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”. Note that you may need to manually change the quality setting on YouTube to HD in order to be able to read the board.

- Lecture 1: What is space? (notes | video1 | video2)
- Lecture 2: Examples of spaces (notes | video1 | video2)
- Lecture 3: Isometries (notes | video1 | video2)
- Lecture 4: Metrics from matrices (notes | video)
- Lecture 5: Minkowski space and special relativity (notes | video1 | video2)
- Lecture 6: Topological spaces (notes | video)
- Lecture 7: Constructing topological spaces (notes | video1 | video2 | video3)
- Lecture 8: Compactness I (notes | video)
- Lecture 9: Compactness II (notes | video1 | video2)
- Lecture 10: Compactness III (notes | video)
- Lecture 11: Hausdorff spaces (notes | video1 | video2 | video3)
- Lecture 12: Function spaces (notes | video1 | video2 | video3 | video4)
- Lecture 13: Metrics on function spaces (notes | video1 | video2)
- Lecture 14: Banach fixed point theorem (notes | video)
- Lecture 15: Picard theorem on ODEs (notes | video)
- Lecture 16: Stone-Weierstrass theorem (notes | video1 | video2)
- Lecture 17: Integrals (notes
*updated 16-10*| video) - Lecture 18: Banach space (notes | video1 | video2 | video3)
- Lecture 19: Duality and Hilbert space (notes | video)
- Lecture 20: Hilbert space (notes
*updated 16-10*| video1 | video2 | video3) - Lecture 21: Coordinates in Hilbert space (notes
*updated 16-10*| video1)

Extension lectures (not examinable, schedule TBA):

- Lecture 22: What is space? Quantum edition
- Lecture 23: Urysohn’s lemma

The course contained the following major theorems:

- Adjunction property for function spaces
- Banach fixed point theorem
- Picard theorem
- Stone-Weierstrass theorem
- Riesz representation 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)
- Tutorial 4: Topological groups (sheet)
- Tutorial 5: Constructing the real numbers
(sheet) (
*Note:*Q4, Q6 are optional). - Tutorial 6: Towards Fourier theory (sheet) (
*Note:*Q4 is optional). - Tutorial 7: Duality and adjoints (sheet | solutions)
- Tutorial 8: Integration (sheet | solutions)
- Tutorial 9: Around Stone-Weierstrass (sheet)
- Tutorial 10: Zorn’s lemma and vector spaces (sheet)
- Tutorial 11: Examples of vectors in L^2 spaces

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

- J. R. Munkres “Topology” (for topological spaces)
- K. Wysocki, H. Rubinstein “Metric and topological spaces“
- N. Bourbaki, “General topology” Chapters 1-4 (topological groups)
- T. Tao “Analysis” Vol. II (for metric spaces, integrals)
- W. Cheney “Analysis for applied mathematics” (for Hilbert spaces)
- W. A. Sutherland “Introduction to metric and topological spaces”
- E. Hewitt, K. Stromberg, “Real and abstract analysis” (Banach spaces)
- E. M. Stein, R. Shakarchi “Real analysis” and “Functional analysis” (Hilbert and Banach spaces)

Various references:

- Dual spaces (notes | video) and double duals (notes)
- The Morse lemma (wiki | notes | for the two-dimensional case see this)
- T. Phillips, “The topology of impossible spaces”, AMS feature column 2014.
- B. Barak “Black holes, paradoxes, and computational complexity”. This is an article by Boaz Barak, a professor of Computer Science at Harvard and author of a popular textbook on computational complexity, on the AdS/CFT correspondence, which hints at (among many other things) a possible role for computational complexity in understanding the “ultimate” nature of spacetime. Note that this is far from settled science!
- Inverse function theorem from the Banach fixed point theorem (notes)
- For the connection between CW-complexes, Morse theory and modern physics see Witten’s paper, Nicolas Mee’s thesis and Morse homology (thanks to Omar Foda for the reference to Mee’s thesis).
- T. Tao on “The uncertainty principle”.
- Background reading on “Harmonic analysis“

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** (Box 191 for Patrick and 192 for Daniel).

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

- Exercise L2-3 (
*by hand proof of arc length metric*) - Exercise L3-3 (
*relations in the isometry group of S^1*), L3-4 (*rotations, translations, reflections*), L3-5 (*orientations*) - Exercise L4-0 (
*d-1 and d-infinity metrics*), L4-4 (*isometries from similar matrices*)

The solutions are generously provided by a student (who prefers to remain anonymous).

**Assignment 2** due by 4pm on 7-9:

- Exercise L5-7 (
*hyperbolic angles and Lorentz boosts*) - Exercise L6-3 (
*Sierpinski is not metrisable*) - Exercise L7-2 (
*the product topology*), L7-10 (*circle as a pushout*), L7-12 (*homeomorphism warmup*), L7-19 (*spheres are quotients of disks*) - Exercise L8-5 (
*unbounded functions exist if they can*) - Exercise L9-5 (
*closed in compact implies compact*)

The solutions are again, generously provided by a student.

**Assignment 3** due by 4pm on 5-10:

- Exercise L11-8 (
*metrisable spaces are normal*), L11-11 (*closed diagonal vs Hausdorff*) - Exercise L12-1 (
*graphs and continuity*), L12-6 (*rotating loops*), L12-10 (*homotopy*), L12-13 (*adjunction property*) - Exercise L13-3 (
*continuity of metric*), L13-8 (*metric products and completeness*)

The bulk of the exam will be questions that are either *exactly the same as* exercises assigned in lectures, or which are very similar. The questions will be at a level of difficulty below the one-star exercises (there will be one *part* of one question at the level of the easiest one-star problem). To be clear, exercises which appear on the exam may not have appeared on any homework assignment, and may not have solutions provided in the lecture notes. Questions assigned in tutorials which do *not* have provided solutions are guaranteed to not appear on the exam.

In addition to these “exercise style” questions, there will be at least one **proof** from the lectures that you will be asked to give in the exam. The following results are explicitly **excluded**:

- Theorem on p.8 of L5 (
*Lorentz group*) - Theorem L8-2 (
*Bolzano-Weierstrass*) - Theorem L9-2 (
*sequentially compact iff. compact*) - Theorem L11-4 (
*finite CW-complexes are compact Hausdorff*) - Theorem L12-4 (
*adjunction property*) - Theorem L12-7 (
*compact-open topology vs. closed graphs*) - Exercise L14-2 (
*Bellman equation*) - Theorem L16-0 (
*Weierstrass approximation*) - Lemma L16-0.5 (
*Bernstein polynomials*) - Theorem L16-3 (
*Stone-Weierstrass*) - Theorem L18-1 (
*L^p-norm and Holder inequality*) - Lemma L19-1 (
*characterisation of finite dimension*) - Lemma L20-11 (
*conspiracy avoidance*)

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 (see Lecture 14).
- 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.

The following topics appear in the lecture notes. Those topics in the lecture notes or exercises but not discussed out loud during lectures are given in italics:

**Lecture 1**: SO(2) acting on R2**Lecture 2**: metric space, arc length metric on S1**Lecture 3**: isometries, classification of isometries of S1,*orientation***Lecture 4**: Rn as metric space, metrics from positive-definite matrices, Cauchy-Schwartz inequality (baby version)**Lecture 5**: postulates of special relativity, derivation of Lorentz group, Minkowski space**Lecture 6**: topological spaces, continuous maps, Sierpinski space, topology associated to a metric, abstract continuity matches with epsilon-delta continuity**Lecture 7**: bases, product spaces, quotient spaces, disjoint unions, homeomorphisms, pushouts, torus, Mobius strip, graphs as topological spaces, finite CW complexes,*projective space***Lecture 8**: closed and bounded sets, Bolzano-Weierstrass theorem, sequences and convergence in metric spaces, continuity as preservation of limits, sequential compactness in metric spaces**Lecture 9**: cover compactness, sequential compactness equals compactness for metric spaces, extreme value theorem for compact spaces, Cauchy sequences in metric spaces**Lecture 10**: preservation of compactness under quotients and finite products, Heine-Borel theorem, finite CW complexes are compact**Lecture 11**: metrisable spaces are Hausdorff, products of Hausdorff spaces are Hausdorff, real line with double point, compact subspaces of Hausdorff spaces are closed, continuous bijections from compact to Hausdorff spaces are homeomorphisms, finite CW complexes are Hausdorff,*normal and regular spaces***Lecture 12**: function space as configuration space, compact-open topology, evaluation and composition are continuous,*path space*,*loop space*,*space of graphs***Lecture 13**: for X compact and Y metrisable the compact-open topology on Cts(X,Y) is metrisable, closure, interior, density, pointwise and uniform convergence, continuity of uniform limits of continuous functions, complete metric spaces, completeness of Cts(X,Y) for X compact and Y a complete metric space.**Lecture 14**: Banach fixed point theorem, contraction mappings, Bellman equation in reinforcement learning.**Lecture 15**: Picard’s theorem on existence and uniqueness of solutions to ordinary differential equations**Lecture 16**: Weierstrass approximation theorem, Stone-Weierstrass theorem, algebras of functions, topological algebras, Bernstein polynomials,*trigonometric polynomials are dense*,*separable spaces*,*one-point compactification*.**Lecture 17**: integrals as positive linear functionals (called here*integral pairs*), topological vector spaces, products, disjoint unions and quotients of integral pairs, Fubini’s theorem via Stone-Weierstrass.**Lecture 18**: normed spaces, L^p norm, Holder inequality, non-completeness of R-valued functions with the L^p norm, Banach spaces, completion of a metric space, completion of a normed space to a Banach space, definition of the L^p Banach space.**Lecture 19**: finite-dimensionality of vector spaces characterised by the isomorphism with the double dual, continuous linear duals of topological vector spaces and normed spaces, bounded operators between normed spaces, bounded and linear equals continuous and linear, duality for L^p-spaces (statement only) and the self-duality of L^2-spaces.**Lecture 20**: inner product spaces, Cauchy-Schwartz inequality, inner product spaces are normed spaces, Hilbert spaces, conjugate vector spaces, orthogonal complement of a subset, decomposition of Hilbert space by a closed vector subspace, Riesz representation theorem, L^2 spaces are Hilbert spaces, self-duality of L^2-spaces,*constructing vectors in L^2-spaces from integrable functions*.**Lecture 21**: complex exponentials give an orthonormal (dense) basis for L^2 of the circle, Bessel’s inequality, equivalent conditions for an orthonormal system to be a dense basis, complete description of the function space of the circle via Fourier coefficients, and*more generally for any integral pair embedded in Euclidean space*.

And in the tutorials:

**Tutorial 3**: Lipschitz equivalence**Tutorial 4**: Topological abelian groups**Tutorial 5**: Real numbers as Cauchy sequences**Tutorial 6**: Cyclic groups, Q/Z and R/Z, direct limits**Tutorial 7**: Dual of a linear transformation, adjoints**Tutorial 8**: Riemann integrals, compactly supported functions

The later tutorials were primarily extracts from lectures.