Courses
Advanced Course - Ordered Vector Spaces and Their Completions
Speaker: Eugene Bilokopytov (ICMAT)Date: Wednesday, 08 July 2026 - 14:30-16:30Place: Aula Gris 1, ICMAT
Abstract:
Date & time: 1 June - 8 July, Mondays & Wednesdays, 14:30 - 16:30 (CEST)
Abstract: The course is meant as an introduction into ordered vector space theory with the focus on various notions of completeness available there, and the corresponding completions. For a given completeness the plan is to:
- present standard construction of the completion;
- characterize the completion as a complete space which contains the given space in a certain way;
- prove various universal properties for the completion.
We will use the concept of the reflective subcategory as the organizing principle for our exposition (this concept formalizes the idea of a completion as the ``complete'' object which satisfies a universal property).
Even though an ordered vector space is not a priori equipped with a topology, the order and algebraic structures on it generate certain topological-like structures (called convergences). Various completeness properties can be stated in terms of these convergences. In the course we will discuss relative uniform convergence, as well as order convergence and its relatives.
The topics to be covered:
- The MacNeille completion of a poset.
- The Dedekind and Riesz completions of a directed Archimedean ordered vector space.
- Order and relative uniform convergences.
- The Dedekind completion vs. order convergence. The relative uniform completion.
- The universal completion of an Archimedean vector lattice.
- Adjoining a unit to an ordered normed space; ice cream spaces.
