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.