Section B.6 Problem Sheet 6
Due 8 November at 3pm, either in class or in the pigeon hole in 4W.
Exercise 2.2.1 and Exercise 2.4.2, like Exercise 2.2.2 on the previous problem sheet, were need in the proofs of Theorem 2.10 and Theorem 2.12, and get you working some more with Cauchy sequences. They should both be relatively easy, which is why they don’t have hints and also why we have five exercises this week instead of four.
Exercise 2.1.4 gets you thinking about what it means to be separable at least once. The first part asks you to show that a subset is not dense, which is a useful thing to be able to do.
Exercise 3.1.1 is a straightforward computational exercise, which follows up on Exercise 1.1.8 from Problem Sheet 2. To save you a bit of work in the first part, I’m releasing solutions to the unassigned Exercise 1.1.7. Being able to calculate and/or estimate \(C^k\) norms is a useful skill in this unit, and also gets you working with these spaces in a very concrete way. For something a bit more theoretical, take a look at Exercise 3.1.2.
We will need Exercise 3.3.1 in Section 3.4, and it’s also useful practice in estimating integrals. In principle you know quite a lot about integrals and how to estimate them from earlier analysis units, but in case you’re rusty the basic facts that we will need are collected in Section 3.3.
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.
