This is the web page of the Functional Analysis course in the Dutch Master Program in Mathematics, see the course description on the home page of the program. This page is under permanent construction. We maintain here the week to week program and assignments.

From the Banach page:

[Banach] would spend most of his days in cafes, not only in the company of others but also by himself. He liked the noise and the music. They did not prevent him from concentrating and thinking. There were cases when, after the cafes closed for the night, he would walk over to the railway station where the cafeteria was open around the clock. There, over a glass of beer, he would think about his problems.

Literature:

Martin Schechter, Principles of Functional Analysis, AMS Graduate Studies in Mathematics Volume 36, 2nd edition 2001, ISBN 0-8218-2895-9

Hours:

Thursday mornings 10:15 - 13:00

Place:

University of Utrecht, Mathematics building (WG), room 611a.

Program

week 1 (Sept 9): Ch 1. This is what Joost did. Not done: 1.3: infinite matrix examples and l_infty; parallelogram law and the demonstration that C([a,b]) is not a Hilbert space. 1.4. Fourierseries, the closure (completion) of a normed space, and the discussion of L^2. The book avoids measure theory here, at the expense of some rigour.
Homework: 1.5: 13, 17. All exercises are recommended, except 21, which I think is wrong.
Look at the contents of the book and inform me by e-mail when sections in say the first half of the book are already familiar, and also when section seem of greater interest to you.

week 2 (Sept 16): Ch 2

Odo: Material covered :
-- recap Ch 1
-- preview Ch 2
-- 2.1 Riesz representation theorem
-- 2.2 Hahn-Banach
-- 2.3 Thm 2.7 and Cor 2.8 formulated (not proved) Thm 2.10 formulated (not proved)
-- 2.4 skipped entirely
Exercises 2.5 : suggested to do 1, 2, 9 and 18 (this is NOT homework that has to be handed in)
NB (Joost). Ex 7 is known as Riesz' lemma, used in the proof that in infinite dimensions bounded sequences do NOT have convergent subsequences (in general),
9 is for all linear functionals, 22 relates 2.1 and 2.2.

week 3 (Sept 23): Ch 3

Material covered :
-- recap Ch 2
-- preview Ch 3 , 3.1-3.4
-- 3.1 Intro
-- 3.2 Adjoints
-- 3.3 Annihilators
Exercises 3.8 : suggested to do 4, 5 (hint : use a consequence of Hahn-Banach), 14 and 15 Of these 4 and 15 have to be handed in on September 30


week 4 (Sept 30): Ch 3

Material covered :
-- (Odo) 3.4, proofs of Baire and Closed Graph Theorem still to be done, read them first
-- (Joost) 3.5
-- exercises suggested: 3.8 1-12, except the ones already done and 7 and 9.



week 5 (Oct 7): Ch 3, conclusion (finally), begin Chapter 4.
Exercises suggested: 3.8 7, 9. Home work: 26, 27.

week 6 (Oct 14): Strike? If so, read 4.1, 4.2 and do these exercises,
which are a nice warming up for the theory of Fredholm operators.
Hand in homework next week. Here are J.J.'s answers.

week 7 (Oct 21) Odo finished Chapter 4.

week 8 (Oct 28): no course.


week 9 (Nov 4): Odo did 5.1 and 5.2


week 10 (Nov 11): This is what Joost did, i(BA)=i(B)+i(A),
plus perturbations and adjoints of Fredholm operators.
Homework to be handed in next time: 5.8. 2,3,4


Odo, Joost.