Norm Attaining Multilinear Forms on 𝐿1(𝝁) - Yousef Saleh (International Journal of Mathematics and Mathematical Sciences)
Given an arbitrary measure 𝜇, this study shows that the set of norm attaining multilinear forms is not dense in the space of all continuous multilinear forms on 𝐿1(𝜇). However, we have the density if and only if 𝜇 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∗ is dense in the space of all bounded linear operators 𝐿(𝑋,𝑋∗). In contrast, the set of norm attaining bilinear forms on 𝑋 is not dense in the space of continuous bilinear forms on 𝑋.