19-22 J.-B. Bru and W. de Siqueira Pedra
Weak* Hypertopologies with Application to Genericity of Convex Sets (325K, pdf) Feb 14, 19
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Abstract. We propose a new class of hypertopologies, called here weak* hypertopologies, on the dual space $\mathcal{X}^{st }$ of a real or complex topological vector space $\mathcal{X}$. The most well-studied and well-known hypertopology on a complete metric space is the Hausdorff metric (hyper)topology. Therefore, we study in detail its corresponding weak$^{st}$ hypertopology, constructed from the Hausdorff distance on the field of the vector space $\mathcal{X}$ and named here the weak$^{st }$-Hausdorff hypertopology. It has not been considered so far and we show that it can have very interesting mathematical connections with other mathematical fields, in particular with mathematical logics. We explicitly demonstrate that weak$^{st }$ hypertopologies are very useful and natural structures by using again the weak$^{st }$-Hausdorff hypertopology in order to study generic convex weak$^{st }$-compact sets in great generality. We show that convex weak$^{st }$-compact sets have generically weak$^{st }$-dense set of extreme points in infinite dimension. An extension of the well-known Straszewicz theorem to Gateaux-differentiability (non necessarily Banach) spaces is also proven in the scope of this application.

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