Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 66, Number 3 (2014), 957-979.
The Bishop-Phelps-Bollobás property for bilinear forms and polynomials
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.
J. Math. Soc. Japan, Volume 66, Number 3 (2014), 957-979.
First available in Project Euclid: 24 July 2014
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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