Trees, linear orders and Gâteaux smooth norms

Abstract

We introduce a linearly ordered set Z and use it to prove a necessity condition for the existence of a Gâteaux smooth norm on C(T), where T is a tree. This criterion is directly analogous to the corresponding equivalent condition for Fréchet smooth norms. In addition, we prove that if C(T) admits a Gâteaux smooth lattice norm then it also admits a lattice norm with strictly convex dual norm.

A different version of this paper is to appear in J. London Math. Soc

Cited by in corpus (1)

• Gruenhage compacta and strictly convex dual norms