Abstract:
In the space C(Q) of real functions that are continuous on the compact set Q, a finite-dimensional subspace P will have a uniformly continuous metric projection if and only if Q is a finite sum of compact sets Qi, and either P is on each Qi a one-dimensional Chebyshev space, or \mathrm{x(t)\equiv0\mathrm} for any x belonging to P. The metric projection onto any finite-dimensional subspace of the space L[a,b] of real integrable functions is not uniformly continuous.
Citation:
V. I. Berdyshev, “Metric projection onto finite-dimensional subspaces of C and L”, Mat. Zametki, 18:4 (1975), 473–488; Math. Notes, 18:4 (1975), 871–879
This publication is cited in the following 6 articles:
A. R. Alimov, I. G. Tsar'kov, “Chebyshev centres, Jung constants, and their applications”, Russian Math. Surveys, 74:5 (2019), 775–849
I. G. Tsarkov, “Ustoichivost otnositelnogo chebyshëvskogo proektora v poliedralnykh prostranstvakh”, Tr. IMM UrO RAN, 24, no. 4, 2018, 235–245
K. V. Chesnokova, “The Linearity Coefficient of Metric Projections onto One-Dimensional Chebyshev Subspaces of the Space $C$”, Math. Notes, 96:4 (2014), 556–562
P. A. Borodin, “The Linearity Coefficient of the Metric Projection onto a Chebyshev Subspace”, Math. Notes, 85:1 (2009), 168–175