Backward adaptive biorthogonalization.
The notion of ball proximinality and the strong ball proximinality were recently introduced in [2]. We prove that a closed * subalgebra A of C(Q) is strongly ball proximinal in C(Q) and the metric projection from C(Q), onto the closed unit ball of A, is Hausdorff metric continuous and hence has continuous selection.
We study Banach spaces X with subspaces Y whose unit ball is densely remotal in X. We show that for several classes of Banach spaces, the unit ball of the space of compact operators is densely remotal in the space of bounded operators. We also show that for several classical Banach spaces, the unit ball is densely remotal in the duals of higher even order. We show that for a separable remotal set E ⊆ X, the set of Bochner integrable functions with values in E is a remotal set in L¹(μ,X).
Let be a nonempty compact subset of a Banach space and denote by the family of all nonempty bounded closed convex subsets of . We endow with the Hausdorff metric and show that there exists a set such that its complement is -porous and such that for each and each , the set of solutions of the best approximation problem , , is nonempty and compact, and each minimizing sequence has a convergent subsequence.
If f is a continuous seminorm, we prove two f-best approximation theorems for functions Φ not necessarily continuous as a consequence of our version of Glebov's fixed point theorem. Moreover, we obtain another fixed point theorem that improves a recent result of [4]. In the last section, we study continuity-type properties of set valued parametric projections and our results improve recent theorems due to Mabizela [11].