Videos

How to work through this material

Each video corresponds (roughly) to a section in Chapter 3 in Spaces. You should:

  • Watch the video (or just skim it, if you think you already know it)
  • Read (or skim) the corresponding section in Spaces
  • Do as many of the accompanying exercises as you can.

If you're stuck, don't worry. Rewatch the video, reread the section in Spaces, reread earlier sections, discuss with a fellow student, ask a question in the padlet, or simply skip the exercise.

When you are certain that you have completed an exercise (or you have given up completely), you can check the solution (not all exercises have worked solutions, unfortunately).

The more exercises you do, the better! If you should run out of exercises, don't be afraid of doing exercises which aren't listed here.

Videos

Section Video PDF
25–29 January
3.1 Metric spaces PDF
3.2 Convergence PDF
3.2 Continuity PDF
3.3 Open and closed sets PDF
1–5 February
3.3 Alternative definitions of continuity PDF
3.4 Completeness PDF
3.4 Banach's fixed point theorem PDF
3.5 Compactness PDF
8–12 February
3.5 Compactness II PDF
  Compactness III PDF
3.6 Compactness IV PDF
3.7 Completion of metric spaces (this one is optional!) PDF
15–19 February
4.1 Modes of continuity PDF
4.2 Modes of convergence PDF
22–26 February
4.3 Integrating sequences of functions PDF
  Differentiating sequences of functions PDF
2.2 Lim inf and lim sup PDF
4.4 Power series PDF
1–5 March
4.5 The space of bounded functions PDF
4.6 The space of bounded, continuous functions PDF
4.7 Ordinary differential equations I PDF
  Ordinary differential equations II PDF
  Ordinary differential equations III PDF
(Note on multiindices) Multiindices I PDF
(Note on multiindices) Multiindices II PDF
8–12 March
4.8 The Arzela–Ascoli theorem PDF
4.9 Convergence of numerical approximations of ODEs PDF
15–19 March
4.10 The Weierstrass approximation theorem I PDF
  The Weierstrass approximation theorem II PDF
5.1 Normed vector spaces I – preliminaries PDF
  Normed vector spaces II – equivalent norms PDF
5.2 Series and bases PDF
5–9 April
5.3 Inner product spaces PDF
5.4 Linear operators I PDF
  Linear operators II – boundedness PDF
  Linear operators III – the space \(\mathcal{L}(V,W)\). PDF
5.5 Invertible linear operators I PDF
  Invertible linear operators II – Neumann series PDF
12–16 April
10.1 Fourier series – Introduction and motivation PDF
10.2 Fourier convergence I PDF
  Fourier convergence II PDF
19–23 April
10.3 The Dirichlet kernel PDF
10.4 Cesàro convergence of Fourier series PDF
26–30 April
6.1 What is a derivative? PDF
  The Fréchet derivative I – Definition PDF
  The Fréchet derivative II – Calculus PDF
6.2 The Gateaux derivative PDF
6.3 The mean value theorem PDF
3–7 May
6.7 The inverse function theorem PDF
6.6 Partial derivatives PDF
6.8 The implicit function theorem I PDF
6.8 The implicit function theorem II PDF

 

Published Jan. 22, 2021 11:43 AM - Last modified Feb. 24, 2023 1:26 PM