Abstract
A theory of best approximation is developed in the normed linear space C( T, E), the space of E-valued bounded continuous functions on the locally compact Hausdorff space T, with the supremum norm. The approximating functions belong to the subspace C F ( T, E) of C( T, E) consisting of those functions which have “limit at infinity” which lies in the subspace F of the normed linear space E. A distance formula is obtained, and a selection for the metric projection onto C f ( T, E) is constructed which has many desirable properties. The theory includes study of best approximation in l ∞ by the subspace c 0, and closely parallels the known theory of best approximation by M-ideals (although our subspace is not an M-ideal, in general).
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