A characterization of the duals of some echelon Köthe spaces of Banach valued functions

In [8] Nyugen Phuong-Cac considers Köthe spaces of vector valued functions in a Banach space $X$. In this work we improve the duality result of [6], [8], restricting ourselves to an echelon Köthe space. We prove that the topological dual of $\Lambda^p(X)$ is the same as its $\alpha$-dual if and only if $X'$ has the Radon-Nikodym property.