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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.

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. American Mathematical Society. 47: 313–317. doi:10.1090/S0002-9904-1941-07451-3.

Functional analysis (topics – glossary)

Hilbert space Banach space Fréchet space topological vector space


Hahn–Banach theorem closed graph theorem uniform boundedness principle Kakutani fixed-point theorem Krein–Milman theorem min-max theorem Gelfand–Naimark theorem Banach–Alaoglu theorem


bounded operator compact operator adjoint operator unitary operator Hilbert–Schmidt operator trace class unbounded operator


Banach algebra C*-algebra spectrum of a C*-algebra operator algebra group algebra of a locally compact group von Neumann algebra

Open problems

invariant subspace problem Mahler's conjecture


Besov space Hardy space spectral theory of ordinary differential equations heat kernel index theorem calculus of variation functional calculus integral operator Jones polynomial topological quantum field theory noncommutative geometry Riemann hypothesis

Advanced topics

locally convex space approximation property balanced set Schwartz space weak topology barrelled space Banach–Mazur distance Tomita–Takesaki theory

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