Pointwise Best Approximation in the Space of Strongly Measurable Functions with Applications to Best Approximation in Lp(μ,X) Original Research Article

The object of this paper is to prove the following theorem: Let Y be a closed subspace of the Banach space X, (S,Σ,μ) a σ-finite measure space, L(S,Y) (respectively, L(S, X)) the space of all strongly measurable functions from S to Y (respectively, X), and p a positive number. Then L(S,Y) is pointwise proximinal in L(S,X) if and only if Lp(μ,Y) is proximinal in Lp(μ,X). As an application of the theorem stated above, we prove that if Y is a separable closed subspace of the Banach space X, p is a positive number, then Lp(μ,Y) is proximinal in Lp(μ,X) if and only if Y is proximinal in X. Finally, several other interesting results on pointwise best approximation are also obtained.