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

 

The distance $ dist (\mathcal{B},X)$ when $ \mathcal{B}$ is a boundary of $ B(X^{**})$


Authors: A. S. Granero, J. M. Hernández and H. Pfitzner
Journal: Proc. Amer. Math. Soc. 139 (2011), 1095-1098
MSC (2010): Primary 46B20; Secondary 46B26
Published electronically: August 6, 2010
MathSciNet review: 2745660
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Abstract: Let $ X$ be a real Banach space and let $ \mathcal{B}$ be a boundary of the unit ball $ B(X^{**})$ of the bidual $ X^{**}$ (which means that for each $ x^*\in X^*$ there is $ b\in \mathcal{B}$ such that $ \langle b,x^*\rangle =\Vert x^*\Vert$). We show that $ dist(\mathcal{B},X)=dist(B(X^{**}),X)$ where $ dist(A,X)$ denotes the sup of all $ dist(a, X)$ with $ a\in A$. Since $ \overline{{co}}^{w^*}(\mathcal{B})=B(X^{**})$ this is in contrast with the fact that in general strict inequality can occur between $ dist(K,X)$ and $ dist(\overline{{co}}^{w^*}(K),X)$ even for a $ w^*$-compact $ K\subset X^{**}$.


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Additional Information

A. S. Granero
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040-Madrid, Spain
Email: AS_granero@mat.ucm.es

J. M. Hernández
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040-Madrid, Spain
Email: juanmanuel_hrl@hotmail.com

H. Pfitzner
Affiliation: Université d’Orléans, BP 6759, F-45067, Orléans Cedex 2, France
Email: Hermann.Pfitzner@univ-orleans.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10529-4
PII: S 0002-9939(2010)10529-4
Keywords: Convex sets, James boundaries, unit ball, distances
Received by editor(s): February 26, 2010
Received by editor(s) in revised form: April 6, 2010
Published electronically: August 6, 2010
Additional Notes: This work was supported in part by grant DGICYT MTM2005-00082, grant UCM-910346 and grant UCM-BSCH PR27/05-14045
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.