Mathematics

The linearity of metric projection operator for subspaces of $L_p$ spaces

Yu. Yu. Druzhinin

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: Let $Y$ be a Chebyshev subspace of a Banach space $X$. Then the single-valued metric projection operator $P_Y:X\to Y$ taking each $x\in X$ to the nearest element $y\in Y$ is well defined. Let $M$ be an arbitrary set and $\mu$ 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 finite dimension and finite codimension the operator $P_Y$ is linear for.

Key words: metric projection, Chebyshev subspace, quasiorthogonal set, linearity criterion.

UDC: 517.982.256

Received: 20.10.2008

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