James's theorem

In mathematics, particularly functional analysis, James' theorem, named for Robert C. James, states that a Banach space $X$ is reflexive if and only if every continuous linear functional's norm on $X$ attains its supremum on the closed unit ball 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 each continuous linear functional on $X$ attains a maximum on $C.$

The hypothesis of completeness in the theorem cannot be dropped.{{harvtxt|James|1971}}

Statements

The space $X$ considered can be a real or complex Banach space. Its continuous dual space is denoted by $X^{\prime}.$ The topological dual of $\mathbb{R}$ -Banach space deduced from $X$ by any restriction scalar will be denoted $X^{\prime}_{\R}.$ (It is of interest only if $X$ is a complex space because if $X$ is a $\R$ -space then $X^{\prime}_{\R} = X^{\prime}.$ )

{{math theorem|name=James compactness criterion|note=|style=|math_statement=

Let $X$ be a Banach space and $A$ a weakly closed nonempty subset of $X.$ The following conditions are equivalent:

$A$ is weakly compact.
For every $f \in X^{\prime},$ there exists an element $a_0 \in A$ such that $\left|f\left(a_0\right)\right| = \sup_{a \in A} |f(a)|.$
For any $f \in X^{\prime}_{\R},$ there exists an element $a_0 \in A$ such that $f\left(a_0\right) = \sup_{a \in A} |f(a)|.$
For any $f \in X^{\prime}_{\R},$ there exists an element $a_0 \in A$ such that $f\left(a_0\right) = \sup_{a \in A} f(a).$

}}

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:

{{math theorem|name=James' theorem|note=|style=|math_statement=

A Banach space $X$ is reflexive if and only if for all $f \in X^{\prime},$ there exists an element $a \in X$ of norm $\|a\| \leq 1$ such that $f(a) = \|f\|.$

}}

History

Historically, these sentences were proved in reverse order. In 1957, James had proved the reflexivity criterion for separable Banach spaces{{harvtxt|James|1957}} and 1964 for general Banach spaces.{{harvtxt|James|1964}} 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.{{harvtxt|Klee|1962}} This was then actually proved by James in 1964.{{harvtxt|James|1964}}

Notes

References

{{citation|last=James|first=Robert C.|title=Reflexivity and the supremum of linear functionals|journal=Annals of Mathematics|year=1957|volume=66|issue=1|pages=159–169|jstor=1970122|mr=0090019|doi=10.2307/1970122}}
{{citation|last=Klee|first=Victor|authorlink=Victor Klee|title=A conjecture on weak compactness|journal=Transactions of the American Mathematical Society|volume=104|issue=3|year=1962|pages=398–402|doi=10.1090/S0002-9947-1962-0139918-7|doi-access=free|mr=139918}}.
{{citation|last=James|first=Robert C.|title=Weakly compact sets|journal=Transactions of the American Mathematical Society|volume=113|year=1964|pages=129–140|doi=10.2307/1994094|issue=1|jstor=1994094|mr=165344|doi-access=free}}.
{{citation|last=James|first=Robert C.|title=A counterexample for a sup theorem in normed space|journal=Israel Journal of Mathematics|volume=9|year=1971|pages=511–512|doi=10.1007/BF02771466|doi-access=free|issue=4|mr=279565}}.
{{citation|last=James|first=Robert C.|title=Reflexivity and the sup of linear functionals|journal=Israel Journal of Mathematics|volume=13|issue=3–4|year=1972|pages=289–300|doi=10.1007/BF02762803|doi-access=free|mr=338742}}.
{{citation|title=An introduction to Banach space theory|volume=183|series=Graduate Texts in Mathematics|first=Robert E. |last=Megginson|authorlink= Robert Megginson |publisher=Springer-Verlag|year=1998|isbn=0-387-98431-3}}

Category:Theorems in functional analysis

James's theorem

Statements

History

See also

Notes

References