dedicated to the memory of professor deeb hussein

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Given any set Γ, let be the family of all finite subsets of . Let f:[0, ∞) → R satisfying: (1) f(x) = 0 if and only if x = 0, (2) f is increasing, (3) f(x + y) ≧ f(x) + f(y) for all x, y ≦ 0, and (4) f is continuous at zero from the right. Such an f is called a modules. Let C be the set of all moduli, and F = {fv ∊ C:v ∊ Γ). Q(Γ) will denote the set of all such F, s. For each F ∊ Q(Γ) letthe summation is taken over Γ, and setIf Γ is countable Q(Γ) will be denoted by Q and LΓ(F) by L(F). LetNote thatsee [4, 5 and 6].

It is the object of this paper is to study the proximinality of L Φ (I, Y ) in L Φ (I, X) for some class of Young's functions Φ, where I is the unit interval. We prove (among other results) that if Y is a separable proximinal subspace of X, then L Φ (I, Y ) is proximinal in L Φ (I, X).

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