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Linearity of metric projections on Chebyshev subspaces inL 1 andC

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Abstract

LetY be a Chebyshev subspace of a Banach spaceX. Then the single-valued metric projection operatorP Y :X → Y taking eachx ∈X to the nearest elementy ∈ Y is well defined. LetM be an arbitrary set, and letμ be aσ-finite measure on someσ-algebra gS of subsets ofM. We give a complete description of Chebyshev subspacesY ⊂L 1(M, Σ,μ) for which the operatorP Y is linear (for the spaceL 1[0, 1], this was done by Morris in 1980). We indicate a wide class of Chebyshev subspaces inL 1(M, Σ,μ), for which the operatorP Y is nonlinear in general. We also prove that the operatorP Y , whereY ⊂C[K] is a nontrivial Chebyshev subspace andK is a compactum, is linear if and only if the codimension ofY inC[K] is equal to 1.

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Authors and Affiliations

  1. M. V. Lomonosov Moscow State University, USSR

    P. A. Borodin

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  1. P. A. Borodin
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Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 812–820, June, 1998.

The author is grateful to E. P. Dolzhenko nd S. V. Konyagin for their attention to the work.

This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01366.

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Borodin, P.A. Linearity of metric projections on Chebyshev subspaces inL 1 andC . Math Notes 63, 717–723 (1998). https://doi.org/10.1007/BF02312764

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  • DOI: https://doi.org/10.1007/BF02312764

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