I'm trying to find an example of a space $V$ which is strictly convex, but has a dual space $V^*$ which is not strictly smooth. Any help please?
As any separable space, $\ell_1$ admits an equivalent strictly convex norm. Indeed, since the identity map from $\ell_1$ to $\ell_2$ is bounded, we can take $\|x\|_1+\|x\|_2$ as such a norm. The dual of this strictly convex space is isomorphic to $\ell_\infty$. But M.M. Day proved in 1955 (see Theorem 9) that $\ell_\infty$ does not admit any equivalent smooth norm.