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
MathSciNet review: 2917084
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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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Richard J. Smith
Affiliation: School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

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.