Let X be an infinite-dimensional uniformly convex Banach space and let BX and SX be its closed unit ball and unit sphere, respectively. The main result of the paper is that the identity mapping on BX can be expressed as the mean of n uniformly continuous retractions from BX onto SX for every n >= 3. Then, the authors observe that the result holds under a property weaker than uniform convexity, satisfied by any complex Banach space, so that the result generalizes that of [A. Jim´enez-Vargas et al., Studia Math. 135 (1999), no. 1, 75–81; MR1686372 (2000b:46025)]. As an application the extremal structure of spaces of vector-valued uniformly continuous mappings is studied.
CAPONETTI, D. (2009). MR2370688 (2009e:46013) Navarro-Pascual, J. C.; Mena-Jurado, J. F.; Sánchez-Lirola, M. G. A two-dimensional inequality and uniformly continuous retractions. J. Math. Anal. Appl. 339 (2008), no. 1, 719--734. (Reviewer: Diana Caponetti) 46B20 (46E40).
MR2370688 (2009e:46013) Navarro-Pascual, J. C.; Mena-Jurado, J. F.; Sánchez-Lirola, M. G. A two-dimensional inequality and uniformly continuous retractions. J. Math. Anal. Appl. 339 (2008), no. 1, 719--734. (Reviewer: Diana Caponetti) 46B20 (46E40)
CAPONETTI, Diana
2009-01-01
Abstract
Let X be an infinite-dimensional uniformly convex Banach space and let BX and SX be its closed unit ball and unit sphere, respectively. The main result of the paper is that the identity mapping on BX can be expressed as the mean of n uniformly continuous retractions from BX onto SX for every n >= 3. Then, the authors observe that the result holds under a property weaker than uniform convexity, satisfied by any complex Banach space, so that the result generalizes that of [A. Jim´enez-Vargas et al., Studia Math. 135 (1999), no. 1, 75–81; MR1686372 (2000b:46025)]. As an application the extremal structure of spaces of vector-valued uniformly continuous mappings is studied.
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