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 that have “limit at infinity” that lie 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).