ) → X with ‖T‖ ≤ α such that ‖Tf (x) — x‖ ≤ γε for all xX, where L(f ) is the closed linear span of f (X). In this article, we study the stability of a pair of Banach spaces (X, Y) when X is a C(K) space. This gives a new positive answer to Qian’s problem. Finally, we also obtain a nonlinear version for Qian’s problem.

Key words

Stability ε-isometry Figiel theorem Banach space 

2010 MR Subject Classification

46B04 46B20 54C60 


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Copyright information

© Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  • Duanxu Dai (戴端旭)
    • 1
    Email author
  • Bentuo Zheng (郑本拓)
    • 2
  1. 1.College of Mathematics and Computer ScienceQuanzhou Normal UniversityQuanzhouChina
  2. 2.Department of Mathematical SciencesUniversity of MemphisMemphisUSA

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