Week 21
- Exam 2018.
- The solution can be found here.
- Gaute will discuss the exam in the weekly exercise session.
Week 20
- Exam 2019.
- The solution can be found here.
- Gaute will discuss the exam in the weekly exercise session.
Week 19
- Exam 2020: Problem 2.
- The exercise session on Thursday, 13 May is cancelled due to a holiday.
- You may send a PDF with your solution to Ole before 16:00 on Friday, 14 May and he will check your work. (This can be considered a small "practice exam".)
- The solution can be found here, but please do not to consult it if you want to submit your work.
Week 18
- The problems are available here.
- Solutions to first two problems, 5.6, 5.15, 5.17, 5.18 and 5.19
Week 17
- The problems are available here.
- Solutions: Corollary proof, 5.9, 5.12
Week 16
- The problems are available here.
- Solutions to: Diagonal operators, 5.1, 5.4, 5.5 and 5.8
Week 15
- The problems are available here (revised).
- Solutions to 4.15, 16, 17, 18, 19, 23, 25 and 27
Week 13 & 14
- ELA Section 4: 9, 13 and 14.
- Problem 5 in this previous exam 2016 exam (additional hint: apply the Riesz representation theorem).
- Suggested solutions to 9, 14 and problem 5
- Week 14 Exercise Session
Week 12
- ELA Section 4: 3, 5, 7 and 10.
- Extra exercises
Exercise session w12
Week 11
- ELA Section 3: 7, 9, 10 c) and d), 12, 13, 15 and 16. (ex. 9 uses the notation + with a dot over, but you can just replace it with the oplus symbol).
Exercise session w11
Week 10
- ELA Section 3: 1-4. (We will take 'infinite dimensional vector space' to mean a vector space that isn't finite dimensional, i.e. does not have a finite basis. One can show that even these spaces have a basis (necessarily infinite), but this is beyond the scope of the course)
Week 9
- ELA Section 2: 4 (b).
- The other problems are here.
Exercises w9 from session
Week 8
- Spaces Section 8.4: 1, 5, 6.
- Mandatory Assignment 2020: Problem 2. (You can find the problem sheet here.)
- ELA Section 2: 2, 4 (a), 5.
- Problem: Prove Lemma 63 from Lecture 14.2.
- Spaces Section 8.5: 4, 5.
Week 7
There are fewer problems than usual this week, in part due to the additional time required for the reading assignment related to Carathéodory's extension theorem for semi-algebras (see Lecture 12.4). It is also possible to start looking at the mandatory assignment.
- Spaces Section 8.2: 4, 5.
- Problem 1: Prove Lemma 52 from Lecture 12.2
- Problem 2: Prove Lemma 54 from Lecture 12.3.
- Spaces Section 8.3: 1, 3.
All exercises w7
Week 6
- ELA Section 2: 2.11.
- Problem: Let \((X,\mathcal{A},\mu)\) be a finite measure space. Prove that uniform convergence almost everywhere implies \(\mathcal{L}^p\)-convergence for every \(1\leq p \leq \infty\).
- Spaces Section 7.8: 1, 2 (there is a misprint — replace 0 with 1), 3, 4.
- Spaces Section 8.1: 1, 2, 3, 4.
- Spaces Section 8.2: 1.
Exercises
Week 5
- ELA Section 2: 2.1, 2.2, 2.9, 2.10, 2.12, 2.13.
- Problem: Suppose that \((X,\mathcal{A},\mu)\) is a finite measure space (as in ELA 2.13). Prove that \(\displaystyle \lim_{p\to\infty} \|f\|_p = \|f\|_\infty\).
- Spaces Section 7.7: 16. (A geometric proof of Young's inequality.)
Solutions 2.13 and the lim p ||f||_p exercise
Week 4
From Spaces:
- Section 7.5: 4, 5, 6, 9, 11, 12, 13, 16.
- Section 7.6: 1, 3, 5, 6, 7.
Exercises pdf
Week 3
From Spaces:
-
Section 7.3: 1, 3, 5, 6, 10, 11, 12, 13, 14.
-
Section 7.4: 1, 2, 3, 4, 5.
Video
Week 2
The exercises for this week are in the first two sections of Spaces:
- Section 7.1: 1, 3, 4, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19.
- Section 7.2: 1, 3, 4, 5, 6.
Video 1 og video 2.