Progress plan

Chapter 1¨C2 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¨CWeierstrass 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.

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 ¨C 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.

We saw two examples of Banach's Fixed Point Theorem (of which the second was a bit of a fiasco). We then introduced subsequences and the concept of compactness. We saw that all compact sets are closed and bounded; and that in ?, all closed and compact sets are compact. We then saw an example of a set which is closed and bounded but not compact: A subset of C([-1,1],?).

On Tuesday we developed the above example further: We saw an argument for the fact that in infinite-dimensional vector spaces, a ball is never compact. We moved on to some consequences of compactness: Continuous functions map compact sets to compact sets; and continuous, real-valued functions on a compact set attain a maximum and a minimum. We then started on Section 4.1, and defined continuity, uniform continuity, and Lipschitz continuity.

We finished Sec. 4.1 by showing that all continuous functions functions on a compact set are automatically uniformly continuous.

We then moved on to Sec. 4.2, introducing pointwise and uniform convergence of sequences of functions, and seeing that uniform convergence is the same as convergence in the supremum metric. We saw several examples, and saw two useful results: Uniform convergence implies pointwise convergence; and uniform convergence of [uniformly] continuous functions implies that the limit is [uniformly] continuous.

On Tuesday we studied the set B(X,Y) of bounded functions between two metric spaces X and Y, and the subset C_b(X,Y) of bounded, continuous functions. The supremum metric ? is a natural metric to put on these sets. We saw in particular that if Y is complete, then both of these metric spaces are complete. (This is Sec. 4.5¨C4.6 in Spaces.)

We covered Sec. 4.3, investigating conditions that ensure that we can interchange limits and integration, and limits and differentiation: \( \lim_{n\to\infty} \int_a^x f_n(y)\,dy = \int_a^x \lim_{n\to\infty} f_n(y)\,dy \) ?and ?\( \lim_{n\to\infty} \frac{d f_n}{dx}(x) = \frac{d}{dx}\lim_{n\to\infty} f_n(x). \) ?We then started on Sec. 4.4, on the radius of convergence of a power series.

On Tuesday we finished Sec. 4.4 (the section on Abel's theorem on power series is not part of the syllabus). We saw that within the radius of convergence, the power series converges pointwise (and uniformly in any strictly smaller subset), that the series is infinitely differentiable, and that the series is actually equal to its own Taylor series. We then moved on to Sec. 4.7: We saw several examples of ODEs, we talked about well-posedness (existence, uniqueness and stability), and we saw examples of ODEs that are not well-posed.