Best approximants from non-Archimedean Stone-Weierstrass subspaces. (English) Zbl 0605.46057

This paper studies approximation theory in non-Archimedean normed spaces. If a normed space \((N,\| \cdot \|)\) and a non-empty subset \(W\subset N\) are given, there are two main problems. The first is to characterize the closure of \(W\) in \(N\); this can lead to Stone-Weierstrass type theorems.
If \(\mathrm{dist}(f,W)>0\) we can ask whether \(f\) has a best approximant in \(W\). More generally, if \(B\) is a bounded subset of \(N\) we can look for a Chebyshev center of \(B\) in \(W\), i.e. a \(g\in W\) such that \[ \sup_{f\in B}\| f- g\| =\inf_{w\in W}\sup_{f\in B}\| f-w\|. \] The paper contains results on these and related questions when \(N\) is a space of continuous functions on a locally compact Hausdorff space and with values in a non-Archimedean normed space.
Reviewer: W.Govaerts


46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46E40 Spaces of vector- and operator-valued functions
46E10 Topological linear spaces of continuous, differentiable or analytic functions
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