I can show that the dual of a surjective isometry between two Banach spaces is surjective by using the Hahn-Banach theorem. However, I am not sure how to prove that the dual is isometric. I would appreciate it if anyone can help with that.
Thanks a lot.
$\sup_{\|x\|\leq 1} |x^{*}(T(x))|=\sup_{\|y\|\leq 1} |x^{*}(y)|$ because $T$ is an isometry. Hence, $\sup_{\|x\|\leq 1} |x^{*}(T(x))|=\|x^{*}\|$. This means $\|T^{*}(x^{*})\|=\|x^{*}\circ T\|=\|x^{*}\|$ so $T^{*}$ is an isometry.
Notations: $X$ is a Banach space, $T: X \to X$ is an isometry. $x^{*}$ is a typical element of the dual space $X^{*}$ and $T^{*}$ is the adjoint (dual) of $T$.