24.01: Did not make any notes.
31.01: Here's what we did, and here are some solution sketches.
07.02: Solutions.
14.02: Solutions.
21.02: Solutions. In the lecture we did 3.5.5, 3.5.12, 3.5.16, 3.6.5 and 3.6.7. (The solution for 3.6.7 might look different from what we did in the lecture, but it is essentially the same.)
28.02: Sketches. Some remarks: There's a typo in 4.1.4 (it should say |x-y| to the right of < ). There is no solution to 4.3.10c, because it seems there is no straightforward way to do it, and I haven't had the time to look closely at it. You can safely skip this exercise for now.
07.03: The session was unfortunately cancelled. Solutions to all the exercises (except 4.4.2 and 4.5.4) can be found here. Some remarks:
- For 4.4.2 consider \(\sum_{n=0}^\infty \frac{x^n}{n^2}\), \(\sum_{n=0}^\infty \frac{x^n}{n}\) and \(\sum_{n=0}^\infty x^n\).
- Here is an alternative solution (to the one in the link above) to exercise 4.5.6: A sequence\(\{x_n\}\) of real numbers is just a function \(x: \mathbb{N} \to \mathbb{R}, \, x(n) := x_n.\) Hence \(c_0 = B(\mathbb{N},\mathbb{R})\) by definition, and this is a complete metric space by Theorem 4.5.3. Note: \(c_0\) usually denotes the space of sequences converging to zero, so the notation in the exercise is not standard.
- Fun fact: The function \(L: C([0,1], \mathbb{R}) \to C([0,1], \mathbb{R})\) in 4.6.3 is an example of a bounded linear operator between Banach spaces, which is a topic in chapter 5.
14.03, 28.03, 04.04: Solutions to all(?) of the exercises can be found the usual places. Feel free to ask me if some are missing.
25.04: Most solutions can be found the usual places. For 10.1.1 and 10.1.9 note that we are using \(\int f(x) dx = \int \mathsf{Re} f(x)dx + i\int \mathsf{Im} f(x) dx\). The solution for 10.1.1 is the solution to 5.3.2 here. For 10.1.9 you can use the product rule and the fundamental theorem of calculus.
02.04: The solutions that are not here, can be found here (sorry about the medium scan quality and the strange order).
16.05: We discussed two "main methods" to prove that a function between normed spaces is continuous. Examples of both are here, as well as answers to the other exercises this week.
23.05: Solutions to the weekly problems (including delayed exam 2016) are here. 6.7 is relevant for the exam, while 6.8 is not (but there are some nice exercises).
There are no more exercise sessions. If you have any questions you can ask per mail or simply see if I'm in my office (715). Good luck on the exam!