Of the problems below, hand in only #s 10, 11, 16, and 20.
- Rosenlicht Chap. 6, exercise 10. Due date: Fri. 1/23/09. Hand in.
- Rosenlicht Chap. 6, exercises 7, 11-12, 15, 16 (see Q&A below). Due date: Mon. 1/26/09. Hand in only #s 11 and 16.
Additionally, in problem 12, the quantity (∫ab f) /(b-a ) is called the average value of f on (or over) the interval [a,b]. Give an explanation of why this terminology makes sense, and for positive f give a geometric interpretation in terms of areas.
- Rosenlicht Chap. 6, exercises 20. Due date: Wed. 1/28/09. Hand in. Note: this problem is not easy. If you have what you think is a quick proof, you are probably overlooking something, making an implicit assumption, etc.
- Rosenlicht Chap. 6, exercises 17, 19. Due date: Wed. 2/4/09.
Q&A (taken from students' questions).Question. "In problem #16 of Chapter VI is there a specific metric we should be using on C([a,b])?"
Answer. Yes, the "uniform metric" (see next paragraph). Whenever Rosenlicht writes "C(E)", where E is a compact metric space (e.g. [a,b] with the standard metric), he means not just the set of real-valued continuous functions on E, but this set as a normed vector space (with the uniform norm), together with the metric that comes from this norm. Unfortunately, Rosenlicht doesn't mention this convention very prominently; it's in the very last paragraph of Chapter IV, without any marking like "Definition" to call your attention to it.
In case you don't already know: the uniform metric is the one defined in that paragraph of Rosenlicht. The uniform norm of f is what you get if you erase "-g(p)" from the formula defining the metric.
In class, when I write something like "C([a,b])", you should be able to tell from context whether I'm referring to this object as a set, as a vector space, or as a metric space. E.g. when I wrote "C([a,b]) ⊂ ℑ([a,b])", I said that the former is a vector subspace of the latter; nothing about metric structure was implied. (We haven't even tried to define a metric on ℑ([a,b]).)
Thanks go to Kyle Huey for getting me the HTML code for the subset-symbol. Thanks go to Frances Tirado and Douglas Whitaker for finding me the code for "ℑ", which seems to be the closest thing in HTML for the "curly I" I've been writing for the space of integrable functions.