If I have to normed vector spaces $A$ and $B$, I was wondering when the topological duals are isomorphic (i.e. $A^* \cong B^*)$ . Is it sufficient that $A \cong B$? Or that $A$ has to be isometric to $B$?
If it is sufficient for $A$ isometric to $B$, given an isometry $f$, what would be the isomorphism mapping between the spaces $A^*$ and $B^*$?
You just need to have $A \simeq B$.
Consider $f$ an isomorphism between $A$ and $B$. By definition, $f$ is linear and continuous and it turns out that linearity and continuity are preserved through composition. If you consider the application that associates to each linear $l$ form in $A^*$ the linear form $l \circ f^{-1}$ in $B^*$ then you get an isomorphism from $A^*$ to $B^*$ (with inverse $l \rightarrow l \circ f$). (You can easily verify that this application between $A^*$ and $B^*$ is also linear and continuous.)