Weak$^{\ast }$ Hypertopologies with Application to Genericity of Convex Sets

TL;DR

This paper introduces weak$^{\ast}$ hypertopologies on dual spaces, explores their mathematical connections, and applies them to study generic properties of convex weak$^{\ast}$-compact sets, including density of extreme points.

Contribution

It proposes a new class of weak$^{\ast}$ hypertopologies, especially the weak$^{\ast}$-Hausdorff hypertopology, and demonstrates their usefulness in analyzing convex sets and their extreme points.

Findings

Weak$^{\ast}$-Hausdorff hypertopology can be constructed from the Hausdorff distance.

Convex weak$^{\ast}$-compact sets have a generic weak$^{\ast}$-dense set of extreme points.

An extension of Straszewicz's theorem to non-Banach spaces is established.

Abstract

We propose a new class of hypertopologies, called here weak$^{\ast }$ hypertopologies, on the dual space $\mathcal{X}^{\ast }$ of a real or complex topological vector space $\mathcal{X}$. 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$^{\ast }$ hypertopology, constructed from the Hausdorff distance on the field (i.e. $\mathbb{R}$ or $\mathbb{C}$) of the vector space $\mathcal{X}$ and named here the weak$^{\ast }$-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$^{\ast }$ hypertopologies are very useful and natural structures\ by using again the weak$^{\ast }$-Hausdorff hypertopology in order to study generic convex weak$^{\ast }$-compact sets in great generality. We show that convex weak$^{\ast }$-compact sets have generically weak$^{\ast }$-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.