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Existence of best simultaneous approximations in L p ( S , Σ , X )

Existence of best simultaneous approximations in L p ( S , Σ , X ),10.1016/j.jat.2011.04.007,Journal of Approximation Theory,Xian-Fa Luo,Chong Li,Hong

Existence of best simultaneous approximations in L p ( S , Σ , X )  
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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.
Journal: Journal of Approximation Theory - JAT , vol. 163, no. 9, pp. 1300-1316, 2011
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