Opuscula Math. 31, no. 4 (2011), 645-650

Opuscula Mathematica

Strengthened Stone-Weierstrass type theorem

Piotr Niemiec

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.

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  • Piotr Niemiec
  • Jagiellonian University, Institute of Mathematics, ul. Łojasiewicza 6, 30-348 Krakow, Poland
  • Received: 2010-07-30.
  • Revised: 2010-12-01.
  • Accepted: 2010-12-12.
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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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