Dual of Köthe–Bochner Spaces and Radon–Nikodým Property

Abstract

The Köthe–Bochner spaces $L_{\rho }\left(X\right)$ are the vector valued version of the scalar Köthe spaces $L_{\rho }$, which generalize the Lebesgue spaces $L^p$, the Orlicz spaces and many other functional spaces. We study the possible identification between the dual ${\left(L_{\rho }\left(X\right)\right)}^{\prime }$ and the associated space $L_{{\rho }^{\prime }}\left(X^{\prime }\right)$ in connection with the Radon–Nikodým property of $X^{\prime }$.