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This paper studies the set of
points which have a unique best approximation from a set M in a Banach space X.
The ideal is for every element of X to have a unique best approximation (M is then
called Chebyshev). Unfortunately, finite dimensional subspaces of L1[0,1] fail to have
this property. To remedy this problem and a similar situation in C(T), A. L. Garkavi
introduced almost Chebyshev subspaces as those for which the set of elements of
X which do not have unique best approximations from M is of the first
category.
A class of subsets is determined, containing all finite dimensional subspaces of
L1(μ;E) where μ is a non-atomic measure and E is a Banach space, which, though
not Chebyshev, are almost Chebyshev.
Next characterizations are given of the finite dimensional almost Chebyshev
subspaces of L1(μ;R) when μ is arbitrary. Finally, these results are applied to C(T)∗,
the Banach space of bounded Borel measures on a compact Hausdorff space T,
determining the finite dimensional almost Chebyshev subspaces of C(T)∗. Scattered
throughout the paper are results on the existence (or nonexistence, as the case may
be) of continuous selections for the metric projections, including a characterization of
the finite dimensional subspaces of C(T)∗ which support lower semi-continuous
metric projections.
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