Exercises

Week 3

• Spaces 7.1: 1, 3, 4, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19. (Solutions)
• Spaces 7.2: 1, 3, 4, 5, 6. (Solutions)
• Mandatory Assignment 2021: Problem 1. (You can find the problem sheet and solution here.)

Week 4

• Spaces 7.3: 1, 3, 5, 6, 10, 11, 12, 13, 14. (Solutions)

• Spaces 7.4: 1, 2, 3, 4, 5. (Solutions)

• Mandatory Assignment 2021: Problem 2. (You can find the problem sheet and solution here.)

Week 5

• Spaces 7.5: 4, 5, 6, 9, 11, 12, 13, 16. (Solutions) NB: As I remark in the solutions, there is a problem with exercise 4d.
• Spaces 7.6: 1, 3, 5, 6, 7. (Solutions)
• Mandatory Assignment 2021: Problem 4. (You can find the problem sheet and solution here.)

Week 6

• Spaces 8.1: 1, 2, 3, 4 (Solutions) NB: Here both the exercise text and my solutions use rho and mu interchangeably- it is meant to refer to the same thing (a measure on \mathcal{R})! The easiest way to avoid confusion is to replace all instances of rho with mu.
• Spaces 8.2: 1, 4, 5.
• Exam 2021: Problem 1. (You can find the problem sheet and solution here.)

Week 7

• Problem 1: Prove Lemma 52 from Lecture 9.2
• Problem 2: Prove Lemma 54 from Lecture 9.3.
• Spaces 8.3: 1, 3. (Solutions)
• Spaces 8.4: 1, 5, 6. (Solutions)
• Mandatory Assignment 2020: Problem 2. (You can find the problem sheet and solution here.)

Week 8

• AMoLM Chapter 1: 1, 2, 3, 4, 5, 6. (Solutions)
• AMoLM Chapter 2: 1, 2, 3, 4, 5, 6, 7. (Solutions)

Week 9

• ELA Chapter 1: 1, 2, 3, 4, 5, 6. (Solutions)

Week 10

• ELA Chapter 2: 1, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16. (Solutions)
• Spaces 7.7: 16.
• The Cesaro operator.

Week 11

• ELA Chapter 3: 1, 2, 4, 5, 6, 7, 8, 9. (Solutions)
• Exam 2021: Problem 2. (You can find the problem sheet and solution here.)
• Suppose that \((V,\|\cdot\|)\) is a normed space which satisfies the parallelogram law, i.e. that \(||x+y||^2 + ||x-y||^2 = 2(\|x\|^2+\|y\|^2)\) holds for every pair \(x,y\in V\). Prove that V is an inner product space. Hint. Polarization.

Week 12

• ELA Chapter 3: 13, 14, 15. (Solution)
• Suppose that \(S\) is a subset of an inner product space \(H\). Show that \(S^\perp=S^{\perp\perp\perp}\). (Solution)
• Let \(H\) be a separable Hilbert space and \(M\subseteq H\) a closed subspace. Show that \(H\) has an orthonormal basis consisting of vectors in \(M \cup M^\perp\). (Solution)
• Prove that \(\ell^2(\mathbb{R})\) is not separable. (Solution)

Week 13

• Prove claims (i), (ii) and (iii) in Theorem 3.4.2.
• ELA Chapter 3: 17, 19, 20, 22, 25, 26, 27, 28, 29. (Solutions.)
• Find a formula for the adjoint of the Cesaro operator from the exercises in week 10.

Week 14

• Prove the following lemma from Lecture 25.5: Let \(X\) and \(Y\) be normed spaces and set \(X_1 = \{x \in X \,:\, \|x\|=1\}.\) Prove that if \(T \colon X \to Y \) is compact, then \(\overline{T(X_1)}\) is a compact subset of \(Y\).
• ELA Chapter 3: 34.
• ELA Chapter 4: 1, 3, 4, 5, 7, 8. (Solutions.)
• Diagonal operators.

Week 16

• Prove that \(\mathcal{F}(H) \subseteq \mathcal{HS}(H)\) to finish the proof of Theorem 4.2.8. (This is also a part of Exercise 4.6 (a) below, so if you plan to do that you can skip this.)
• ELA Chapter 2: 11. (Solution)
• ELA Chapter 3: 18. (Solution)
• ELA Chapter 4: 6, 10. (Solution)
• Consider the Diagonal operators from week 14 and add the following subproblem. (j) For which sequences \(\lambda\) is the operator \(T_\lambda\) Hilbert¨CSchmidt?
• Hankel operators.

Week 17

• ELA Chapter 3: 21. (Solutions)
• ELA Chapter 4: 11, 12, 13, 14, 16. (Solutions)

Week 18

• ELA Chapter 4: 9 (also covered in Lecture 28), 17, 18, 19, 20. (Solutions)

Week 19

• Exam 2018. (You can find the problem sheet and solution here.)

Week 20

• Exam 2019. (You can find the problem sheet and solution here.)

Week 21

• Exam 2021. (You can find the problem sheet and solution here.)