Abstract
The closed subspaces of finite codimension of the space C(X) of all continuous real-valued functions on a compact Hausdorff space X, for which the set of elements of best approximations of every function f ε C(X) is nonempty and compact, are characterized. It is shown that if the compact Hausdorff space X is infinite, then C(X) has no subspace of a finite Codimension n > 1 which has a nonempty set of elements of the best approximation for an arbitrary function f 6 ε(X) and which has an upper-semicontinuous metric projection.
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Literature cited
V. L. Klee, “Convexity of Chebyshev sets,” Math. Ann.,142, No. 3, 292–304 (1961).
N. V. Efimov and S. B. Stechkin, “Some properties of Chebyshev sets,” Dokl. Akad. Nauk SSSR,118, No. 1, 17–19 (1958).
L. P. Vlasov, “Approximation properties of sets in normed linear spaces,” Usp. Matem. Nauk,28, No. 6, 3–66 (1973).
E. A. Michael, “Topologies on spaces of subsets,” Trans. Amer. Math. Soc.,71, 152–182 (1951).
N. Dunford and J. T. Schwartz, Linear Operators, Vol. I, Wiley, New York (1958).
A. L. Garkavi, “The Helly problem and the best approximation in the space of continuous functions,” Izv. Akad. Nauk SSSR, Ser. Matem.,31, No. 3, 641–656 (1967).
J. Blatter, P. D. Morris, and D. E. Wulbert, “Continuity of the set-valued metric projection,” Math. Ann.,178, No. 1, 12–24 (1968).
P. D. Morris, “Metric projections onto subspaces of finite codimension,” Duke Math. J.,35, No. 4, 797–808 (1968).
E. V. Oshman, “The characterization of subspaces with a continuous metric projection in a normed linear space,” Dokl. Akad. Nauk SSSR,207, No. 2, 292–295 (1972).
B. Brosowski and F. Deutsch, “Radial continuity of set-valued metric projections,” J. Approximation Theory,11, No. 3, 236–253 (1974).
A. L. Garkavi, “On compacta admitting Chebyshev systems of measures,” Matem. Sb.,74, No. 2, 209–217 (1967).
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Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 531–539, April, 1976.
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Oshman, E.V. The continuity of the metric projection on a subspace of finite codimension in the space of continuous functions. Mathematical Notes of the Academy of Sciences of the USSR 19, 324–328 (1976). https://doi.org/10.1007/BF01156791
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DOI: https://doi.org/10.1007/BF01156791