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ArticleOriginal scientific text

Title

Averages of uniformly continuous retractions

Authors 1, 2, 3, 1

Affiliations

  1. Departamento de Álgebra y Análisis Matemático, Universidad de Almería, 04120 Almería, Spain
  2. Departamento de Análisis, Matemático Universidad de Granada, 18071 Granada, Spain
  3. Department of Mathematics, Technion, Haifa 32000, Israel

Abstract

Let X be an infinite-dimensional complex normed space, and let B and S be its closed unit ball and unit sphere, respectively. We prove that the identity map on B can be expressed as an average of three uniformly retractions of B onto S. Moreover, for every 0≤ r < 1, the three retractions are Lipschitz on rB. We also show that a stronger version where the retractions are required to be Lipschitz does not hold.

Keywords

ENG
uniformly retraction, Lipschitz retraction, extreme point

Bibliography

  1. Y. Benyamini and Y. Sternfeld, Spheres in infinite dimensional normed spaces are Lipschitz contractible, Proc. Amer. Math. Soc. 88 (1983), 439-445.
  2. V. Bogachev, J. F. Mena-Jurado and J. C. Navarro-Pascual, Extreme points in spaces of continuous functions, ibid. 123 (1995), 1061-1067.
  3. J. Cantwell, A topological approach to extreme points in function spaces, ibid. 19 (1968), 821-825.
  4. A. Jiménez-Vargas, J. F. Mena-Jurado and J. C. Navarro-Pascual, Complex extremal structure in spaces of continuous functions, J. Math. Anal. Appl. 211 (1997), 605-615.
  5. P. K. Lin and Y. Sternfeld, Convex sets with the Lipschitz fixed point property are compact, Proc. Amer. Math. Soc. 93 (1985) 633-639.
  6. J. C. Navarro-Pascual, Extreme points and retractions in Banach spaces, Israel J. Math. 99 (1997), 335-342.
  7. B. 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), 861-864.
  8. N. T. Peck, Extreme points and dimension theory, Pacific J. Math. 25 (1968), 341-351.
Studia Mathematica
1999/135/1
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Pages:
75-81
Main language of publication
English
Received
1998-04-28
Accepted
1998-11-26
Published
1999
Exact and natural sciences