In this paper, the imbedding of the normed linear space R into its weighted direct sumspace R_ω~P is applied to the discussion on the characterization theorems of functional and variat-ional forms of the best simultaneous approximation by using the elements of an arbitrary convexset G in the space R. The main result is as follows. Theorem. Let G be a convex subset in a normed linear space. Assume that a sequence{x_n} R consisting of distinct points or x is not in G. Then y_0∈G ({x_n}) if and only ifthere exists a sequence {f_n} of bounded linear functionals such thatfor p=1:(1)||f_n||=1, N=1, 2, …;(2)Re sum form n=1 to ∞α_nf_n(y-y_0)≥0, y∈G;(3)Ref_n(x_n-y_0)=||x_n-y_0||, n=1, 2, ….for p>1:(1)||f_n||=(||x_n-y_0||~(p/q))/(sum form n=1 to ∞ α_n||x_n-y_0||~p)~(1/q), N=1, 2, …;(2)Re sum from n=1 to ∞ f_n(y_0-y)≥0, y∈G; (3)Ref_n(x_n-y_0)=||f_n||·||x_n-y_0||=(||x_n-y_0||~p/(sum form n=1 to ∞ α_n||x_n-y_0||~p)~(1/q),As an example, we give the characterization of the best simultaneous L_p-approximation.