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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the Lipschitz classification of normed spaces, unit balls, and spheres
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by Ronny Nahum PDF
Proc. Amer. Math. Soc. 129 (2001), 1995-1999 Request permission

Abstract:

For every normed space $Z$, we note its closed unit ball and unit sphere by $B_Z$ and $S_Z$, respectively. Let $X$ and $Y$ be normed spaces such that $S_X$ is Lipschitz homeomorphic to $S_{X \oplus R}$, and $S_Y$ is Lipschitz homeomorphic to $S_{Y \oplus R}$. We prove that the following are equivalent: 1. $X$ is Lipschitz homeomorphic to $Y$. 2. $B_X$ is Lipschitz homeomorphic to $B_Y$. 3. $S_X$ is Lipschitz homeomorphic to $S_Y$. This result holds also in the uniform category, except (2 or 3) $\Rightarrow$ 1 which is known to be false.
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Additional Information
  • Ronny Nahum
  • Affiliation: Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel
  • Email: ronnyn@techunix.technion.ac.il
  • Received by editor(s): April 20, 1998
  • Received by editor(s) in revised form: October 20, 1999
  • Published electronically: December 13, 2000
  • Additional Notes: This paper is a part of the author’s Ph.D. thesis, prepared at the University of Haifa under the supervision of Prof. Y. Sternfeld.
  • Communicated by: Dale Alspach
  • © Copyright 2000 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 129 (2001), 1995-1999
  • MSC (2000): Primary 46B20
  • DOI: https://doi.org/10.1090/S0002-9939-00-05782-8
  • MathSciNet review: 1825907