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 Mat. Zametki: Year: Volume: Issue: Page: Find

 Mat. Zametki, 1998, Volume 63, Issue 6, Pages 812–820 (Mi mz1351)

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.

DOI: https://doi.org/10.4213/mzm1351

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

Bibliographic databases:

UDC: 517.982.256
Revised: 05.03.1997

Citation: P. A. Borodin, “Linearity of metric projections on Chebyshev subspaces in $L_1$ and $C$”, Mat. Zametki, 63:6 (1998), 812–820; Math. Notes, 63:6 (1998), 717–723

Citation in format AMSBIB
\Bibitem{Bor98} \by P.~A.~Borodin \paper Linearity of metric projections on Chebyshev subspaces in $L_1$ and $C$ \jour Mat. Zametki \yr 1998 \vol 63 \issue 6 \pages 812--820 \mathnet{http://mi.mathnet.ru/mz1351} \crossref{https://doi.org/10.4213/mzm1351} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1679213} \zmath{https://zbmath.org/?q=an:0917.41018} \transl \jour Math. Notes \yr 1998 \vol 63 \issue 6 \pages 717--723 \crossref{https://doi.org/10.1007/BF02312764} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000076726600023} 

• http://mi.mathnet.ru/eng/mz1351
• https://doi.org/10.4213/mzm1351
• http://mi.mathnet.ru/eng/mz/v63/i6/p812

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This publication is cited in the following articles:
1. P. A. Borodin, V. M. Tikhomirov, “Kriterii gilbertovosti banakhova prostranstva, svyazannye s teoriei priblizhenii”, Matem. prosv., ser. 3, 3, MTsNMO, M., 1999, 189–207
2. I. A. Pyatyshev, “Operations on Approximatively Compact Sets”, Math. Notes, 82:5 (2007), 653–659
3. P. A. Borodin, “The Linearity Coefficient of the Metric Projection onto a Chebyshev Subspace”, Math. Notes, 85:1 (2009), 168–175
4. P. A. Borodin, “$2$-Chebyshev Subspaces in the Spaces $L_1$ and $C$”, Math. Notes, 91:6 (2012), 770–781
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