Exercise set 10
Exercises about Fredholm operators.
Exercise set 9
Exercises involving the Borel functional calculus and trace class operators.
Exercise set 8
These exercises cover Section 4.3, 4 and 5.
Exercise set 7
Update: Thanks to a student for notifying me, 8d) was to my surprise much harder to prove than I had anticipated, so I had to simplify it.
These exercises are about the basic theory of *-algebras and C*-algebras, and cover the first few sections of Chapter 4. We have already used the results of 2, 5 and 8 in the lectures. I recommend beginning with 1 and 2, and then moving on to the others which might be more challenging. 3 and 7 in particular show some remarkable properties about C*-algebras and *-homomorphisms between them.
Exercise set 6
These problems cover sections 3.2 and 3.3 in the notes.
Exercise set 5
(Update 01.03: Small typo in 3d) corrected)
These problems cover sections 3.1 and 3.2 in the notes. Exercises 1-5 cover the basics of Banach algebras, while 6-9 covers the Stone-Weierstrass theorem. I recommend that you do some problems from both of the sections, if you have time. Also, exercise 10 goes through the construction of a quotient space and the quotient norm, for those of you who have not seen this before.
Exercise set 4
(Update 26.03: Another correction: In 7b), X should be assumed a Banach space.)
(Update 24.03: Turns out I mentioned the wrong exercise in the below update. The mistake is in 5d), not 6d). 5d) works fine in the locally convex setting, it is in 6d) one needs X to be normed. Sorry about that!)
(Update 19.02: I fixed a couple of mistakes in the exercise set. There was a typo in exercise 2 in the definition of the functions. Also 5d) was wrong, bit with the assumption that X is normed it should be right.
These problems cover sections 2.5 and 2.6 in the notes.
Exercise set 3
(Update 12.02: There was a mistake in exercise 6 b), one should assume that C is symmetric here)
This exercise set has problems involving weak topologies determined by semi-norms and the Hahn-Banach separation theorems. Exercise 6 is a bonus problem, one of the big theorems about complete metric spaces might be useful.
Exercise set 2
(Update 01.02: A mistake in exercise 5b) was mixed. U is now required to contain 0)
(Update: The first exercise was still incorrect but should now be correct)
(Update: Corrected the first exercise and added proving Proposition 2.1.11 to the second exercise).
This exercise set is meant to cover the basics of convex sets, semi-norms and topological vector spaces, corresponding to Sections 2.1 and 2.2 in the notes. I would recommend starting with 1, 2 and 3. Exercise 4 should be fun and tests your intuition about sums of sets in topological vector spaces. Exercise 5 is a handy fact that we will use at least once in the lectures. The same goes with 8, which is a little harder to show. Exercise 7 will be very important later for determining when a topological vector space is Hausdorff. The final exercise, 9, is more for fun and we will probably not use it in the course.
Exercise set 1
(Notes: 1 should be a nice exercise to start with. I also highly recommend doing 3. 2b) is probably a little hard. 5 showcases a very general class of spaces where one can (often) use sequences instead of nets. 6 is all about a space where sequences are not enough to describe the topology, i.e. one needs nets)
(Note: I have sneakily updated exercise 3, making it slighly more comprehensive but also more useful)