Part II Analysis of Functions

I have taught the Part II Analysis of Functions course from 19-21. Below are fairly comprehensive notes. For problem sheets, please see the DPMMS webpage, or the course Moodle.
Ch 1: Lebesgue integration theory
Ch 2: Banach and Hilbert Space Analysis
Ch 3: Test Functions and Distributions
Ch 4: The Fourier Transform and Sobolev Spaces
Appendix 1: Functional Analysis
Appendix 2: Measure Theory
Complete notes