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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Spheres in infinite-dimensional normed spaces are Lipschitz contractible


Authors: Y. Benyamini and Y. Sternfeld
Journal: Proc. Amer. Math. Soc. 88 (1983), 439-445
MSC: Primary 46B20; Secondary 57N17
MathSciNet review: 699410
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X$ be an infinite-dimensional normed space. We prove the following:

(i) The unit sphere $ \{ x \in X:\left\Vert x \right\Vert = 1\} $ is Lipschitz contractible.

(ii) There is a Lipschitz retraction from the unit ball of $ X$ onto the unit sphere.

(iii) There is a Lipschitz map $ T$ of the unit ball into itself without an approximate fixed point, i.e. $ \inf \{ \left\Vert {x - Tx} \right\Vert:\left\Vert x \right\Vert \leqslant 1\} > 0$.


References [Enhancements On Off] (What's this?)

  • [1] C. Bessaga and A. Pełczyński, Selected topics in infinite dimensional topology, PWN, Warsaw, 1975.
  • [2] Mahlon M. Day, Normed linear spaces, 3rd ed., Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21. MR 0344849 (49 #9588)
  • [3] K. Goebel, On the minimal displacement of points under Lipschitzian mappings, Pacific J. Math. 45 (1973), 151–163. MR 0328708 (48 #7050)
  • [4] K. Goebel and W. A. Kirk, A fixed point theorem for transformations whose iterates have uniform Lipschitz constant, Studia Math. 47 (1973), 135–140. MR 0336468 (49 #1242)
  • [5] Bogdan Nowak, On the Lipschitzian retraction of the unit ball in infinite-dimensional Banach spaces onto its boundary, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 11-12, 861–864 (1981) (English, with Russian summary). MR 616177 (82g:58008)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0699410-7
PII: S 0002-9939(1983)0699410-7
Article copyright: © Copyright 1983 American Mathematical Society



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