TITLE: Isomorphisms of Banach Spaces 
QUESTION [21 upvotes]: Suppose $X$ and $Y$ are Banach spaces whose dual spaces are isometrically isomorphic.  It is certainly true that $X$ and $Y$ need not be isometrically isomorphic, but must it be true that there is a continuous (not necessarily isometric) isomorphism of $X$ onto $Y$?

REPLY [6 votes]: One may also remark that $c_{0}$ $\times$ (ℓ∞$/c_{0}$) is a predual of (ℓ∞)$^*$.                      
However, $c_{0}$ $\times$ (ℓ∞$/c_{0}$) is not isomorphic to ℓ∞.