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JOURNALS // Matematicheskie Zametki // Archive

Mat. Zametki, 1998 Volume 63, Issue 6, Pages 812–820 (Mi mzm1351)

This article is cited in 6 papers

Linearity of metric projections on Chebyshev subspaces in $L_1$ and $C$

P. A. Borodin

M. V. Lomonosov Moscow State University

Abstract: Let $Y$ be a Chebyshev subspace of a Banach space $X$. Then the single-valued metric projection operator $P_Y\colon X\to Y$ taking each $x\in X$ to the nearest element $y\in Y$ is well defined. Let $M$ be an arbitrary set, and let be a-finite measure on some $\sigma$-algebra $gS$ of subsets of $M$. We give a complete description of Chebyshev subspaces $Y\in L_1(M,\Sigma,\mu)$ for which the operator $P_Y$ is linear (for the space $L_1[0,1]$, this was done by Morris in 1980). We indicate a wide class of Chebyshev subspaces in $L_1(M,\Sigma,\mu)$, for which the operator $P_Y$ is nonlinear in general. We also prove that the operator $P_Y$, where $Y\subset C[K]$ is a nontrivial Chebyshev subspace and $K$ is a compactum, is linear if and only if the codimension of $Y$ in $C[K]$ is equal to 1.

UDC: 517.982.256

Received: 13.05.1996
Revised: 05.03.1997

DOI: 10.4213/mzm1351


 English version:
Mathematical Notes, 1998, 63:6, 717–723

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