Exercises

Some important problems will be boldfaced. The solution to these problems will appear on this page. Even if the solutions appear early, we stress that you should first attempt the exercises on your own.

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, 34 (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, 2829. (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, 45, 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–Schmidt?
  • 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.)
Published Jan. 8, 2022 1:08 PM - Last modified Feb. 27, 2023 1:56 PM