Trees, linear orders and G\^ateaux smooth norms

Richard J. Smith
Arxiv ID: 0710.4230Last updated: 6/14/2022
We introduce a linearly ordered set Z and use it to prove a necessity condition for the existence of a G\^ateaux smooth norm on C(T), where T is a tree. This criterion is directly analogous to the corresponding equivalent condition for Fr\'echet smooth norms. In addition, we prove that if C(T) admits a G\^ateaux smooth lattice norm then it also admits a lattice norm with strictly convex dual norm.

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