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.
[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
University of Utrecht, Mathematics building (WG), room 611a.
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
plus perturbations and adjoints of
Homework to be handed in next time: 5.8. 2,3,4