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Bulletin of the London Mathematical Society Advance Access published online on December 15, 2006
Bulletin of the London Mathematical Society, doi:10.1112/blms/bdl022
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© 2006 London Mathematical Society

Trees, Gâteaux norms and a problem of Haydon

Richard J. Smith

Queens' College, Cambridge CB3 9ET, United Kingdom rjs209{at}cam.ac.uk

Received 27 June 2005. Revision received 22 November 2005.

We give a sufficient condition for the existence of an equivalent Gâteaux smooth norm on C0{Upsilon}, where {Upsilon} is a tree. Using this, we prove that if C0{Upsilon} admits an equivalent strictly convex norm, then it admits an equivalent Gâteaux smooth norm. This resolves an open problem from Haydon's study of trees in renorming theory.


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