James's theorem

James compactness criterion — Let ${\displaystyle X}$  be a Banach space and ${\displaystyle A}$  a weakly closed nonempty subset of ${\displaystyle X.}$  The following conditions are equivalent:

• ${\displaystyle A}$  is weakly compact.
• For every ${\displaystyle f\in X^{\prime },}$  there exists an element ${\displaystyle a_{0}\in A}$  such that ${\displaystyle \left|f\left(a_{0}\right)\right|=\sup _{a\in A}|f(a)|.}$ 
• For any ${\displaystyle f\in X_{\mathbb {R} }^{\prime },}$  there exists an element ${\displaystyle a_{0}\in A}$  such that ${\displaystyle f\left(a_{0}\right)=\sup _{a\in A}|f(a)|.}$ 
• For any ${\displaystyle f\in X_{\mathbb {R} }^{\prime },}$  there exists an element ${\displaystyle a_{0}\in A}$  such that ${\displaystyle f\left(a_{0}\right)=\sup _{a\in A}f(a).}$ 

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