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 x(t)≡0 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.
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Translated from Matematicheskie Zametki, Vol. 18, No. 4, pp. 473–488, October, 1975.
The author expresses his gratitude to S. B. Stechkin for his interest.
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Berdyshev, V.I. Metric projection onto finite-dimensional subspaces of C and L. Mathematical Notes of the Academy of Sciences of the USSR 18, 871–879 (1975). https://doi.org/10.1007/BF01153037
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DOI: https://doi.org/10.1007/BF01153037