Gå til hovedinnhold
0

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

Diameter two properties is the recurring theme throughout the thesis.

André Martiny

PhD Candidate

André Martiny of the Faculty of Engineering and Science at the University of Agder has submitted his thesis entitled «The big slice phenomenon in Banach spaces. Diameter 2 properties, Daugavet- and delta-points» and will defend the thesis for the PhD-degree Friday 17 September 2021.

He has followed the PhD-programme at the Faculty of Engineering and Science at the University of Agder, with Specialisation in Mathematical Sciences, scientific field Mathematics.

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

The thesis consists of four articles, linked to the part of mathematics called Functional Analysis, and studies Diameter 2 properties, Daugavet- and delta-points.

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.

## Disputation facts:

The trial lecture and the public defence will take place online, via the Zoom conferencing app (link below)

Head of Department Ingvald Erfjord, Department of Mathematical Sciences, Faculty of Engineering and Science at the University of Agder, will chair the disputation.

#### Public defence Friday 17 September 2021 at 12:30 hours

Given topic for trial lecture«The Radon-Nikodym property»

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

Search for the thesis in AURA - Agder University Research Archive, a digital archive of scientific papers, theses and dissertations from the academic staff and students at the University of Agder.

### The thesis is available here:

#### https://hdl.handle.net/11250/2767791

Article 4 is removed until it is published.

Complete thesis:

The CandidateAndré Martiny (1991, Trollhättan, Sweden,growing up in Herøy, Nordland, Norway) Bachelor degree as Teacher for school years 5-10, UiA (2015 ), Masters degree in Mathematics Education, UiA (2016). Present position: Assistant Professor at the Department of Mathematical Sciences, UiA.

### Opponents:

First opponent: Professor Dirk Werner, Free University Berlin, Germany

Second opponent: Professor Antonio Avilés López, University of Murcia, Spain

Professor emeritus Leiv Storesletten, Department of Mathematical Sciences, University of Agder, is appointed as the administrator for the assessment committee.

Supervisors in the doctoral work were Professor Trond A. Abrahamsen, UiA (main supervisor), Professor Olav Dovland, UiA and Professor Vegard Lima, UiA (co-supervisors)

### Opponent ex auditorio:

The chair invites members of the public to pose questions ex auditorio in the introduction to the public defense, with deadlines. It is a prerequisite that the opponent has read the thesis. Questions can be submitted to the chair Ingvald Erfjord on e-mail ingvald.erfjord@uia.no