MA30252: Advanced Real Analysis

Due 22 November at 3pm, either in class or in the pigeon hole in 4W.

We spent a lot of time in Chapter 3 proving that things were Banach spaces. Exercise 3.2.5 gives you some experience running these sorts of arguments yourself. One option is to recapitulate the proofs from class with the necessary modifications. Another, faster, option is to use those results to do as much of the work for you as possible. Last year students found this exercise significantly more difficult than I had anticipated, and so I have added an extra hint/warning.

Exercise 3.4.3 gives you some practice taking a limit in \(L^2([a,b])\text{,}\) and makes the important point that this is in general quite different from taking a pointwise limit. It should be relatively straightforward.

It is not unreasonable to view Exercise 4.1.1 as a primary motivation for Section 4.1, and by extension all of Chapter 4. This is one of the basic existence theorems in analysis, and has a huge number of applications. Working through the proof yourself should hopefully give you a bit more of a feeling for how sequential compactness can be useful.

By combining Theorem 4.22 and Proposition 4.12, we know that all subsets of \(\R\) or \(\R^n\) which are closed and bounded are sequentially compact. This is not true in general metric spaces, and Exercise 4.3.5 gives a concrete example. It can be quite useful to have such an example in mind when trying to organise the various results and definitions in this chapter. The exercise is also good practice at showing the negations of many of these definitions.

Please feel free to email me, or drop by office hours (Tuesdays 1:15–2:05 in 4W 1.12), with any questions about these problems whatsoever. I am also more than happy to meet one-on-one or in a small group.