Functional Analysis Seminar Spring 2012

Text: R.A. Ryan, "Introduction to Tensor Products of Banach Spaces", Springer, 2002.
Given two Banach spaces X and Y, one supplies the usual algebraic tensor product of these vector spaces with a norm. After completing a Banach space results: a so-called Banach space tensor product of X and Y. Naturally, the final Banach space critically depends on the choice of the norm introduced on the algebraic tensor product, and various well-known and/or interesting spaces can be obtained by chosing appropriate X, Y, and norm. The author writes in his preface that "Our viewpoint is that tensor products are a natural and productive way to understand many of the themes of modern Banach space theory and that "tensorial thinking" yields insight into many otherwise mysterious phenomena. We hope to convince the reader of this belief".
The author is probably right and certainly he is in good company: before moving from functional analysis to algebraic geometry, Grothendieck did groundbreaking work on these tensor products in the fifties. We will not see all of his work in this seminar, and certainly not the more difficult and technical parts, which are not in this book, but fortunately that is not at all necessary to start appreciating these Banach space tensor products as a means for a better understanding of Banach spaces and operators between Banach spaces.
We will not be able to cover the whole book, but the first 125 (Chapters 1-5) or 155 (Chapters 1-6) pages will already suffice to give a good introduction to the field.

Intended for: Students, PhD students and staff.

Prerequisites: This is not a mathematical sequel to the national Functional Analysis course as taught by André Ran and myself. Strictly speaking, introductory courses in functional analysis and measure theory (almost) suffice as prerequisites, but this is not recommended. This seminar does not build mathematically on the national FA-course, but a comparable functional analytical maturity and fluency in the functional analytic terminology help a great deal.

Venue: Mathematical Institute, Leiden University, Niels Bohrweg 1 (Snellius building), Leiden, room 412 (except April 20).

Date and time: Friday afternoons, 14.00-17.00hr at the latest (except June 8).

Lecture 1: February 10, 2012: Marten Wortel (Leiden)
Chapter 1

Lecture 2: March 2, 2012: Jan van Waaij (Leiden)
2.1 (Here is a proof that every Banach space is a quotient of an l¹-space, used in the proof of Proposition 2.8.)

Lecture 3: March 9, 2012: Nikita Moryakov (Delft)
2.2 and 2.3

Lecture 4: March 16, 2012: Frejanne Ruoff (Leiden)
2.4 and 2.5

Lecture 5: March 23, 2012: Willem van Zuylen (Nijmegen)
2.6 and 3.1

Lecture 6: April 13, 2012: Marcel de Jeu (Leiden)
3.2 and 3.3

Lecture 7: Apr 20, 2012: Jan Rozendaal (Delft), room B1 (!)
3.4 and 3.5

Lecture 8: Apr 27, 2012: Björn de Rijk (Leiden)
4.1

Lecture 9: May 11, 2012: Chris Groothedde (Utrecht)
4.2 and 4.3

Lecture 10: May 25, 2012: Florian Kluck (Utrecht)
5.1

Lecture 11: June 1, 2012: Bas Jordans (Nijmegen)
5.2 and 5.5

Lecture 12: June 8, 2012: Vaya Vos (Leiden) (morning meeting 10.00-13.00!)
5.3

Lecture 13: June 15, 2012: Ron Hoogwater (Leiden)
5.4

EC: 6 for attendance and delivering an afternoon filling lecture with break(s). These lectures are not public exams before an audience: the atmosphere in this seminar has been very informal in the past years and we will keep it that way. Think of it as a group of people who like functional analysis and who are learning a new subject together.

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 not from Leiden, contact your study advisor about the eligibility of this seminar for your own programme beforehand, in order to prevent unwanted surprises. If your institution should require this, then, although this is not the preferred method, a grade could be supplied instead of a "pass".

Contact: Marcel de Jeu (mdejeu@math.leidenuniv.nl, tel.: 071 527 7118).