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Stone-Weierstrass and extension theorems in the nonlocally convex case. (English) Zbl 1401.46026

In the literature, some Stone-Weierstrass-type theorems for spaces of continuous functions with values in locally convex spaces are available. In the case where the range space is not locally convex but a general topological vector space, less seems to be known. Motivated by a result of A. H. Shuchat [Proc. Am. Math. Soc. 31, 97–103 (1972; Zbl 0247.41026)] concerning the approximation of continuous vector-valued functions by functions with finite-dimensional range, the authors consider spaces of continuous functions defined on a Hausdorff topological space with the property that all its compact sets have finite Lebesgue covering dimension. In this setting, two general Stone-Weierstrass theorems are proved. The article also contains a series of interesting applications and corollaries to these theorems.
This article can be seen as a continuation of the work by the third author [Proc. Edinb. Math. Soc., II. Ser. 22, 35–41 (1979; Zbl 0429.46023); J. Math., Univ. Punjab 12–13 (1979), 11–14 (1980; Zbl 0502.46021); Period. Math. Hung. 30, No. 1, 81–86 (1995; Zbl 0821.46029)].

MSC:

46E40 Spaces of vector- and operator-valued functions
46E10 Topological linear spaces of continuous, differentiable or analytic functions
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