Section B.10 Problem Sheet 10
Due 8 December at 3pm, either in class or in the pigeon hole in 4W.
Exercise 4.5.2 is the star of the show this week. It is both an example of a typical application of Theorem 4.29 and also an extremely useful result in its own right. It is also not particularly difficult.
Exercise 4.5.3 connects Section 4.5 with Exercise 4.3.5 from Problem Sheet 8, and gives you more practice with equicontinuity. One could ask a similar question about the functions in, for instance, Exercise 3.4.3. It is not particularly difficult.
Exercise 4.5.4 gets you applying Theorem 4.29 (or more specifically Corollary 4.31) in a somewhat more concrete setting. It is similar to part of a question on the 2020 exam.
Finally, Exercise 5.1.1 gives you some practice applying Theorem 5.1. One consequence of the exercise is a stronger version of this theorem, saying that that polynomial \(p\) in can always be chosen to have rational coefficients. Another consequence of the exercise is that \(C^0([a,b])\) is separable, a fact which may be important in some of the Year 4 Analysis units. The first and last parts of the exercise require some thought, but there are very big hints in case you are stuck and/or pressed for time.
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.
