## Basic Functional Analysis

You will need this for the following Part III courses:

• Functional Analysis
• Analysis of PDEs
• Ergodic Theory
• Elliptic PDEs
• Inverse Problems
• (Topics in Mathematics of Information)

Reality check

• Familiarity with vector spaces. See e.g. Ib Linear Algebra. You can check if you are at the required level by going through the Linear Algebra example sheets (available on the page linked to here).
• Understanding of basic topological spaces (concepts such as separability, completeness and compactness). See e.g. Ib Metric and Topological Spaces. You can check if you are at the required level by going through the given example sheets.
• Undergraduate Analysis, see e.g. Ia Analysis I and Ib Analysis II. You can check if you are at the required level by looking through the given example sheets.

Prerequisites

You will need knowledge of the topics listed in the Part II Analysis of Functions course above. You will also need some results from the Linear Analysis course mentioned above; in particular, you need to be able to work with the spaces l2, lp and C(K). For example, you should know:

• Basic definitions in Banach spaces and duals.
• Results about separable Hilbert spaces, for example the Riesz-Representation theorem, existence of projections, the spectral theorem for compact normal operators.
• Results about the lp spaces, for example the dual spaces of lp (sequence variants)
• Results about C(K) spaces, for example Urysohn's Lemma, Tietze Extension theorem and the Arzela Ascoli theorem. (particularly helpful for PDE theory)
• Baire Category Theorem.

You can use the following exercises to check your present level and to guide your study. All these are from example sheets for the Part II course Linear Analysis from 2015/16, which you can find on the lecturer's homepage.

Useful books and resources

• The book Analysis by Lieb and Loss is an excellent (and celebrated) account of the core analysis of undergraduate / advanced undergraduate level.
• Notes from 2015 catch-up workshop which give a good summary, provided by James Kilbane (thank you!).
• Resumè on Hilbert Spaces and Spectral Theory, meant for catch-up reading, on András Zsák's teaching page.
• The book Linear Analysis by B. Bollobas covers everything in terms of functional analysis needed here.

## Measure Theory

If you are coming from a degree which was not solely mathematics, make sure you look up and get familiar with basic common mathematical notation, for example for set inclusions, intersections and unions, etc.

You will need this for the following Part III courses:

• Analysis of Partial Differential Equations
• Functional Analysis
• Ergodic Theory
• Elliptic PDEs
• Stochastic Calculus
• Percolation and Related Topics

Reality check

• Reasonably firm understanding of undergraduate analysis. See e.g. Ia Analysis I, Ib Metric and Topological Spaces, some concepts from Ib Analysis II;
In particular concepts like:
• topological spaces, continuity;
• convergence;
• norms.

You can check if you are at the required level by going through the example sheets for those courses.

• Basic linear algebra. See e.g. Ib Linear Algebra; You can check if you are at the required level by going through the example sheets for this course.
• (Less important but still useful) Banach spaces and Hilbert spaces. See e.g. Part II Linear Analysis. You can check if you are at the required level by going through the example sheets for this course and finding the questions concerning Banach and Hilbert spaces.

Prerequisites

Results in measure theory, in particular: definition of a measure space; Borel σ-algebras; π-systems and d-systems; an intuitive idea of Lebesgue measure; the measure theoretic formulation of probability; measurable functions; random variables; convergence of measurable functions; some introduction to Lebesgue integration; convergence of integrals; product measure and Fubini's Theorem; Lebesgue spaces and some useful inequalities relating to measure theory or Lebesgue spaces. You can see these topics in the workshop notes given below.

You can use the following exercises to check your present level and to guide your study.

• Measure Theory exercises and solutions, provided by Jo Evans (thank you!). It goes without saying that you learn an awful lot more by trying the exercises really seriously yourself and not just looking up solutions. It is your own preparation so it's your responsibility.

Useful books and resources

• The book Analysis by Lieb and Loss is an excellent (and celebrated) account of the core analysis of undergraduate / advanced undergraduate level.
• Notes from previous catch-up workshop on Measure Theory, provided by Jo Evans (thank you!).
• Notes on basic measure theory, meant for catch-up reading, on András Zsàk's teaching page.
• Lecture notes for the Part II Probability and Measure course by James Norris.
• D. Williams, Probability with martingales, CUP 1991.