Strengthened Stone-Weierstrass type theorem

The aim of the paper is to prove that if L is a linear subspace of the space 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 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.