Logic and Topology in Banach spaces

The project is devoted to the study of topological and geometric properties of Banach spaces and their duals, aiming at a better understanding of their structure. Topological properties of the weak topology often imply important geometric properties of the Banach space in question. On the other hand, geometric properties of the Banach space often give information about its weak topology. Similar statements are true for duals of Banach spaces with the weak-star topology. We are going to explore this interplay in detail. The main project goals are: 1) Developing new tools for constructing and studying Banach spaces, using techniques from set theory and category theory. 2) Exploring different types of networks and related concepts in weak topologies, determining connections with renorming theory. Results of Goal 1 will lead to new examples, settling some of the problems concerning interplay between geometric and topological properties of non-separable Banach spaces. Goal 2 will lead to a better understanding of the weak topology and its relations to the geometric structure of a Banach space.

Logic and Topology in Banach spaces Abstract for Public Relations The project was devoted to the study of topological and geometric properties of Banach spaces and their duals, aiming at a better understanding of their structure. Topological properties of the weak topology often imply important geometric properties of the Banach space in question. On the other hand, geometric properties of the Banach space often give information about its weak topology. Similar statements are true for duals of Banach spaces with the weak-star topology. We have explored several instances of this interplay, especially implications between various convergence types of measures on boolean algebras. The main project achievements are: 1) Developing new tools for constructing and studying Banach spaces, using techniques from set theory like forcing and guessing principles. 2) Exploring different types of networks and related concepts in weak topologies, determining structural properties of Banach spaces. Results related to 1) have led to new examples, settling some of the problems concerning interplay between convergence and topological properties of non-separable Banach spaces. Results obtained in the framework of 2) have revealed the combinatorial nature of those properties, leading to better comprehension of the topology of normed spaces.