@Article{Niemiec2011,
author = "Niemiec, Piotr",
title = "Strengthened Stone-Weierstrass type theorem",
journal = "Opuscula Math.",
year = "2011",
volume = "31",
number = "4",
pages = "645--650",
doi = "http://dx.doi.org/10.7494/OpMath.2011.31.4.645",
url = "http://dx.doi.org/10.7494/OpMath.2011.31.4.645",
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",
issn = "1232-9274"
}