In mathematics, the Milman–Pettis theorem states that every uniformly convex Banach space is reflexive.
The theorem was proved independently by D. Milman (1938) and B. J. Pettis (1939). S. Kakutani gave a different proof in 1939, and John R. Ringrose published a shorter proof in 1959.
Mahlon M. Day (1941) gave examples of reflexive Banach spaces which are not isomorphic to any uniformly convex space.
References edit
- S. Kakutani, Weak topologies and regularity of Banach spaces, Proc. Imp. Acad. Tokyo 15 (1939), 169–173.
- D. Milman, On some criteria for the regularity of spaces of type (B), C. R. (Doklady) Acad. Sci. U.R.S.S, 20 (1938), 243–246.
- B. J. Pettis, A proof that every uniformly convex space is reflexive, Duke Math. J. 5 (1939), 249–253.
- J. R. Ringrose, A note on uniformly convex spaces, J. London Math. Soc. 34 (1959), 92.
- Day, Mahlon M. (1941). "Reflexive Banach spaces not isomorphic to uniformly convex spaces". Bull. Amer. Math. Soc. 47. American Mathematical Society: 313–317. doi:10.1090/S0002-9904-1941-07451-3.