Spring 2019

* 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 *l*^{p}-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 *L*^{p}-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 (due to circumstances, the seminar was shorter than usual this year):*

Friday afternoons, 14.00-17.00hr (at the latest), on:

15 February

22 February

1 March

8 March

22 March

12 April

26 April

10 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.

Pictures of the blackboards are here.

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

Chapter 7.

Pictures of the blackboards are here.

*Lecture 8:* 10 May 2019 (room B03): Alex Spieksma

Chapter 9.

Pictures of the blackboards are here.