Dual of Köthe-Bochner spaces and Radon-Nikodým property
Let \(\rho\) be a saturated function norm on a \(\sigma\)-finite measure space. Suppose that both \(\rho\) and its associated function norm \(\rho'\) are of absolutely continuous type. Let \(X\) be a Banach space such that its dual~\(X'\) has the Radon-Nikodým property. Under the previous assumptions, the main result of the paper under review states that the dual of the Köthe-Bochner space \(L_\rho(X)\) coincides with \(L_{\rho'}(X')\). There are general results on the representation of the dual of a Köthe-Bochner space in the absence of the Radon-Nikodým property, based on lifting theory. See, e.g., Theorem 3.2.4 in [\textit{P.-K. Lin}, Köthe-Bochner function spaces. Boston, MA: Birkhäuser (2004; Zbl 1054.46003)].
scientific article; zbMATH DE number 7393755