Abstract
Let $\operatorname{WCC}(X)$ be the collection of all non-empty, weakly compact, convex subsets of a Banach space $X$ endowed with the Hausdorff metric $h$. Weak topology $\mathcal{T}_{w}$ will be defined on $\operatorname{WCC}(X)$. We shall prove that every weakly compact ($\mathcal{T}_{w}$-compact) convex subset $\mathcal{K} \subset (\operatorname{WCC}(X), \mathcal{T}_{w})$ has an extreme point. We also show that there exists strongly bounded ($h$-bounded), closed ($h$-closed) convex subsets which are not weakly closed (i.e., not $\mathcal{T}_{w}$-closed).
Citation
Jennifer Shueh-Inn Hu. Thakyin Hu. "Krein-Milman's Extreme Point Theorem and Weak Topology on Hyperspace." Taiwanese J. Math. 20 (3) 629 - 638, 2016. https://doi.org/10.11650/tjm.20.2016.6411
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