The big slice phenomenon in Banach spaces. Diameter 2 properties, Daugavet- and delta-points

Abstract

Preliminary theory will be presented prior to each result. We begin, in Subsection 1.2.1, by discussing Müntz spaces, which is the focus of the two first papers, “Two properties of Müntz spaces” and “Octahedrality and Müntz spaces”. In Subsection 1.2.2, we then discuss diameter two properties which is the recurring theme throughout the thesis. We end the summary, with Subsection 1.2.3, by presenting the results related to Daugavet- and delta-points, which form the focus of the papers “Daugavet- and delta-points in Banach spaces with unconditional bases” and “Delta-points in Banach spaces generated by adequate families”. All the results are stated without proofs, but their origin is referenced where their proofs can be found in full detail. The notation and terminology used throughout the thesis is standard (see e.g. [AK06]). If X is a Banach space, then Bx, Sx and X* denote the unit ball, unit sphere and topological dual space, respectively. The convex hull of A of a subset of X is denoted conv(A) and the linear span by span(A). The norm- and weak-closure of A will be denoted A and Aw, respectively.