Let $X$ and $Y$ be Banach Spaces. I have to prove if $X'$ and $Y'$ are isomorphic and $X$ is reflexive, then $X$ and $Y$ are isomorphic.
My idea. Since $X$ is reflexive we have that $X'$ is also reflexive, and since $X'$ and $Y'$ are isomorphic, $Y'$ is also reflexive and $Y$ will be reflexive. But I'm stuck here.