[Banach] Abstract of a paper by Richard J. Smith

This is an announcement for the paper "Gruenhage compacta and strictly convex dual norms" by Richard J. Smith.

Abstract: We prove that if K is a Gruenhage compact space then C(K)* admits an equivalent, strictly convex dual norm. As a corollary, we show that if X is a Banach space and X* is the |.|-closed linear span of K, where K is a Gruenhage compact in the w*-topology and |.| is equivalent to a coarser, w*-lower semicontinuous norm on X*, then X* admits an equivalent, strictly convex dual norm. We give a partial converse to the first result by showing that if T is a tree, then C(T)* admits an equivalent, strictly convex dual norm if and only if T is a Gruenhage space. Finally, we present some stability properties satisfied by Gruenhage spaces; in particular, Gruenhage spaces are stable under perfect images.

Archive classification: math.FA math.GN

Mathematics Subject Classification: 46B03; 46B26

The source file(s), arxiv29-10-07.tex: 67073 bytes, is(are) stored in gzipped form as 0710.5396.gz with size 19kb. The corresponding postcript file has gzipped size 112kb.

Submitted from: rjs209@cam.ac.uk

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/0710.5396

or

http://arXiv.org/abs/0710.5396

or by email in unzipped form by transmitting an empty message with subject line

uget 0710.5396

or in gzipped form by using subject line

get 0710.5396

to: math@arXiv.org.