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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Strictly convex norms, $ G_\delta$-diagonals and non-Gruenhage spaces


Author: Richard J. Smith
Journal: Proc. Amer. Math. Soc. 140 (2012), 3117-3125
MSC (2010): Primary 46B03, 54G12
Published electronically: January 4, 2012
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Abstract: We present an example in ZFC of a locally compact, scattered Hausdorff non-Gruenhage space $ D$ having a $ G_\delta $-diagonal. This means that Gruenhage spaces are not necessary for the construction of strictly convex dual norms on dual Banach spaces, answering a question posed by Orihuela, Troyanski and the author. In addition, we show that the Banach space of continuous functions $ C_0(D)$ admits a $ C^\infty $-smooth bump function.


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Additional Information

Richard J. Smith
Affiliation: School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland
Email: richard.smith@ucd.ie

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11142-6
PII: S 0002-9939(2012)11142-6
Received by editor(s): February 4, 2011
Received by editor(s) in revised form: March 21, 2011
Published electronically: January 4, 2012
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.