On the residuality of the set of norms having Mazur’s intersection property. (English) Zbl 0760.46007

Georgiev, Pando Gr.

Math. Balk., New Ser. 5, No. 1, 20-26 (1991).

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.

Reviewer: S.J.Sidney (Storrs)

MSC:

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

Keywords:

Mazur’s intersection property; Baire category argument; \(w^*\)-Mazur’s intersection property for dual norms; Fréchet differentiable