On the residuality of the set of norms having Mazur’s intersection property.

*(English)*Zbl 0760.46007A 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.

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 |