Vol. 53, No. 2, 1974

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Almost Chebyshev subspaces of L1(μ; E)

Edward Ralph Rozema

Vol. 53 (1974), No. 2, 585–604

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.

Mathematical Subject Classification 2000
Primary: 41A65
Secondary: 46E40
Received: 18 January 1973
Revised: 30 May 1973
Published: 1 August 1974
Edward Ralph Rozema