Journal of the Mathematical Society of Japan

The Bishop-Phelps-Bollobás property for bilinear forms and polynomials

María D. ACOSTA, Julio BECERRA-GUERRERO, Yun Sung CHOI, Domingo GARCÍA, Sun Kwang KIM, Han Ju LEE, and Manuel MAESTRE

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Abstract

For a $\sigma$-finite measure $\mu$ and a Banach space $Y$ we study the Bishop-Phelps-Bollobás property (BPBP) for bilinear forms on $L_1(\mu)\times Y$, that is, a (continuous) bilinear form on $L_1(\mu)\times Y$ almost attaining its norm at $(f_0,y_0)$ can be approximated by bilinear forms attaining their norms at unit vectors close to $(f_0,y_0)$. In case that $Y$ is an Asplund space we characterize the Banach spaces $Y$ satisfying this property. We also exhibit some class of bilinear forms for which the BPBP does not hold, though the set of norm attaining bilinear forms in that class is dense.

Article information

Source
J. Math. Soc. Japan, Volume 66, Number 3 (2014), 957-979.

Dates
First available in Project Euclid: 24 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1406206979

Digital Object Identifier
doi:10.2969/jmsj/06630957

Mathematical Reviews number (MathSciNet)
MR3238324

Zentralblatt MATH identifier
1293.32031

Subjects
Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 46B22: Radon-Nikodým, Kreĭn-Milman and related properties [See also 46G10] 46B25: Classical Banach spaces in the general theory

Keywords
Banach space Bishop-Phelps-Bollobás Theorem norm attaining bilinear form polynomial

Citation

ACOSTA, María D.; BECERRA-GUERRERO, Julio; CHOI, Yun Sung; GARCÍA, Domingo; KIM, Sun Kwang; LEE, Han Ju; MAESTRE, Manuel. The Bishop-Phelps-Bollobás property for bilinear forms and polynomials. J. Math. Soc. Japan 66 (2014), no. 3, 957--979. doi:10.2969/jmsj/06630957. https://projecteuclid.org/euclid.jmsj/1406206979


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