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 ${\displaystyle f(a)=\|f\|.}$