Functional Analysis

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Functional analysis is a field of mathematics where one studies vector spaces
equipped with a vector topology (e.g. a topology generated by an inner
product or a norm) and linear continuous functions on these spaces. Vector
spaces equipped with a norm, and where the vector spaces are complete
under this norm, are called Banach spaces. Banach space theory is thus a
subfield of functional analysis.

More about the Functional Analysis group

The functional analysis group has its roots tracing back to Professor
Åsvald Lima at Agder Distrikshøgskole (the predecessor to Universitetet i
Agder) in the 1980s. Lima’s research was related to geometry in Banach
spaces, including approximation properties, ideal structures, and the connection
between them. The group continues its research on geometry in
Banach spaces, focusing on the shape (e.g., roundness and corners) of the
unit sphere in such spaces. The group’s recent focus has been on diameter
two properties, but research has also been done on M-ideals and other ideal
structures in Banach spaces, as well as on approximation properties and their
connection with ideal structures.

Diameter two properties are a relatively new research area within Banach
space theory. If all sets in a given class of subsets of the unit ball of a Banach
space have diameter two, we say that the space has a diameter two property
relative to this class. A typical such class of subsets is the class of all slices.
The fact that a Banach space has a diameter two property relative to the
class of all slices suggests that the unit ball in the space has a certain form
of ’square’ geometry. Diameter two properties are closely related to classical
geometric Banach space properties such as octahedrality and the Radon-
Nikodym property. The FA-group was the first to conduct a systematic
study of diameter two properties in Banach spaces.

Collaboration and network

Abrahamsen, Dovland, Lima, and Martiny collaborate with colleagues
at the University of Tartu, Freie Universität Berlin, University of
Murcia, Polytechnical University in Prague, University of Eastern Finland,
Saint Louis University, University of Granada, Ben-Gurion University of the
Negev, Bulgarian Academy of Sciences, University of Franche-Comté in Besancon,
Polytechnic University of Valencia, and University College Dublin in
Belfield. They actively contribute to the Estonian doctoral school in Mathematics
and Statistics.

Selected publications

Publications

Scientific articles and book chapters

Abrahamsen, Trond A.; Hájek, Petr; Lima, Vegard & Troyanski, Stanimir (2026). Strictly convex norms and the local diameter two property. Revista Matemática Complutense. ISSN 1139-1138. 39, p. 857–877. doi: 10.1007/s13163-025-00559-z. Full text in Research Archive Show summary
We introduce and study a strict monotonicity property of the norm in solid Banach lattices of real functions that prevents such spaces from having the local diameter two property. Then we show that any strictly convex 1-symmetric norm on $$c_0(\Gamma )$$ c 0 ( Γ ) possesses this property. In the opposite direction, we show that any Banach space which is strictly convex renormable and contains a complemented copy of $$c_0({\mathbb {N}}),$$ c 0 ( N ) , admits an equivalent strictly convex norm for which the space has the local diameter two property. In particular, this enables us to construct a strictly convex norm on $$c_0(\Gamma ),$$ c 0 ( Γ ) , where $$\Gamma $$ Γ is uncountable, for which the space has a 1-unconditional basis and the local diameter two property.
Haller, Rainis; Kaasik, Jaan Kristjan & Ostrak, Andre (2026). The local diameter two property and the diameter two property in spaces of Lipschitz functions. Journal of Mathematical Analysis and Applications. ISSN 0022-247X. 556(1). doi: 10.1016/j.jmaa.2025.130178. Full text in Research Archive Show summary
We separate the local diameter two property from the diameter two property and their weak-star counterparts from each other in spaces of Lipschitz functions. We characterise the w⁎-LD2P, the w⁎-D2P, the LD2P, and the SD2P in these spaces. We introduce a generalised version of cyclical monotonicity to study functionals on spaces of Lipschitz functions.
Haller, Rainis; Kaasik, Jaan Kristjan & Ostrak, Andre (2025). Separating diameter two properties from their weak-star counterparts in spaces of Lipschitz functions. Studia Mathematica. ISSN 0039-3223. 280 (1), p. 87–102. doi: 10.4064/sm240612-9-9. Full text in Research Archive Show summary
We address some open problems concerning Banach spaces of real-valued Lipschitz functions. Specifically, we prove that the diameter 2 properties differ from their weak-star counterparts in these spaces. In particular, we establish the existence of dual Banach spaces lacking the symmetric strong diameter 2 property but possessing its weakstar counterpart. We show that there exists an octahedral Lipschitz-free space whose bidual is not octahedral. Furthermore, we prove that the Banach space of real-valued Lipschitz functions from any infinite subset of ℓ1 possesses the symmetric strong diameter 2 property. These results are achieved by introducing new sufficient conditions, providing new examples and clarifying the status of known ones.

View all works in NVA

Previous participants

Andre Ostrak, Postdoc (Jan 2024– Jan 2026). Now at: University of Tartu.

Participants

Published Sep. 2, 2024 2:01 PM - Last modified Apr. 13, 2026 12:39 PM