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The linearity of metric projection operator for subspaces of \(L_{p}\) spaces. (English. Russian original) Zbl 1304.41019

Mosc. Univ. Math. Bull. 65, No. 1, 28-33 (2010); translation from Vest. Mosk. Univ. Mat. Mekh. 65, No. 1, 36-41 (2010).
Summary: Let \(Y\) be a Chebyshev subspace of a Banach space \(X\). Then the single-valued metric projection operator \(P_{Y}:X \in Y\) taking each \(x \in X\) to the nearest element \(y \in Y\) is well defined. Let \(M\) be an arbitrary set and \(\mu\) be a \(\sigma\)-finite measure on some \(\sigma\)-algebra \(\Sigma\) of subsets of \(M\). We give a description of Chebyshev subspaces \(Y \subset L_{p}(M, \Sigma, \mu)\) with a finite dimension and a finite codimension which the operator \(P_{Y}\) is linear for.

MSC:

41A50 Best approximation, Chebyshev systems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B20 Geometry and structure of normed linear spaces
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References:

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