The final syllabus is as follows:
- chapter 1, except 1.5.
- chapter 2.
- chapter 3, except 3.7.
- chapter 4, except 4.8, 4.9, 4.11.
- chapter 5, except 5.6, 5.7.
- chapter 6: only 6.1, 6.2, 6.3, 6.6, 6.7, 6.8.
- chapter 10: except 10.7.
In section 4.10 only "Proof 2" is pensum, based on the notion of "Good kernel/Dirac sequence". This notion, which appears also in Fourier analysis, is fundamental. It is covered in the lectures, week 9. Alternatively see Lang "Undergraduate analysis" chapter XI, sections 1 and 2.
Fourier analysis is covered only in the complex-valued case.
Important norms to know are the sup-norm on continuous functions (eg as it appears in the Weierstrass M-test) and the operator norm on continuous operators (eg as it appears in the Neumann series). In Fourier analysis the L^2 norm is also important.
Basic work horses are the Banach fixed point theorem and the inverse function theorem. The Neumann series and the study of perturbations of the identity by a contraction occupy an intermediate position. See week 13.
The theory of Riemann integration is assumed known for functions from an interval to R^n, and applied only to piecewise continuous functions.
The inverse function theorem is more important than the implicit function theorem. For the latter only Corollary 6.8.3 is pensum.