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
The proof of Proposition 2.1 in the book is not entirely convincing: here is one I wrote down based on Lindenstrauss' 1964 Memoir, and here is a proof of Auerbach's Lemma, which is needed to produce projections with a norm estimate.

Lecture 5: March 23, 2012: Willem van Zuylen (Nijmegen) 12.30-15.30 room 313 (!)
2.6 and 3.1. His speaking notes (also containing some details there is no time to elaborate on) are here.

Lecture 6: April 13, 2012: Marcel de Jeu (Leiden)
3.2 and 3.3. My notes, containing the details (including a correct proof of the key Corollary 3.10) and also what could not be covered, are here. I have added some extra material explaining how Example 3.4, on the injective tensor product of l_1 and X, fits into the theory of the Pettis integral. The starting point is to realize that the Pettis integrable functions on N can be identified with the weakly subseries summable sequences and that, according to the Orlicz-Pettis theorem, these are precisely the unconditionally summable sequences. The norm in Example 3.4 on the space l_1[X] (i.e., on the Pettis integrable functions), is then precisely the Pettis norm. There is a technical complication because the counting measure on N is not finite, so that Corollary 3.10 does not apply, but in this particular case one can show "by hand" that l_1[X] is complete and that the image of the canonical isometric embedding as in Proposition 3.13 is still dense. Hence the injective tensor product of l_1 and X is isometrically isomorphic with the X-valued Pettis integrable functions on N in the Pettis norm, which is the analogue of Proposition 3.13 as applicable in this particular case.

Lecture 7: Apr 20, 2012: Björn de Rijk (Leiden) room B1 (!)
3.4 and 3.5

Lecture 8: Apr 27, 2012: Jan Rozendaal (Delft)
4.1. Here are his notes.

Lecture 9: May 11, 2012: Chris Groothedde (Utrecht)
4.2 and 4.3. Here are his notes.

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

Lecture 11: June 1, 2012: Bas Jordans (Nijmegen)
5.2 and 5.5. Here are his notes.

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

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 (, tel.: 071 527 7118).