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On the residuality of the set of norms having Mazur’s intersection property. (English) Zbl 0760.46007
A Banach space $$(E,p)$$ has Mazur’s intersection property if every closed bounded convex subset of $$E$$ is the intersection of the closed balls that contain it. Similarly, if $$p^*$$ is the norm on $$E^*$$ dual to $$p$$, $$(E^*,p^*)$$ has $$w^*$$-Mazur’s intersection property if every $$w^*$$-compact convex subset of $$E^*$$ is the intersection of all the closed balls that contain it.
The author continues his own work on the subject [Proc. Am. Math. Soc. 104, No. 1, 157-164 (1988; Zbl 0673.46005)] by using a well-constructed Baire category argument to show that if $$(E,\|\cdot\|)$$ has Mazur’s intersection property and $$P$$ is the set of all norms on $$E$$ equivalent to $$\|\cdot\|$$ (in the usual uniform topology of real-valued functions on the unit ball of $$E$$), then $$P$$ has a dense $$G_ \delta$$- subset $$P_ 0$$ such that $$(E,p)$$ has Mazur’s intersection property for each $$p$$ in $$P_ 0$$. The analogous theorem for $$w^*$$-Mazur’s intersection property for dual norms on $$E^*$$ is also true, by (essentially) the same proof. Finally, when $$(E,p)$$ has Mazur’s intersection property and $$(E^*,p^*)$$ has $$w^*$$-Mazur’s intersection property, where $$p^*$$ is the dual norm of $$p$$, then $$p$$ and $$p^*$$ are Fréchet differentiable on dense $$G_ \delta$$-subsets of $$E$$ and $$E^*$$ respectively; this involves deep results from convex analysis. The author concludes by discussing various known results from the viewpoint of the proofs in this paper.

MSC:
 46B03 Isomorphic theory (including renorming) of Banach spaces 46B20 Geometry and structure of normed linear spaces