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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
Posted: January 4, 2012
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Abstract: We present an example in ZFC of a locally compact, scattered Hausdorff non-Gruenhage space 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 admits a $C^\infty$ -smooth bump function.

References

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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
Posted: January 4, 2012
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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