Proximinality inLp(μ, X)

Measurable selectors for the metric projection

“…which properly extends the known results about proximinality of L p (µ, Y ) in L p (µ, X) proved by Light and Cheney in [12] -for Y finite dimensional subspace of L 1 (µ), by Khalil in [8] -for reflexive subspaces Y , by Khalil and Saidi in [9] -for separable quasireflexive proximinal subspaces Y and by Mendoza in [13] -for separable proximinal subspaces Y , because all these subspaces Y are proximinal and weakly K-analytic. Let us stress that after Theorem 4, L p (µ, Y ) is proximinal in L p (µ, X) even for all proximinal subspaces Y when X = L 1 (µ) and for all quasireflexive proximinal subspaces Y of X without the separability assumption in [9].…”

“…On the other hand, Mendoza has built in [13] a Banach space X with a proximinal subspace Y such that L p (µ, Y ) is not proximinal in L p (µ, X) where 1 ≤ p ≤ ∞. This implies, in particular, that the selection Theorem 3.3 is not true in general for arbitrary proximinal subspaces.…”

Best Simultaneous Approximation in Lp(I,E)

“…For some of the strongest results on this question, we refer the reader to [12] and [8]. Therefore, one can obtain several corollaries from Theorem 2.…”

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

“…In particular, in the case when (S, Σ , µ) is a finite measure space, it was proved in [4] that L 1 (S, Σ , Y ) is proximinal in L 1 (S, Σ , X ) if Y is reflexive and in [5] that L p (S, Σ , Y ) is proximinal in L p (S, Σ , X ) if and only if L 1 (S, Σ , Y ) is proximinal in L 1 (S, Σ , X ). These results have been extended to the case when (S, Σ , µ) is a σ -finite measure space in [13], where it was further proved for a closed separable subspace Y that L p (S, Σ , Y ) is proximinal in L p (S, Σ , X ) if and only if Y is proximinal in X .…”

“…In the case when Y is a closed subspace of X , the problem whether L p (S, Σ , Y ) is proximinal in L p (S, Σ , X ) has been studied deeply and extensively, see for example [4][5][6]11,13,15,18]. In particular, in the case when (S, Σ , µ) is a finite measure space, it was proved in [4] that L 1 (S, Σ , Y ) is proximinal in L 1 (S, Σ , X ) if Y is reflexive and in [5] that L p (S, Σ , Y ) is proximinal in L p (S, Σ , X ) if and only if L 1 (S, Σ , Y ) is proximinal in L 1 (S, Σ , X ).…”