Functional Analysis Seminar
Spring 2019

Banach space theory


Organisers
Marcel de Jeu (mdejeu@math.leidenuniv.nl)
Onno van Gaans (vangaans@math.leidenuniv.nl)

Contact
If you are interested please contact Marcel de Jeu or Onno van Gaans.

Topic
The study of Banach spaces started in the early years of functional analysis and has been an active area of research ever since. The aim is to understand the structure of Banach spaces, in order to be able to answer natural questions about commonly occurring Banach spaces and also in order to obtain a better understanding of operators between Banach spaces. For example, there do not appear to be natural isometries between the lp-spaces for different p, but that does not show that such isometries do not exist. Is it, in fact, true that these spaces are pairwise non-isometric? If two of these spaces are not isometric, are they then perhaps still isomorphic? The cases where p equals 1 or infinity are easily set aside by considering reflexivity or separability, but how about the others? Can one of these spaces be isometric/isomorphic to a closed subspace of an other space in the family? How is this when also the classical Lebesgue Lp-spaces on the unit interval are taken into consideration, or spaces of continuous functions on compact Hausdorff spaces? Studying such questions has led to an attractive and well-developed theory with many deep results.
We intend to cover aspects of this theory on basis of the second edition of the book by Albiac and Kalton on the subject. The late Nigel Kalton was one of the grandmasters of the field, and this very well-written book has quickly become a standard reference.

Intended for
Students, PhD students, and faculty.

Prerequisites
Proficiency in the functional analytic language at the level corresponding to a solid `pass' for the national functional analysis course in the Mastermath programme. An introductory course in functional analysis is usually not sufficient.

Literature
Fernando Albiac and Nigel J. Kalton, Topics in Banach space theory. Second edition, Graduate Texts in Mathematics Vol. 233, Springer, 2016. ISBN: 978-3-319-31555-3 (hardcover); 978-3-319-81063-8 (softcover); 978-3-319-31557-7 (e-book).
When working on a computer at the Mathematical Institute in Leiden, it is possible to order a softcover copy for 25 euros via SpringerLink. This may also be the case at other institutes.

Venue
Mathematical Institute, Leiden University, Niels Bohrweg 1 (Snellius building), Leiden. See here for directions.
Most lectures, but not all, are in room 405.

Dates and time
Friday afternoons, 14.00-17.00hr (at the latest), on:
15 February
22 February
1 March
8 March
22 March
12 April
26 April
3 May
10 May
17 May

EC
6 EC for participation and delivering an afternoon filling lecture.

Grade
It is enevitable that some topics are more suitable for an attractive presentation than others, so, as in previous years, there will be no grades but simply a `pass'.

Please note
If you are a student, but not from Leiden, contact your study advisor or exam committee beforehand about the eligibility of this seminar for your own programme, in order to prevent unwanted surprises. If your institute should require this, then, although this is not the preferred method, a grade could be supplied instead of a `pass'.


Programme

Lecture 1: 15 February 2019 (room 408): Onno van Gaans
Chapter 1.
Pictures of the blackboards are here.

Lecture 2: 22 February 2019 (room 405): Mayke Straatman
Chapter 2.
Pictures of the blackboards are here.

Lecture 3: 1 March 2019 (room 403): Ivan Yaroslavtsev
Chapter 3.
Pictures of the blackboards are here.

Lecture 4: 8 March 2019 (room 405): Yang Deng
Chapter 4.
Pictures of the blackboards are here.

Lecture 5: 22 March 2019 (room 405): Koen Keijzer
Chapter 5.
Pictures of the blackboards are here.

Lecture 6: 12 April 2019 (room 405): Mark van den Bosch
Chapter 6.

Lecture 7: 26 April 2019 (room 405): Emiel Lorist
Chapter 7.

Lecture 8: 3 May 2019 (room 412): Stefan van Lent
Chapter 8.

Lecture 9: 10 May 2019 (room B03): Alex Spieksma
Chapter 9.

Lecture 10: 17 May 2019 (room 405): Marcel de Jeu
Chapter 10 or 11/12.