Milman–Pettis theorem

Milman–Pettis theorem 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 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. hide vte Functional analysis (topics – glossary) Spaces BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraic/topological)locally convexreflexiveseparable Theorems Hahn–BanachRiesz representationclosed graphuniform boundedness principleKakutani fixed-pointKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operators adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebras Banach algebraC*-algebraspectrum of a C*-algebraoperator algebragroup algebra of a locally compact groupvon Neumann algebra Open problems invariant subspace problemMahler's conjecture Applications Hardy spacespectral theory of ordinary differential equationsheat kernelindex theoremcalculus of variationsfunctional calculusintegral operatorJones polynomialtopological quantum field theorynoncommutative geometryRiemann hypothesisdistribution (or generalized functions) Advanced topics approximation propertybalanced setChoquet theoryweak topologyBanach–Mazur distanceTomita–Takesaki theory Categories: Banach spacesTheorems in functional analysis

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