## Journal of the Mathematical Society of Japan

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

#### 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

#### 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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