Existence of best simultaneous approximations in L p ( S , Σ , X )
Let (S,Σ,μ) be a complete positive σ-finite measure space and let X be a Banach space. We are concerned with the proximinality problem for the best simultaneous approximations to two functions in Lp(S,Σ,X). Let Σ0 be a sub-σ-algebra of Σ and Y a nonempty locally weakly compact convex subset of X such that spanY¯ and its dual have the Radon–Nikodym property. We prove that Lp(S,Σ0,Y) is N-simultaneous proximinal in Lp(S,Σ,X) (with the additional assumption that (S,Σ,μ) be finite for the case when p=1). Furthermore, for the special case when Σ0=Σ, we show that the assumption that the dual of spanY¯ has the Radon–Nikodym property can be removed.