Vol. 24, No. 1, 1968

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Interposition and approximation

Bernard Robert Kripke and Richard Bruce Holmes

Vol. 24 (1968), No. 1, 103–110

Let (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 (X). The question whether an element f of (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.

Mathematical Subject Classification
Primary: 41.40
Received: 6 August 1966
Published: 1 January 1968
Bernard Robert Kripke
Richard Bruce Holmes