James' Theorem
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, particularly
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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, James' theorem, named for Robert C. James, states that a
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) 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 vectors and ...
X is reflexive if and only if every continuous
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of ...
's norm on X attains its
supremum In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
on the closed
unit ball Unit may refer to: General measurement * Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law **International System of Units (SI), modern form of the metric system **English units, histo ...
in X. A stronger version of the theorem states that a weakly closed subset C of a Banach space X is weakly compact if and only if the
dual norm In functional analysis, the dual norm is a measure of size for a continuous function, continuous linear function defined on a normed vector space. Definition Let X be a normed vector space with norm \, \cdot\, and let X^* denote its continuous d ...
each continuous linear functional on X attains a maximum on C. The hypothesis of completeness in the theorem cannot be dropped.


Statements

The space X considered can be a real or complex Banach space. Its
continuous dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V,'' together with the vector space structure of pointwise addition and scalar multiplication by const ...
is denoted by X^. The topological dual of \mathbb-Banach space deduced from X by any restriction scalar will be denoted X^_. (It is of interest only if X is a complex space because if X is a \R-space then X^_ = X^.) A Banach space being reflexive if and only if its closed unit ball is weakly compact one deduces from this, since the norm of a continuous linear form is the upper bound of its modulus on this ball:


History

Historically, these sentences were proved in reverse order. In 1957, James had proved the reflexivity criterion for separable Banach spaces and 1964 for general Banach spaces. Since the reflexivity is equivalent to the weak compactness of the unit sphere, Victor L. Klee reformulated this as a compactness criterion for the unit sphere in 1962 and assumes that this criterion characterizes any weakly compact quantities. This was then actually proved by James in 1964.


See also

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Notes


References

* * . * . * . * . * {{Functional analysis Theorems in functional analysis