Banach-Alaoglu theorem states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology, while Milman–Pettis theorem states that every uniformly convex Banach space is reflexive.
Let $(E, |\cdot|)$ be a Banach space and $B_E$ its closed unit ball. Let $J:E \to E^{**}$ be the canonical linear isometry. The proof of Milman–Pettis theorem in Brezis' Functional Analysis relies on the fact that $J(B_E)$ is closed in $E^{**}$ w.r.t. the norm topology. This fact follows from the fact that $E$ is complete.
Could you provide an example of a uniformly convex normed vector space which is not reflexive?
Let $(H, \langle \cdot, \cdot \rangle)$ be an incomplete inner product space and $|\cdot|$ its induced norm. By parallelogram law, $(H, |\cdot|)$ is uniformly convex. Because $H$ is incomplete, $H$ is not reflexive.
I have provided an explicit example of such space in this thread.