Vladimir Troitsky, University of Alberta
Convergence structures in vector lattices
There are important convergences that appear naturally in analysis are not topological including, for example, convergence almost everywhere. Several common convergences in the theory of vector lattices not topological. These include order, unbounded order, and relative uniform convergences of nets. However, they fit into the framework of the theory of convergence structures. This theory was originally formulated in terms of filters, which made it impractical for applications in analysis. However, it was recently reformulated in the language of nets. In the talk, I will present an overview of the theory of convergence structures in terms of nets. I will revisit order, unbounded order, and relative uniform convergences in vector lattices from the point of view of convergence structures. I will discuss locally solid convergence structures; this concept is the natural extension of a locally solid topology to the framework of convergence structures. I will also discuss several interesting modifications of convergence structures: Mackey, unbounded, bounded, and Choquet.