Topics in Analysis (Cambridge Part II 2016-17 Course)



Frankie Higgs


This course covers, as the name might suggest, some topics in analysis. It assumes some basic familiarity with analysis and a little experience with metric spaces.

Some of the eponymous topics are:
1. Metric spaces, definitions of Cauchy sequences, completeness, compactness, etc.
2. Compact sets in Euclidean space, proof of fundamental theorem of algebra
3. Laplace's equation, uniqueness and existence of a solution on the disc
4. Brouwer's fixed point theorem: proof in the plane, via no-retraction theorem and Sperner's lemma
5. Non-cooperative games and Nash equilibrium points
6. Negotiation games
7. Approximation by polynomials (on an interval, then on certain sets in complex plane)
8. More compact set stuff
9. Irrational and transcendental numbers, Liouville's theorem
10. Continued fractions (not much of this section made it into these cards, since it was mostly in the number theory ones already)
11. Winding numbers

These cards contain definitions, statements and proofs of theorems from the course.

I wrote these only to aid with remembering facts for the exam - I've provided them here in the hope that they'll be helpful to someone taking an equivalent course in the same position.
They may be of interest to someone who isn't taking a similar course, but since I've not included any of the explanation, motivation or diagrams that accompanied the theorems, you'll be better off learning the material first from a textbook.
I don't think I've ever read an analysis textbook, so I can't recommend you one. Maybe Prof. Körner's "A Companion to Analysis". If anyone can recommend one they've found good for a third-year (or equivalent) undergraduate, mention it in the comments.

This may go without saying, but a fellow student has had a little trouble along these lines before: if you are taking a course in this subject, these notes are not a substitute for attending lectures.

A note on the cards: I found the LaTeX renderer used by Mnemosyne to be ugly and difficult to read, so the cards are in fact PNG images rendered from LaTeX. This does mean they are a pain to edit, and may not fit nicely on mobile screens. If anyone would like the source files to render themselves, email [email protected] and I will provide them.


Based on my lecture notes from the Cambridge University part II course "Coding and Cryptography", lectured in Lent term 2017 by Prof. Tom Körner

There are notes on the lecturer's homepage:

Mnemosyne Compatibility: 

  • 2.1+

Card Set File: