BEST SIMULTANEOUS APPROXIMATION IN METRIC SPACES

For a banach space x and an increasing subadditive continuous function ϕ on [0,∞) with ϕ (0) = 0, let us denote by l^ϕ (i,x), the space of all x-valued ϕ - integrable functions f: i→x on a certain positive complete σ-finite measure space (i, σ , μ, ) with ∫_i^(φ ) ii f(t) ii dμ(t) < ∞ and l^ϕ (x) =[ (xk) : ∑_(k=1) ^(∞ ) ϕii x_k < ∞, x_k ϵx] . the aim of this paper is to prove that for a closed separable subspace g of x, l^ϕ (i,g) is simultaneously proximinal in l^ϕ (i,x) if and only if g is simultaneously proximinal in x. other result on simultaneous approximation of l^ϕ (g) in l^ϕ (x) is presented.