Functional Analysis, spring 2005, J. Hulshof

Credits: 6 ECTS

Examination: homework and oral exam. The homework can be mailed directly to Guit Jan Ridderbos,
mailbox in the staff room, who will grade it.

Week 14-20. Hours: Tuesdays, 13.30-16.15 (R239), Wednesdays 13.30-16.15 (R232).
From week 15 on the last 2 hours on Tuesdays are exercise class.

Prerequisites: a general background in analysis and linear algebra.

Laste year I did more or less the first 3 chapters, see last year.
The programme of last year is a first approximation of this year's programme.

The first week I did most of Chapter 1, see here.
I forgot to shoot the last part showing that x is well defined and PHI is surjective.
Not done yet: Fourier series. Parallelogram law.

The second week I will talk about Chapter 2 and duality. On Wednesday in the exercise class we will do the exercise from Chaper 1.
Suggested: 1, 2 was done in the course, 3, 15, 17 was done 4, 5 was done, 6 is about completion, which I have not done yet,
7 was done in the course (de Hoofdstelling), 8,9,10,11,18,20.
Homework to be handed in on the first Friday after the exercise class (and graded by GJ before the next exercise class?): 13 and 20.

The second week I did the most of Ch2. No pictures from Tuesday when I discussed
(briefly) Fourier series, the completion of a normed space, and
projection on closed convex subsets of Hilbert spaces. Wednesday
I did Riesz, Hahn-Banach and discussed a bit some examples of dual spaces.
See here. As for the exercises in 2.5. 1,2 are very easy. 3,4,
5 I did,6 we skip,7 and 8 we do, 9 I did,10,skip 11,14 fills in a detail,15 (he means F defined
on the whole of X, later on we/he will consider F defined only on part of X),16 I did
(no pics), 17 shows that 16 is optimal in some sense, 18 (' notation for elements of X'=X^*),
19,20,21,22,23,24. Homework 10,15,19.
Exercise 20 is wrong, Replace by: if F satisfies the upper estimate, then
it also satisfies the lower estimate, which is EASY. Of course the proof of 21
is to easy for the statement to be acceptable.....

The third week I finished Chapter 2. See here. Next week we do these exercises and start with Chapter 3 and 4. Hand in homework: 8,9,10,11 of these exercises.
The 4th week I did Section 3.1-3.4. No pictures of Wednesday, only of Tuesday, when I prepared for the exercises above.
The 5th week you can do from 3.8: 2,3,5,6,7,10,11(!),12,14,15,16, hand in home work 1,4,8,26.
In the course I finished Chapter 3. No pictures due to camera failure.

The 6th week you can do from 3.8: 9,14,15,16(any->some),17,18(first N(A)={0}, R(A)=Y, second N(A)={0}
and use ex 9, the first one plus characterisation of N(A'), to apply first step to A:X->R(A), finally the general case),20(X must be Banach),23,24,28.
Hand in home work 19 (first assume N(A)={0}, see hint on home page of last year),21,22.
In the course I went through Chapters 4 and 5, see (pfff) the pics.. I also discussed the spectral theorem for compact selfadjoint operators on a real Hilbert space.
The last week you can still do 3.8. 18,25,28. Also 4.5 3,8,4(one to one is not needed),5,12,14,15.
Finally for hand in home work, refering to pic1, pic2, pic3, fill in the details of the spectral theorem for compact selfadjoint operators.
In particular, prove that the sup is attained, and that maximizers are eigenvectors.
I will conclude the course with a discussion of (6.25).

Literature: Principles of Functional Analysis, Martin Schechter,
American Mathematical Society, Graduate Studies in Mathematics, ISBN: 0-8218-2895-9,
available from STORM.