Milman–Pettis theorem
Encyclopedia
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Milman–Pettis theorem states that every uniformly convex
Uniformly convex space
In mathematics, uniformly convex spaces are common examples of reflexive Banach spaces. The concept of uniform convexity was first introduced by James A...
Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
is reflexive
Reflexive space
In functional analysis, a Banach space is called reflexive if it coincides with the dual of its dual space in the topological and algebraic senses. Reflexive Banach spaces are often characterized by their geometric properties.- Normed spaces :Suppose X is a normed vector space over R or C...
.
The theorem was proved independently by D. Milman
David Milman
David Pinhusovich Milman was a Soviet and later Israeli mathematician specializing in functional analysis. He was one of the major figures of the Soviet school of functional analysis...
(1938) and B. J. Pettis (1939). S. Kakutani
Shizuo Kakutani
was a Japanese-born American mathematician, best known for his eponymous fixed-point theorem.Kakutani attended Tohoku University in Sendai, where his advisor was Tatsujirō Shimizu. Early in his career he spent two years at the Institute for Advanced Study in Princeton at the invitation of the...
gave a different proof in (1939), and John R. Ringrose published a shorter proof in 1959.