PHOTOS: Doctoral defence of Triinu Veeorg "Daugavet and Delta-points in Banach spaces: Lipschitz-free spaces, their duals, and renormings"

On 10 October at Delta Centre, Triinu Veeorg defended her thesis "Daugavet and Delta-points in Banach spaces: Lipschitz-free spaces, their duals, and renormings".

Supervisors:

Professor Rainis Haller, University of Tartu

Professor Vegard Lima, University of Agder

Opponents:

Professor Miguel Martín Suárez, University of Granada (Spain)

Eva Pernecká, PhD, Czech Technical University in Prague (Czech Republic)

Summary

Daugavet and Delta-points in Banach spaces: Lipschitz-free spaces, their duals, and renormings

Daugavet and Delta-points were introduced as pointwise versions of the Daugavet property and the diametral local diameter-two property, respectively. Both of these properties belong to the class of diameter-two properties, meaning that every slice of the unit ball has diameter 2. In contrast, the mere existence of Daugavet or Delta-points does not imply such strong global properties. Indeed, there exist Banach spaces that contain Daugavet or Delta-points, but also admit slices of the unit ball with arbitrarily small diameter.

In this thesis, we study Daugavet and Delta-points in various classes of Banach spaces, among these Lipschitz-free spaces and their duals. We provide characterizations for certain subclasses of these spaces, including a complete characterization of Daugavet points in Lipschitz-free spaces. Furthermore, we present an example of a metric space whose Lipschitz-free space has the Radon—Nikodým property and a Daugavet point. We also show that this Lipschitz-free space is a dual space, isomorphic to ℓ1. The thesis further investigates renormings of Banach spaces that admit Daugavet or Delta-points. As a result, we show that the classical spaces ℓp for pϵ[1,∞) can be renormed to contain a Daugavet point. This provides the first examples of reflexive spaces admitting Daugavet points. Finally, we introduce several classes of Banach spaces that cannot contain Daugavet or Delta-points.