Journal of Convex Analysis 28 (2021), No. 3, 795--802
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 \par\vskip2mm \centerline{$x_0^* \in \{x^* \in X^*: \sup_{a \in A} x^*(a) <0\}\ \ \ \text{and}\ \ \ \mathrm{argmax}(y^*|_A) \not= \varnothing$} \par\vskip2mm 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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