## 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

### MSC:

 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
Full Text:

### References:

 [1] E. Michael : Selected selection theorems , Amer. Math. Monthly 63 (1956) 233-238. · Zbl 0070.39502 [2] C. Olech : Approximation of set-valued functions by continuous functions , Colloquium Mathematicum 19 (1968) 285-303. · Zbl 0183.13603 [3] J.B. Prolla and S. Machado : Stone-Weierstrass theorems for set-valued mappings , Journal of Approximation Theory, Vol. 36 (1982) 1-15. · Zbl 0493.41042 [4] J.B. Prolla : Topics in Functional Analysis over Valued Division Rings , North-Holland, Mathematics Studies; 77 (1982). · Zbl 0506.46059 [5] M.Z.M.C. Scares : Non-Archimedean Nachbin Spaces , Portugaliae Mathematica, Vol. 39 (1980) to appear. · Zbl 0538.46058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.