More generally, uniformly convex spaces are reflexive, by the Milman-Pettis theorem.

In mathematics, uniformly convex spaces are common examples of reflexive Banach spaces.

Every uniformly convex space is strictly convex.

Mahlon M. Day (1941) gave examples of reflexive Banach spaces which are not isomorphic to any uniformly convex space.

B. J. Pettis, A proof that every uniformly convex space is reflexive, Duke Math.