This is the class webpage for the course MAST30026: Metric and Hilbert spaces. Office hours will be 10am-11:45am on Mondays and 3:15pm-4:30pm on Thursdays, in Room 168. There will be three assignments due at regular intervals during semester amounting to a total of 20%, and a 3-hour written examination in the examination period worth 80%.

Lecture notes will be posted here before class, and the videos will be posted once they are edited (usually within a day or two). Note that the “lectures” below are coherent units of content which may end up spread over one or more actual lectures, hence the multiple videos per “lecture”.

- Lecture 1: What is space? (lecture notes | video1 | video2)
- Lecture 2: Examples of spaces (lecture notes | video1 | video2)
- Lecture 3: Isometries (lecture notes | video1 | video2)
- Lecture 4: Metrics from matrices (lecture notes | video)
- Lecture 5: Minkowski space and special relativity (lecture notes | video1 | video2)
- Lecture 6: Topological spaces (lecture notes | video)
- Lecture 7: Constructing topological spaces (lecture notes
*updated 13-8*| video1) - Lecture 8: Compact spaces (lecture notes)

Rough future schedule:

- Week 5: Compactness
- Week 6: Completeness (including the fixed point theorem)
- Week 7: Urysohn’s lemma and metrisation
- Week 8: Tychonoff’s theorem and Stone-Cech compactification
- Weeks 9-11: Hilbert and Banach spaces
- Week 12: Fourier transform and Wigner’s theorem

My aim is to prove the following theorems by the end of the semester:

- Banach fixed point theorem
- Urysohn’s lemma
- Urysohn’s metrisation theorem
- Tychonoff’s theorem
- Stone-Cech compactification
- Riesz representation theorem
- If we have time, I would like to prove the Stone-Weierstrass theorem, but we’ll see how we go.
- Wigner’s theorem

Tutorials:

- Tutorial 1: Permutations and orientations (sheet | solutions)
- Tutorial 2: Sylvester’s law of inertia (sheet | solutions)
- Tutorial 3: Products, disjoint unions and Lipschitz (sheet)

Various references:

- Dual space (notes | video)
- Double dual (notes)
- Morse lemma (wiki | notes | for the two-dimensional case see this)

There is no set text, but the following books may be useful:

- T. Tao “Analysis” Vol. II (for metric spaces)
- J. R. Munkres “Topology” (for topological spaces)

In every lecture there will be assigned exercises and the three graded assignments will be largely made up of questions from those exercises. Ergo, keep up with the exercises and you’ll be in good shape for the assessment. Assignments are to be **submitted to the assignment box of your tutor**.

**Assignment 1** due by 4pm on 17-8 (Week 4):

- Exercise L2-3
- Exercise L3-3, L3-4, L3-5
- Exercise L4-0, L4-4

**Assignment 2** will be assigned on 20-8 and due 7-9

**Assignment 3** will be assigned on 10-9 and due 5-10

The focus of the class will be on important examples of spaces and proving theorems, but this material is widely used outside of pure mathematics and I want to do justice to that fact. To that end, in the last lecture of the first week I asked you all to write down your areas of interest. As I fill in the material of the course, I’ll try to keep a running list here of topics (that we have either covered already, or that we plan to cover in future lectures) relevant to each of the areas of interest that you’ve elected:

- Applied mathematics: Existence of solutions to differential equations, Calculus of variations.
- Physics: Special relativity, Wigner’s theorem, … well, Hilbert spaces!
- Computer science: Bellman equations used in reinforcement learning.
- Probability and statistics: Kullback-Leibler divergence and information metrics.

Unfortunately I do not know any specific applications in biology or chemistry, although there is obviously plenty of quantum mechanics in chemistry and applied math in biology (please let me know if you know more domain-specific applications). You might also find spaces of programs interesting, although this is outside the scope of this course.