Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 28 (2021), No. 3, [final page numbers not yet available]Copyright Heldermann Verlag 2021 An Application of the Generalised James' Weak Compactness Theorem David J. Farrell Department of Mathematics, The University of Auckland, Auckland 1142, New Zealand Warren B. Moors Department of Mathematics, The University of Auckland, Auckland 1142, New Zealand w.moors@auckland.ac.nz [Abstract-pdf] We provide a short proof of following theorem, due to Delbaen and Orihuela and independently, P\'erez-Aros and Thibault. Let $A$ be a nonempty closed and bounded convex subset of a Banach space $(X,\|\cdot\|)$ and let $W$ be a nonempty weakly compact subset of $(X, \|\cdot\|)$. If we have\\[2mm] \centerline{$x_0^* \in \{x^* \in X^*: \sup_{a \in A} x^*(a) <0\}\ \ \ \text{and}\ \ \ \mathrm{argmax}(y^*|_A) \not= \varnothing$}\\[2mm] for each $y^* \in \{x^* \in X^*: \sup_{a \in A} x^*(a) <0$ and $\sup_{w \in W} |(x^*-x^*_0)(w)|<1\}$, then $A$ is weakly compact. Keywords: Weak compactness, James' weak compactness theorem. MSC: 46B20; 46B26, 49A50, 49A51. [ Fulltext-pdf  (104  KB)] for subscribers only.