Part II Analysis of Functions, Lent 2026

This page contains information for students taking my lectures on Analyis of Functions in Lent 2025. The best way to contact me outside lectures is by email: pv270@dpmms.cam.ac.uk. (I do not monitor Moodle.)

Lecture Notes. (Last updated: 17 March 2026.) The notes will be updated in the course of the term. Please check back, and see the changelog below.

Example sheets:

• Example sheet 1. (4 February 2026.)
• Example sheet 2. (26 February 2026.)
• Example sheet 3. (5 March 2026.)

Lecture recordings are available via Panopto. However, I expect that most students will find the notes more useful.

Changelogs

Notes

• Version of 22 January 2026: Contains most of the material related to measure theory.
• Version of 26 January 2026: Added the Riesz representation theorem.
• Version of 4 February 2026: Added section on L^p spaces.
• Version of 5 February 2026: Added section on convolution.
• Version of 10 February 2026: Added section on Hahn-Banach.
• Version of 15 February 2026: Added section on weak and weak-* topologies.
• Version of 19 February 2026: Added section on Fourier transform.
• Version of 26 February 2026: Added section on Fourier coefficients and the Poisson summation formula.
• Version of 1 March 2026: Added some material on distributions.
• Version of 3 March 2026: Added more material on distributions.
• Version of 5 March 2026: Added tempered and periodic distributions.
• Version of 10 March 2026: Added a few pages on Sobolev spaces.
• Version of 12 March 2026: Completed section on Sobolev spaces.
• Version of 17 March 2026: Notes complete. Minor and major corrections are coming soon.

Example Sheet 1

• Version of 4 February 2026: Typos corrected in questions 2, 4, 8 and 10. Hint added to question 6.

Example Sheet 2

• Version of 22 February 2026: Correction to Q5: If \(p=1\), the functions in the sequence must be supported in a fixed bounded set. Corection to Q7: the sequence of elements in the Hilbert space needs to be bounded.
• Version of 26 February 2026: Correction to Q11: Also in part (a), the measure is probability.