The distance $dist (\mathcal {B},X)$ when $\mathcal {B}$ is a boundary of $B(X^{**})$
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- by A. S. Granero, J. M. Hernández and H. Pfitzner PDF
- Proc. Amer. Math. Soc. 139 (2011), 1095-1098 Request permission
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 $\langle b,x^*\rangle =\|x^*\|$). 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 $\overline {\mathrm {co}}^{w^*}(\mathcal {B})=B(X^{**})$ this is in contrast with the fact that in general strict inequality can occur between $dist(K,X)$ and $dist(\overline {\mathrm {co}}^{w^*}(K),X)$ even for a $w^*$-compact $K\subset X^{**}$.References
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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
- MR Author ID: 333993
- Email: Hermann.Pfitzner@univ-orleans.fr
- 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
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 1095-1098
- MSC (2010): Primary 46B20; Secondary 46B26
- DOI: https://doi.org/10.1090/S0002-9939-2010-10529-4
- MathSciNet review: 2745660