I asked myself the following question when I was student just for curiosity. I asked a bit around (my professor, some researchers that I know), but nobody was able to give me an answer. So maybe it is just that nobody thought enough about that, or maybe it is not a stupid question.
Question: Do there exist two Banach spaces, one separable and one non-separable, having isomorphic dual spaces?
Note: isomorphic in the sense that there exists a linear homeomorphism between the two.