Norm attaining multilinear forms on \(L_{1}(\mu )\) (Q938543)

Summary: Given an arbitrary measure \(\mu \), this study shows that the set of norm attaining multilinear forms is not dense in the space of all continuous multilinear forms on \(L_{1}(\mu )\). However, we have the density if and only if \(\mu \) is purely atomic. Furthermore, the study presents an example of a Banach space \(X\) in which the set of norm attaining operators from \(X\) into \(X^{\ast }\) is dense in the space of all bounded linear operators \(L(X,X^{\ast })\). In contrast, the set of norm attaining bilinear forms on \(X\) is not dense in the space of continuous bilinear forms on \(X\).