Averages of uniformly continuous retractions

A. Jiménez-Vargas; J. Mena-Jurado; R. Nahum; J. Navarro-Pascual

Studia Mathematica (1999)

  • Volume: 135, Issue: 1, page 75-81
  • ISSN: 0039-3223

Abstract

top
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.

How to cite

top

Jiménez-Vargas, A., et al. "Averages of uniformly continuous retractions." Studia Mathematica 135.1 (1999): 75-81. <http://eudml.org/doc/216644>.

@article{Jiménez1999,
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.},
author = {Jiménez-Vargas, A., Mena-Jurado, J., Nahum, R., Navarro-Pascual, J.},
journal = {Studia Mathematica},
keywords = {uniformly retraction; Lipschitz retraction; extreme point; uniformly continuous; averages; retractions; continuous retractions},
language = {eng},
number = {1},
pages = {75-81},
title = {Averages of uniformly continuous retractions},
url = {http://eudml.org/doc/216644},
volume = {135},
year = {1999},
}

TY - JOUR
AU - Jiménez-Vargas, A.
AU - Mena-Jurado, J.
AU - Nahum, R.
AU - Navarro-Pascual, J.
TI - Averages of uniformly continuous retractions
JO - Studia Mathematica
PY - 1999
VL - 135
IS - 1
SP - 75
EP - 81
AB - 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.
LA - eng
KW - uniformly retraction; Lipschitz retraction; extreme point; uniformly continuous; averages; retractions; continuous retractions
UR - http://eudml.org/doc/216644
ER -

References

top
  1. [1] Y. Benyamini and Y. Sternfeld, Spheres in infinite dimensional normed spaces are Lipschitz contractible, Proc. Amer. Math. Soc. 88 (1983), 439-445. 
  2. [2] V. Bogachev, J. F. Mena-Jurado and J. C. Navarro-Pascual, Extreme points in spaces of continuous functions, ibid. 123 (1995), 1061-1067. Zbl0832.46030
  3. [3] J. Cantwell, A topological approach to extreme points in function spaces, ibid. 19 (1968), 821-825. Zbl0175.13403
  4. [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. Zbl0920.46028
  5. [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. Zbl0566.47039
  6. [6] J. C. Navarro-Pascual, Extreme points and retractions in Banach spaces, Israel J. Math. 99 (1997), 335-342. Zbl0901.46015
  7. [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. Zbl0472.54008
  8. [8] N. T. Peck, Extreme points and dimension theory, Pacific J. Math. 25 (1968), 341-351. Zbl0157.29603

NotesEmbed ?

top

You must be logged in to post comments.