Trees, linear orders and Gâteaux smooth norms

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.

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