It is known that in a reflexive Banach space, if the norm is strictly convex, then its dual will be smooth Banach space, and if the norm is smooth, then the dual norm is strictly convex.

We can find an equivalent norm on $\mathbb{R}^2$ such that the norm is not strictly convex; however, we can see that the space is smooth. However, I am still looking for an infinite-dimensional reflexive Banach space such that the space is smooth, but its norm is not strictly convex. I appreciate any help you can provide.