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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.

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