Let $(X,Y)$ be an interpolation couple, i.e. there exists a Hausdorff topological vector space $U$ s.t. both $X$ and $Y$ embed continuously into $U$. We define $X\cap Y$ and $X+Y$ in $U$ and turn them into Banach spaces using the maximum of the respective norms and a norm defined in terms of the infimum over decompositions. (for further information, take any book on interpolation theory)
My question is: Under the hypothesis that $X\cap Y$ is dense in both $X$ and $Y$, why is $(X^\prime, Y^\prime)$ again an interpolation couple.
Background of this question is that there is a duality theorem for complex interpolation which states $$[X^\prime,Y^\prime]_\theta=[X,Y]_\theta^\prime$$ in the above situation where additionally one of the spaces needs to be reflexive. I have looked in several sources, e.g. the original article by Calderon and the book by Bergh/Löfström, but all of them don't address this issue.
There is an article by Watbled (http://www.ams.org/mathscinet-getitem?mr=1795744) where the question is at least raised, but necessary identifications for sum and intersection spaces are only claimed.