Opuscula Math. 31, no. 4 (2011), 645-650
http://dx.doi.org/10.7494/OpMath.2011.31.4.645

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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