Cursos
Nets in vector lattices and functional analysis
Ponente: Youssef Azouzi (Université de Carthage, Tunisia)Fecha: lunes 03 de noviembre de 2025 - jueves 06 de noviembre de 2025 - 15:00 -- 17:00 (CET)Lugar: Aula Naranja, ICMAT
Online: https://us02web.zoom.us/j/85130029931?pwd=ixGU1hBTb0V6bumIHG2X0b4rlcfEef.1 (ID: 851 3002 9931; Passcode: 715157)
Resumen:
This course is dedicated to the study of nets, with the goal of introducing students to this powerful and efficient tool and helping them become confident in using it across various contexts.
The first part offers a quick, yet relatively complete, overview of the topic. It presents the basic and foundational concepts of nets, along with a comparison to filters. In this section, nets are considered only within the topological setting. Several new and concise proofs of well-known results will be provided to illustrate the strength and utility of nets.
The second part extends the discussion to more general frameworks, using nets to define a convergence structure developed in parallel with a similar approach via filters. Particular attention will be given to some types of
convergence commonly used in vector lattices and Banach lattices---most notably, order convergence and uniform convergence and their unbounded modifications. In addition to covering standard and foundational material, the course will present recent advancements and applications of nets. Once again our major objective is to help students become familiar with nets and develop the ability to think and reason with them. To support this, three main approaches will be emphasized:
- A rich selection of carefully studied examples,
- Elegant and insightful proofs,
- An overview of recent research, which at times will present results without full proofs.
A key focus of the course will be on two versions of Cantor's diagonal process adapted to nets: one constructive and related to countable properties, and another more general but non-constructive.
The course will conclude with a list of open questions, aimed at inspiring further exploration and engagement with the subject.
