James' theorem
Encyclopedia
In mathematics
, particularly functional analysis
, James' theorem, named for Robert C. James, states that a Banach space
B is reflexive
if and only if every continuous
linear functional
on B attains its maximum on the closed unit ball in B.
A stronger version of the theorem states that a weakly closed subset C of a Banach space B is weakly compact
if and only if each continuous linear functional on B attains a maximum on C.
The hypothesis of completeness in the theorem cannot be dropped .
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...
, particularly functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, James' theorem, named for Robert C. James, states that 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...
B 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...
if and only if every continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
linear functional
Linear functional
In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars. In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...
on B attains its maximum on the closed unit ball in B.
A stronger version of the theorem states that a weakly closed subset C of a Banach space B is weakly compact
Weak topology
In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...
if and only if each continuous linear functional on B attains a maximum on C.
The hypothesis of completeness in the theorem cannot be dropped .