Trees, linear orders and Gâteaux smooth norms
R. Jeffrey Smith · Journal of the London Mathematical Society · 2007
We introduce a linearly ordered set Z and use it to prove a necessity condition for the existence of a Gâteaux smooth norm on 𝒷0(Υ), where Υ is a tree. This criterion is directly analogous to the corresponding equivalent condition for Fréchet smooth norms. In addition, we prove that if 𝒷0(Υ) admits a Gâteaux smooth lattice norm then it also admits a lattice norm with strictly convex dual norm.