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Journal of the London Mathematical Society Advance Access published online on November 12, 2007
Journal of the London Mathematical Society, doi:10.1112/jlms/jdm085
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© 2007 London Mathematical Society

Trees, linear orders and gâteaux smooth norms

Richard J. Smith

Queens’ College
Cambridge
CB3 9ET
United Kingdom

We introduce a linearly ordered set Z and use it to prove a necessity condition for the existence of a Gâteaux smooth norm on C0({Upsilon}), where {Upsilon} is a tree. This criterion is directly analogous to the corresponding equivalent condition for Fréchet smooth norms. In addition, we prove that if C0({Upsilon}) admits a Gâteaux smooth lattice norm then it also admits a lattice norm with strictly convex dual norm.

2000 Mathematics Subject Classification 46B03 (primary), 46B26 (secondary).

Received August 22, 2006; revised May 23, 2007;
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