James' compactness criterion

The compactness criterion of James (by Robert C. James ) is a mathematical theorem from the branch of functional analysis . This theorem characterizes the compact sets of a Banach space with respect to the weak topology and results in James' theorem on reflexive Banach spaces.

A non-empty weakly closed set is weakly compact if and only if every continuous linear functional from the dual space assumes the absolute maximum on this set. This sentence reads more precisely: ${\displaystyle X\,'}$

Compactness criterion from James : Let be a Banach space and a non-empty weakly-closed set. Then the following statements are equivalent: ${\displaystyle X}$ ${\displaystyle A\subset X}$

• ${\displaystyle A}$ is weakly compact.
• For each there is one with . ${\displaystyle f\in X\,'}$ ${\displaystyle x_{0}\in A}$ ${\displaystyle f(x_{0})=\sup\{|f(x)|;\,x\in A\}}$
• For each there is one with . ${\displaystyle f\in X_{\mathbb {R} }'}$ ${\displaystyle x_{0}\in A}$ ${\displaystyle f(x_{0})=\sup\{|f(x)|;\,x\in A\}}$
• For each there is one with . ${\displaystyle f\in X_{\mathbb {R} }'}$ ${\displaystyle x_{0}\in A}$ ${\displaystyle f(x_{0})=\sup\{f(x);\,x\in A\}}$

Here stands for the real vector space, which is created by restricting the scalar multiplication to . This part of the sentence is only interesting for -Banach rooms. A consequence of the above sentence is: ${\displaystyle X_{\mathbb {R} }}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$

James theorem : For a Banach space are equivalent: ${\displaystyle X}$

• ${\displaystyle X}$ is reflexive.
• For everyone there is one with , so that . ${\displaystyle f\in X\,'}$ ${\displaystyle x\in X}$ ${\displaystyle \|x\|\leq 1}$ ${\displaystyle f(x)=\|f\|}$

This follows immediately from the above compactness criterion if one uses that a Banach space is reflexive if and only if the unit sphere is weakly compact, and that the supremum on the unit sphere is by definition the same for one . ${\displaystyle f\in X\,'}$ ${\displaystyle \|f\|}$

Historically, these theorems have been proven in reverse order. First, in 1957, James proved the reflexivity criterion for separable Banach spaces and in 1964 for general Banach spaces. Since reflexivity is equivalent to the weak compactness of the unit sphere, Victor L. Klee reformulated this as the compactness criterion for the unit sphere in 1962 and assumed that this criterion characterizes any weakly compact sets. This was actually proven by RC James in 1964.

Individual evidence

1. ^ Robert E. Megginson: An Introduction to Banach Space Theory , Springer New York (1998), ISBN = 0-387-98431-3, sentence 2.9.3
2. ^ Robert E. Megginson: An Introduction to Banach Space Theory , Springer New York (1998), ISBN = 0-387-98431-3, sentence 2.9.4
3. ^ RC James: Reflexivity and the Supremum of Linear Functionals , Annals of Mathematics (2) 66 (1957), pp. 159-169
4. ^ RC James: Characterization of Reflexivity , Studia Mathematica 23 (1964), pp. 205-216
5. ^ VL Klee: A conjecture on weak compactness , Trans. Amer. Math. Soc. 104, pp. 398-402 (1962)
6. ↑ RC James: Weakly Compact Sets , Trans. Amer. Math. Soc. 113, pp. 129-140 (1964)