This is an overview of what I plan to do in future lectures, and what I have done in past lectures.
| Date | Syllabus | Contents |
|---|---|---|
| 19/1–25/1 | 1–2 | Chapter 1–2 is mostly repetition from things you already known. I will assume that you know much of Ch. 1 already; we will return to families of sets later. We spent the lectures by investigating the differences between ? and ?. We saw that ? is countable and ? is not (Sec. 1.6). But more importantly, we saw that ? is complete, which ? is not (Sec. 2.2). We also visited the Bolzano–Weierstrass theorem, which we will revisit later.? Finally, I gave a rough sketch of the construction of ? by the technique of Dedekind cuts. This is not part of the curriculum. |
| 26/1–1/2 | 3.1–3.2 | We defined metric spaces and saw several examples. In particular, every norm on a vector space gives rise to a metric in a natural way. We then defined convergence of sequences in metric spaces, and again looked at examples. We defined continuity of functions between metric spaces, looked at examples, and finally, proved that continuity is the same as ???????sequential continuity: the function of any convergent sequence is also convergent.? |
| 2/2–8/2 | 3.3–3.4 | In order to define what open, closed and boundary means for a set E, we introduced the concepts of internal points, external points and boundary points of E. We saw that the open/closed balls are open/closed; that the set of internal points is open; that the closure is closed; and so on. We also found equivalent definitions of being open or closed, in terms of the complement of the set, or in terms of converging sequences. On Tuesday we saw what a Cauchy sequence is, and defined completeness of a metric space – Cauchy sequences are automatically convergent. We saw several examples of (in)completeness. We then proved one of the main theorems of the course, Banach's Fixed Point Theorem, which exploits completeness. |
| 9/2–15/2 | 3.5, 4.1–4.2 | We will see an example of Banach's Fixed Point Theorem; introduce compactness; and start on Chapter 4, which is on the metric space consisting of continuous functions. |
| 16/2–22/2 | ? | ? |
| 23/2–1/3 | ? | ? |
| 2/3–8/3 | ? | ? |
| 9/3–15/3 | 5 | ? |
| 16/3–22/2 | ? | ? |
| 23/3–29/3 | 6 | ? |
| 30/3–5/4 | ? | Easter holiday! |
| 6/4–12/4 | 7 | No lecture on Monday. |
| 13/4–19/4 | ? | ? |
| 20/4–26/4 | ? | ? |
| 27/4–3/5 | 8 | ? |
| 4/5–10/5 | ? | ? |
| 11/5–17/5 | ? | Repetition/exercises |
| 18/5–24/5 | ? | Repetition/exercises |