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
DOI:
https://doi.org/10.1090/S0002-9947-2013-05881-3
Published electronically:
July 2, 2013
MathSciNet review:
3091270
Full-text PDF
Abstract | References | Similar Articles | Additional Information
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 satisfying a version of the Bishop-Phelps-Bollobás Theorem for bilinear forms on
is also obtained. As a consequence of this characterization, we obtain positive results for finite-dimensional normed spaces, uniformly smooth spaces, the space
of continuous functions on a compact Hausdorff topological space
and the space
of compact operators on a Hilbert space
. On the other hand, the Bishop-Phelps-Bollobás Theorem for bilinear forms on
fails for any infinite-dimensional
, a result that was known only when
.
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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.