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The Bishop-Phelps-Bollobás Theorem for bilinear forms


Authors: María D. Acosta, Julio Becerra-Guerrero, Domingo García and Manuel Maestre
Journal: Trans. Amer. Math. Soc. 365 (2013), 5911-5932
MSC (2010): Primary 46B20; Secondary 46B25, 46B28
Published electronically: July 2, 2013
MathSciNet review: 3091270
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Abstract: In this paper we provide versions of the Bishop-Phelps-Bollobás Theorem for bilinear forms. Indeed we prove the first positive result of this kind by assuming uniform convexity on the Banach spaces. A characterization of the Banach space $ Y$ satisfying a version of the Bishop-Phelps-Bollobás Theorem for bilinear forms on $ \ell _1 \times Y$ is also obtained. As a consequence of this characterization, we obtain positive results for finite-dimensional normed spaces, uniformly smooth spaces, the space $ \mathcal {C}(K)$ of continuous functions on a compact Hausdorff topological space $ K$ and the space $ K(H)$ of compact operators on a Hilbert space $ H$. On the other hand, the Bishop-Phelps-Bollobás Theorem for bilinear forms on $ \ell _1 \times L_1 (\mu )$ fails for any infinite-dimensional $ L_1 (\mu )$, a result that was known only when $ L_1 (\mu ) = \ell _1$.


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Additional Information

María D. Acosta
Affiliation: Departamento de Análisis Matemático, Universidad de Granada, Facultad de Ciencias, 18071-Granada, Spain
Email: dacosta@ugr.es

Julio Becerra-Guerrero
Affiliation: Departamento de Análisis Matemático, Universidad de Granada, Facultad de Ciencias, 18071-Granada, Spain
Email: juliobg@ugr.es

Domingo García
Affiliation: Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, 46100 Burjasot Valencia, Spain
Email: domingo.garcia@uv.es

Manuel Maestre
Affiliation: Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, 46100 Burjasot Valencia, Spain
Email: manuel.maestre@uv.es

DOI: https://doi.org/10.1090/S0002-9947-2013-05881-3
Keywords: Bishop-Phelps-Bollob\'as Theorem, bilinear form, Banach space
Received by editor(s): February 6, 2012
Published electronically: July 2, 2013
Additional Notes: The first author was supported by MICINN and FEDER Project MTM-2009–07498 and Junta de Andalucía \com{Proyecto de Excelencia} P09-FQM–4911
The second author was supported by Junta de Andalucía grants FQM 0199 and FQM 1215, and MTM-2011-23843
The third and fourth authors were supported by MICINN Project MTM2011-22417
The fourth author was also supported by Prometeo 2008/101 and MEC PR2010-0374.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.