Section B.2 Problem Sheet 2
Due 11 October at 3pm, either in class or in the pigeon hole in 4W.
Exercise 1.1.5 and Exercise 1.1.9 are probably more difficult than Exercise 1.1.6 and Exercise 1.1.8. If you are looking for easier warm up problems I would recommend the unassigned Exercise 1.1.2 and Exercise 1.1.4.
Exercise 1.1.5 introduces you to some alternative norms on the familiar vector space \(\R^2\text{.}\) These are often used instead of the more familiar Euclidean norm, and of course give rise to different normed spaces. All parts of the exercise involve a bit of creativity in manipulating inequalities. The inequalities in the last part are particularly useful, and will come up several times in the lectures and problem sheets.
Exercise 1.1.6 and Exercise 1.1.8 get you working with the supremum norm from Example 1.7. Working with this norm, and with suprema more generally, is an important skill for this unit. Sometimes we can calculate suprema exactly (see Exercise 1.1.7), but often we can’t, and in any case estimating them is often ‘good enough’ and much, much faster.
Exercise 1.1.9 gives you practice showing that something is a metric space, and will be used throughout the unit. Getting the inequalities to work out is a bit tricky, but take a look at the hints. Writing out things like \(d_{X \times Y}((x_1,y_1),(x_2,y_2))\) is a pain, but there is no real shortcut — in previous years most students who tried to abbreviate things eventually got confused.
There are a lot of other good exercises in Section 1.1. Especially since this is supposed to mostly be review from Analysis 2A, I would encourage you to do any additional exercises that seem like they might be useful.
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.
