Section B.4 Problem Sheet 4
Due 25 October at 3pm, either in class or in the pigeon hole in 4W.
Exercise 1.3.3 gets you working with Cauchy sequences, and is used in the proof of Theorem 1.40. Exercise 1.3.4 gives you even more practice working with Cauchy sequences, and will be needed in Chapter 2.
Exercise 1.4.1 is some practice showing that various maps are continuous, something we’ll be doing often in the unit. The solution to the first part of Exercise 1.3.1 is helpful.
Exercise 1.4.2 is one of many ways that our definition of continuity ‘plays well’ with the definitions in Section 1.2. Indeed, the unassigned Exercise 1.4.3 shows that we could alternatively have defined continuity solely in terms of open sets (as they do in the Topology unit!).
I have decide not to assign Exercise 1.5.1 this year, but, since we needed it in the proof of Theorem 1.50, I have released solutions instead.
Exercise 1.6.1 gives you some practice – perhaps more than you really need – with these proliferating brackets which appear when talking about mappings between spaces of functions. Once you’ve reminded yourself of the notation for these things, you might try Example 1.61, which is closer to how we will work with such objects in this unit.
For a question on isometries, you could try Exercise 1.7.1, which is a follow-up to Exercise 1.1.5 on the previous problem sheet.
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.
