A note on coapproximation. (English) Zbl 0749.41034

Soni, D. C.; Bahadur, Lal

Indian J. Pure Appl. Math. 22, No. 7, 579-582 (1991).

Let \(S\) be a set of a normed space \(N\). An element \(s_ 0\in S\) is said to be a coapproximation of \(x\in N\) if \(\| s-s_ 0\|\leq\| x- s\|\) for all \(s\in S\). The authors prove the following results. If \(S\) is a zero hyperplane of \(N\) and \(u\) is single valued, where \(u(x)=\{s_ 0\in S:\;\| s-s_ 0\|\leq\| x-s\|\;\forall s\in S\}\) for \(x\in N\), then \(u\) is linear and \(\| u\|=1\); If \(S\) is a convex subset, then \(u(x)\) is a convex subset of \(S\); If \(N\) is a smooth normed space and \(S\) is a subspace of \(N\), then \(u(x)\) is single valued whenever \(u(x)\neq\emptyset\).

Reviewer: Zhang Ganglu (Dongying)

MSC:

41A65	Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
41A50	Best approximation, Chebyshev systems

Keywords:

coapproximation