Norm attaining bilinear forms on \(L^ 1[0,1]\)
The Bishop-Phelps theorem says that the set of norm attaining bounded linear functionals on a Banach space is dense in the set of all bounded linear functionals on the space. The author gives an explicit counterexample to this result for bounded bilinear functionals on \(L^1[0,1]\). Recall that the norm of a bounded bilinear functional is \(|\phi|=\sup\{|\phi(x,y)|:|x|\leq 1\), \(|y|\leq 1\}\), and a functional is norm attaining if there are \(x,y\in X\), \(|x|=1\), \(|y|=1\) such that \(|\phi|=|\phi(x,y)|\). The set of bounded bilinear functionals is denoted \(B(X)\) and the norm attaining bounded bilinear functionals by \(B_{na}(X)\). It was shown recently by Acosta, Aguirre and Payá that \(B_{na}(G)\) is not dense in \(B(G)\) for a particular Banach space \(G\). On the positive side it is known that if \(X\) satisfies the Radon-Nikodym property, if \(X\) satisfies property \(\alpha\), or if \(X\) has a monotone shrinking basis and the Dunford-Pettis property, then \(B_{na}(X)\) is dense in \(B(X)\). The author characterizes norm attaining bounded bilinear functionals on \(L^1[0,1]\) and proves that \(B_{na}(L^1[0,1])\) is not dense in \(B(L^1[0,1])\).
norm attaining bounded linear functionals
norm attaining bounded bilinear functionals
scientific article; zbMATH DE number 1046885