On Some New Characterizations Of Locally (Weakly) Compact Sets Via Proximinality

First, we characterize locally weakly compactness of a closed convex subset C in a Banach space in terms of the proximinality. By the equivalent descriptions on locally weakly compactness of closed convex sets and the James theorem, we prove that C is locally weakly compact if and only if C is super proximinal, i.e. C is proximinal in every Banach space containing it. This generalizes the classical result which says that Banach space X is reflexive if and only if X is proximinal in every superspace of it; and when C is bounded, we have:C is weakly compact if and only if C is super proximinal. In the view of renormings, we obtain a new characterization of weakly compact sets:a bounded closed convex set C of a Banach space X is weakly compact if and only if for every equivalent norm|·|on X, C is proximinal in (X,|·|). We also construct an equivalent norm|·|on l∞such that the closed unit ball is not proximinal in (l∞,|·|).Secondly, we characterize local compactness of a closed convex subset C in a Banach space in terms of the strongly proximinality. In the way of constructing a new norm by making use of a basic sequence, we prove that C is locally compact if and only if C is super strongly proximinal, i.e. C is strongly proximinal in every Banach space containing it. This generalizes the classical result which says that Banach space X is finite dimensional if and only if X is strongly proximinal in every superspace of it; and when C is bounded, we have:C is compact if and only if C is super strongly proximinal. We also characterize local compactness in terms of the upper semi-continuity of metric projections:C is locally compact if and only if for every Banach space Y(?)C, the metric projection Pc:Yâ†'C is nonempty set-valued and upper semi-continuous. Finally, we obtain another characterization on the local compactness:C is locally compact if and only if every closed subset of C is proximinal.