Final Syllabus

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.

 

Publisert 22. mai 2023 14:22 - Sist endret 22. mai 2023 15:48