Due date: Monday 2/24/14
Last updated Feb 19 23:57 EDT 2014
You are required to do all of the problems below. You will not be required
to hand them all in. I’ve indicated below which ones you do have to hand in.
Don’t make the mistake of thinking that I’m collecting only the problems I
think are important. The “due date” above is the date that your written-up
problems should be handed in, but don’t wait to get started on the assignment.
You should always get started on problems as soon as we cover the relevant
material in class.
- A. Rosenlicht Chap. 6, exercises 1–4, 8, 9. Of the Rosenlicht problems,
hand in only 2, 3, 6 (see below), 8, 9. You are permitted to use the result
of B3 (which is also a hand-in problem) in #9, OR to use the result of #9 in B3,
BUT NOT BOTH (i.e. circular reasoning is not allowed). 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. - A (continued) Rosenlicht Chap. 4 (yes, Chap. 4), exercises 22-23.
These are assigned because exercise 6 of Chap. 6, assigned below,
relies on Chap. 4 #23, which in turn relies on Chap. 4 #22. The facts
stated in Chap. 4 #23 are very important; no math major should
graduate without having learned them.Note: (1) You may find that in the process of proving 22(c), you have
essentially already done #23. (2) Two norms on a vector space are
called equivalent if there exist positive numbers m,M such that the
criterion in line 4 of Chap. 4 #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.) - A (continued) Rosenlicht Chap. 6/ 6. Rosenlicht Chap. 6, exercise 6.
Hand this in (all five parts). Interpret part (a) as meaning: “Give the
definition that generalizes the integral (and integrability) from real-valued
functions to V-valued functions.” In part (d), do not forget to show that the
integral on the right-hand side exists. Also, answer this: if “continuous” in
part (d) were replaced by “integrable”, where would your proof for the
continuous case run into trouble? In part (e), assume the result of problem
23 of Chapter IV. (That problem is still part of your homework; it’s just not
one of the hand-in problems.)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 to 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 codomain V is
finite- or infinite-dimensional. - B. Click here for non-book problems.Of these, hand in only B3.
Q&A. Below is a question and answer from a previous time that I taught
this class. 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 Rosenlicht Chapter 6, 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. 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.)