The Bishop-Phelps-Bollobás theorem fails for bilinear forms on \(l_{1}\times l_{1}\)
In [Bull.\ Am.\ Math.\ Soc.\ 67, 97--98 (1961; Zbl 0098.07905)], \textit{E.\,Bishop} and \textit{R.\,Phelps} proved that for every Banach space \(X\), every functional in its dual space \(X^*\) can be approximated by functionals which attain their norms. Over the years, many manuscripts appeared showing whether or not this density can be proved for operators instead of functionals. In [Bull.\ Lond.\ Math.\ Soc.\ 2, 181--182 (1970; Zbl 0217.45104)], \textit{B.\,Bollobás} sharpened this result giving a quantitative version of the Bishop-Phelps theorem. Motivated by this version, \textit{M.\,D.\thinspace Acosta, R.\,M.\thinspace Aron, D.\,García} and \textit{M.\,Maestre} [J.~Funct.\ Anal.\ 254, No.\,11, 2780--2799 (2008; Zbl 1152.46006)] defined the so-called Bishop-Phelps-Bollobás property as a quantitative way in which the set of norm attaining operators can be dense in the space of all operators between two Banach spaces. In the present paper, the authors propose a Bishop-Phelps-Bollobás property for bilinear maps in the spirit of the analogous property for operators. The authors show that the Bishop-Phelps-Bollobás property fails for bilinear forms on \(\ell_1 \times \ell_1\) (and therefore for \(n\)-linear forms). The main interest for this example is that the Bishop-Phelps property holds for bilinear forms on \(\ell_1 \times \ell_1\) and also that the Bishop-Phelps-Bollobás property holds for operators from \(\ell_1\) to \(\ell_\infty\). Recall that the space of bilinear forms on \(\ell_1 \times \ell_1\) can be identified with the space of operators from \(\ell_1\) to \(\ell_\infty\).
scientific article; zbMATH DE number 5626949