Let ℬ(X) be the space of all
bounded real-valued functions on a set X, with the norm ∥f∥ =sup{|f(x)| : x ∈ X}, and
let K be any nonempty subset of ℬ(X). The question whether an element f of ℬ(X)
has a best approximation g in K (such that ∥f −g∥ = δ(f) =inf{∥f −h∥ : h ∈ K})
can be formulated as the problem of interposing a function g in K between two
functions, L(⋅,f) and U(⋅,f), which are constructed out of K by certain
lattice operations. If K is closed with respect to these lattice operations, or
has a certain interposition property, the best approximation will always
exist.
