Opuscula Mathematica
Opuscula Math. 31, no. 4 (), 645-650
http://dx.doi.org/10.7494/OpMath.2011.31.4.645
Opuscula Mathematica

Strengthened Stone-Weierstrass type theorem


Abstract. The aim of the paper is to prove that if \(L\) is a linear subspace of the space \(\mathcal{C}(K)\) of all real-valued continuous functions defined on a nonempty compact Hausdorff space \(K\) such that \(\min(|f|, 1) \in L\) whenever \(f \in L\), then for any nonzero \(g \in \overline{L}\) (where \(\overline{L}\) denotes the uniform closure of \(L\) in \(\mathcal{C}(K)\)) and for any sequence \((b_n)_{n=1}^{\infty}\) of positive numbers satisfying the relation \(\sum_{n=1}^{\infty} b_n = \|g\|\) there exists a sequence \((f_n)_{n=1}^{\infty}\) of elements of \(L\) such that \(\|f_n \|= b_n\) for each \(n \geq 1\), \(g = \sum _{n=1}^{\infty} f_n \) and \(|g|= \sum _{n=1}^{\infty} |f_n| \). Also the formula for \(\overline{L}\) is given.
Keywords: Stone-Weierstrass theorem, function lattices.
Mathematics Subject Classification: 41A65, 54C30, 54C40.
Cite this article as:
Piotr Niemiec, Strengthened Stone-Weierstrass type theorem, Opuscula Math. 31, no. 4 (2011), 645-650, http://dx.doi.org/10.7494/OpMath.2011.31.4.645
 
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ISSN 1232−9274, e-ISSN 2300−6919, DOI http://dx.doi.org/10.7494/OpMath
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