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Boundaries and polyhedral Banach spaces


Authors: V. P. Fonf, R. J. Smith and S. Troyanski
Journal: Proc. Amer. Math. Soc. 143 (2015), 4845-4849
MSC (2010): Primary 46B20
Published electronically: May 8, 2015
MathSciNet review: 3391042
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Abstract: We show that if $ X$ and $ Y$ are Banach spaces, where $ Y$ is separable and polyhedral, and if $ T:X\to Y$ is a bounded linear operator such that $ T^*(Y^*)$ contains a boundary $ B$ of $ X$, then $ X$ is separable and isomorphic to a polyhedral space. Some corollaries of this result are presented.


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Additional Information

V. P. Fonf
Affiliation: Department of Mathematics, Ben Gurion University of the Negev, Beer-Sheva, Israel
Email: fonf@math.bgu.ac.il

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

S. Troyanski
Affiliation: Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Espinardo (Murcia), Spain — and — Institute of Mathematics and Informatics, Bulgarian Academy of Science, bl.8, acad. G. Bonchev str. 1113 Sofia, Bulgaria
Email: stroya@um.es

DOI: https://doi.org/10.1090/proc/12644
Keywords: Polyhedral norms, renormings, boundaries, polytopes
Received by editor(s): March 31, 2004
Received by editor(s) in revised form: September 8, 2014
Published electronically: May 8, 2015
Additional Notes: The first author was supported by Israel Science Foundation, Grant 209/09. The second and third authors were supported financially by Science Foundation Ireland under Grant Number ‘SFI 11/RFP.1/MTH/3112’. The third author was also supported by FEDER-MCI MTM2011-22457 and by the Bulgarian National Scientific Fund DFNI-I02/10.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2015 American Mathematical Society