Section B.7 Problem Sheet 7
Due 15 November at 3pm, either in class or in the pigeon hole in 4W.
This is not a particularly easy problem sheet! I’ve listed the questions non-chronologically so that the easiest two are first.
Exercise 3.2.6 is hopefully relatively straightforward when combined with the hint, and gives you some practice working with these Hölder norms. Exercise 3.4.2 which gives you some practice calculating \(L^2\) norms, and asks you to reinterpret the Part c of Exercise 3.3.1 in the way it is used in the proof of Lemma 3.11.
Exercise 3.2.2 gives you some classic examples of functions which lie in some of these spaces but not in others. Even treating this relatively simple example involves some fancy footwork estimating various suprema from above and below; some of the hints are relatively big. Exercise 3.2.3 proves Theorem 3.6, which we did not talk about in detail during lecture. As with Exercise 3.2.2, one must think relatively carefully about estimates, this time using some basic theorems from calculus. You’ll also need to think carefully about what it means to be a metric (or normed) subspace.
We are skipping over Exercise 3.4.1, but if you are feeling rusty with the material in Section 3.3, it may be worth attempting as a warm up. Since we talked about the nonseparability of \(B(\R)\) in the problems class (see Example 7), the unassigned exercise Exercise 3.2.4 is probably not quite as difficult as it would otherwise be.
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.
