In mathematics, the Milman–Pettis theorem states that every

uniformly convex In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive space, reflexive Banach spaces. The concept of uniform convexity was first introduced by James A. Clarkson in 1936. Definition A uniformly conv ...

Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...

is reflexive. The theorem was proved independently by D. Milman (1938) and

B. J. Pettis Billy James Pettis (1913 – 14 April 1979), was an American mathematician, known for his contributions to functional analysis. See also *Dunford–Pettis property *Dunford–Pettis theorem *Milman–Pettis theorem *Orlicz–Pettis theorem *Petti ...

(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. * {{DEFAULTSORT:Milman-Pettis theorem Banach spaces Theorems in functional analysis fr:Théorème de Milman-Pettis