Journal of Convex Analysis 29 (2022), No. 1, 013--060
Copyright Heldermann Verlag 2022
Weak* Hypertopologies with Application to Genericity of Convex Sets
Jean-Bernard Bru
Departamento de Matemáticas, Universidad del País Vasco, and: BCAM - Basque Center for Applied Mathematics, Bilbao, Spain
and: IKERBASQUE, Basque Foundation for Science, Bilbao, Spain
Walter de Siqueira Pedra
Departamento de Física Matemática, Universidade de Sao Paulo, Brazil
and: BCAM - Basque Center for Applied Mathematics, Bilbao, Spain
wpedra@if.usp.br
We propose a new class of hypertopologies, called here weak* hypertopologies, on the dual space X* of a real or complex topological vector space X. The most well-studied and well-known hypertopology is the one associated with the Hausdorff metric for closed sets in a complete metric space. Therefore, we study in detail its corresponding weak* hypertopology, constructed from the Hausdorff distance on the field (i.e. R or C) of the vector space X and named here the weak*-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* hypertopologies are very useful and natural structures by using again the weak*-Hausdorff hypertopology in order to study generic convex weak*-compact sets in great generality. We show that convex weak*-compact sets have generically a weak*-dense set of extreme points in infinite dimensions. An extension of the well-known Straszewicz theorem to Gateaux-differentiability (non necessarily Banach) spaces is also proven in the scope of this application.
Keywords: Hypertopology, Hausdorff distance, convex analysis.
MSC: 54B20, 46A03, 52A07, 46A55.
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