TY  - JOUR
ID  - Niemiec2011
LB  - Niemiec2011
AU  - Niemiec, Piotr
TI  - Strengthened Stone-Weierstrass type theorem
ST  - Strengthened Stone-Weierstrass type theorem
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2011
VL  - 31
IS  - 4
SP  - 645
EP  - 650
DO  - http://dx.doi.org/10.7494/OpMath.2011.31.4.645
UR  - http://dx.doi.org/10.7494/OpMath.2011.31.4.645
AB  - 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.
KW  - Stone-Weierstrass theorem
KW  - function lattices
SN  - 1232-9274
ER  -