This is an announcement for the paper "Trees, linear orders and G^ateaux smooth norms" by Richard J. Smith.

Abstract: 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.

Archive classification: math.FA

Mathematics Subject Classification: 46B03; 46B26

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

The source file(s), arxiv12-10-07.tex: 60917 bytes, is(are) stored in gzipped form as 0710.4230.gz with size 18kb. The corresponding postcript file has gzipped size 102kb.

Submitted from: rjs209@cam.ac.uk

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