Conic James' Compactness Theorem

Abstract

Our main result is the following: {\it Let $E$ be a Banach space and $D$ be a weakly compact subset of $E$ with $0\notin D$. If $A$ is a bounded subset of $E$ such that every $x^{\ast }\in E^{\ast }$ with $x^{\ast }(D)>0$ attains its supremum on $A$, then $A$ is weakly relatively compact.}

8 pages