Note to new students (and reminder to continuing students): On every homework assignment, you are expected to work on ALL the problems listed. In order for you to have time to write up the assignment, and for me to grade it, I require you to turn in only a certain subset of the assigned problems. Nonetheless I expect you to seriously attempt ALL the problems, including the ones I don't ask you to hand in, and will hold you responsible for the material in them.
- Go to the miscellaneous handouts page. Read the proof-writing handouts and take the proof-writing quiz. Due date: Fri. 1/8/09.
- (Added 1/16/09.) Do new non-book problem A1. Due date: Wed. 1/21/09; don't hand this one in. This problem should make Rosenlicht's Chap. 6 #1 easier, if you haven't been able to do it yet.
- Rosenlicht Chap. 6, exercises 1-4. Due date: Mon. 1/12/09. You may use either the definition of integrability, or the lemma stated at the end of proved in class on Monday involving Lδ(f), Uδ(f). Hand in #2 and #3 Wed. 1/21/09. Before you start working on #2, read the Q&A near the bottom of this page.
It goes without saying that in a yes/no question such as #3, you are expected to prove your answer. You will probably find #3 rather challenging. A proof of the correct answer can be written in half a page, once you figure out the key idea, but finding the key idea is not so easy. I doubt you'll be able to do #3 until you've figured out a way to do #2, which itself is not so easy. Neither of these is a "just turn the crank" problem.
- Rosenlicht Chap. 6, exercise 8. Due date: Wed. 1/14/09.
- Do non-book problem A2 (called A1 until 1/16/09, when new A1 was added). Due date: Wed. 1/21/09. Hand this in. You may use the Proposition on p. 120, which we'll complete the proof of on Wed. 1/21.
- Rosenlicht Chap. 6, exercise 9. Due date: Wed. 1/21/09. Hand this in. You are permitted to use the result of non-book problem A1 to do this problem (whether or not you succeeded in proving that result), but not the other way around.
- Rosenlicht Chap. 4 (yes, Chap. 4), exercises 22-23. These are assigned because VI/#6, assigned below, relies on IV/#23, and IV/#23 relies on IV/#22. The facts stated in IV/#23 are very important; no math major should graduate without having learned them.
Note: Two norms on a vector space are called equivalent if there exist positive numbers m,M such that the criterion in line 4 of IV/#23 is satisfied. The first conclusion in #23 is often stated as "All norms on Rn are equivalent." Why is this statement equivalent to Rosenlicht's statement? (Sorry, two different uses of the word "equivalent" here.)
- Rosenlicht Chap. 6, exercise 6. Due date: Wed. 1/21/09. Note: the characterization of integrability in terms of upper and lower sums does not extend well to general vector-valued integration (integration of V-valued functions, where V is a complete normed vector space), although with some effort it can be extended finite-dimensional vector-valued integration. This exercise is one instance in which Rosenlicht's definition of integrability has a definite advantage: it generalizes easily to the vector-valued case, regardless of whether the target space V is finite- or infinite-dimensional.
Q&A. A student asked the question below about a particular exercise. A similar question can occur in many other problems. The answer is always the same (except, of course, for specific reference to the particular exercise).Question. "For exercise 2 in chapter 6 of the text may we assume that the integral exists and use that assumption to show that it is equal to 0, or do we have to prove existence first and then prove that it equals 0?"
Answer. You absolutely may not assume that the integral exists. Any time you're asked to prove "this = that", and "this" does not automatically exist, it's implicit that what you're being asked to prove is "this exists AND this=that". Another example of this sort of thing is "Prove that a certain limit = 1". It should be obvious that if you were asked this, you would not be allowed to assume from the start that the limit exists. If the writer of the exercise wanted you to make such an assumption, he/she would say so explicitly.
However, often in these cases, you end up exhibiting the value of "this" (or the limit, in my other example) at the same time that you prove existence. For example, you usually prove that a limit exists by intelligently guessing the value of the limit, then showing that your function, sequence, or whatever, approaches that value. (Note: Very often "intelligent guesswork" does involve a step of the type "Hmm, suppose the limit (or whatever) existed. What would it have to be?" That's perfectly good thinking, and it's by no means against the rules of proof, because you are not making the assumption in your proof.)