Abstract
We develop a theory of best simultaneous approximation for closed convex sets in C ℝ(Q), the space of all real-valued continuous functions on a compact topological space Q endowed with the usual operations and with the norm ‖x‖ = max q∈Q |x(q)|. We give necessary and sufficient conditions for the existence of best simultaneous approximation in a conditionally complete Banach lattice X with a strong unit 1 by elements of the hyperplanes. We study best simultaneous approximation by elements of closed convex sets in C ℝ(Q) and give various characterizations of best simultaneous approximation.
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