Strengthened Stone-Weierstrass type theorem

• Main Author: Piotr Niemiec
• Format: Article
• Language: English
• Published: AGH Univeristy of Science and Technology Press 2011-01-01
• Series: Opuscula Mathematica
• Subjects: Stone-Weierstrass theorem; function lattices
• Online Access: http://www.opuscula.agh.edu.pl/vol31/4/art/opuscula_math_3143.pdf

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.