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(/ƒ/, 1) ∈ L whenever fnof; ∈ L, then for any nonzero g ∈ L (where L denotes the uniform closure of L in C(K)) and for any sequence (bn) ∞/n=1 of positive numbers satisfying the relation [formula] there exists a sequence [formula] of elements of L such that //ƒn// = bn for each n ≥ 1, g = [formula] and /g/| = [formula]. Also the formula for L is given.