Conic James' Compactness Theorem

TL;DR

This paper proves a new compactness criterion in Banach spaces, showing that certain bounded sets are weakly relatively compact if all functionals positive on a weakly compact set attain their supremum on it.

Contribution

It introduces a novel weak compactness criterion involving functionals attaining their supremum, extending James' classical theorem.

Findings

Established a new weak compactness criterion in Banach spaces.

Extended James' compactness theorem to broader conditions.

Provided conditions under which bounded sets are weakly relatively compact.

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 $x^*\in E^*$ with $x^*(D) >0$ attains its supremum on $A$, then $A$ is weakly relatively compact.}