Best approximations in $L_p (S, X)$
Bull. Korean Math. Soc. 1999 Vol. 36, No. 3, 589-597
Mun Bae Lee, Sung Ho Park, and Hyang Joo Rhee
Sogang University, Sogang University, Duksung Women's University
Abstract : Let $G$ be a closed subspace of a Banach space $X$ and let
$(S, \Omega, \mu)$ be a $\sigma$-finite measure space.
It was known that
$L_1(S,G)$ is proximinal in $L_1(S,X)$
if and only if
$L_p(S,G)$ is proximinal in $L_p(S,X)$ for $1 In this article we show that this result remains true when "proximinal" is replaced by
"Chebyshev".
In addition, it is shown that if $G$ is a proximinal subspace of $X$ such that
either $G$ or the kernel of the metric projection $P_G$ is separable then, for
$0

Keywords : Chebyshev, proximinal, pointwise proximinal, proximity map, strongly measurable
MSC numbers : 41A65, 46B20
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