Trees, linear orders and Gâteaux smooth norms

Details Trees, linear orders and Gâteaux smooth norms Trees, linear orders and Gâteaux smooth norms

2007-10-23

arxiv.org/abs/0710.4230v1

Mathematics

Functional Analysis

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

Scientific paper

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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