Last updated Wed Jan 15 19:41 EST 2020
Due-date: Wednesday, 1/22/20 (first class after MLK Day).
You are required to do all of the problems below. You will not be required to hand them all in. I will announce later which ones you do have to hand in on the due-date. 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 Chapter IV/ 17, 18, 21, 32, 34–37. (In yes/no questions like 34, 36, and the last part of 37, you are expected to prove your answers.) Of these, hand in only the following: 17, 21, 32, 34ac (i.e. the first and third sequences in the problem). If you are unable to do #21 completely, then for partial credit you may do a simplified version of the problem, in which you assume (additionally) that the complement of \(S\) in \(E\) consists of just a single point.
Some notes about these problems:
- Reminder: for sequences of functions, Rosenlicht's "converges", absent the modifier "uniformly", is my "converges pointwise" (see p. 83), and similarly his "limit function" is my "pointwise limit function".
- If you do not find #21 harder than the other proof-problems in the list above, you are probably doing something wrong. Here are some hints:
- Show that, even if the uniform-continuity hypothesis in the problem is weakened to "just plain" continuity, there exists at most one continuous function \(\tilde{f}:S\to E'\) that extends \(f\). Thought of another way: the conclusion of problem 21 involves both an existence statement and a uniqueness statement. The hypothesis that \(f\) is uniformly continuous is really needed only for the existence of a continuous extension, not uniqueness. The remaining hints are aimed at the existence question.
- Given metric spaces \( X,Y \), a function \(f:X\to Y\), and a sequence \( (x_n)_{n=1}^\infty\) in \(X\), we can form the sequence \( (f(x_n))_{n=1}^\infty\) in \(Y\). If \((x_n)\) is Cauchy, we can ask whether the sequence \((f(x_n))\) automatically has any nice properties. For an arbitrary function \(f\), or even a continuous function \(f\), the answer is generally "no". But if \(f\) is uniformly continuous, this changes. Figure out what changes, and then (in problem 21) apply this with \(X=S\) and \(Y=E'\).
- In your MAA 4211 final exam, a relation \(\sim\) on the set of sequences in a metric space was introduced in problem 11. This relation is relevant for problem 21. Part (c) of the exam-problem is particularly relevant (more so than parts (a) and (b)), but in the context of problem 21 this relation has other relevant features not mentioned on the exam. In particular, it will pay for you to consider how the relations "\(\sim_X\)" and "\(\sim_Y\)" interact with a uniformly continuous function \(f:X\to Y\).
- In #32, better wording to indicate the sequence of functions would be "the sequence of functions \( (f_n:[0,\infty)\to{\bf R})_{n=1}^\infty\) defined by \( f_1(x)=\sqrt{x},\ f_2(x)=\sqrt{x+\sqrt{x}},\ f_3(x)=\sqrt{x+\sqrt{x+\sqrt{x}}}, \dots\)". A similar comment applies to problems 34–36.
- #32 is tricky and (somewhat) hard. It is relatively easy to figure out, for each \(x\geq 0\), the only possible number to which the sequence \(f_n(x)\) could converge, where \(f_n(x)\) is the \(n^{\rm th}\) term of the sequence indicated in the book. The tricky part is that you are likely to (initially) find two possible values, and discard one of them as being impossible, but the one you discard will actually be the right answer for one value of \(x\). The hardest part of the problem, once you figure out the only possible limit-function \(f\), is proving that the sequence actually does converge (pointwise) to \(f\). Remember that you can't prove that something is true by assuming it's true.
- B. Click here for non-book problems. (Note: As of about 9:00 p.m. 1/13/20 there are problems B2 and B3, in addition to B1. If you downloaded these problems before 9:11 p.m. 1/13/20, download them again; there were a couple of typos that I found and fixed shortly after posting the new problems.) Of these, hand in only B1b, B2a, B3ace.
- C. Start reading the handout Some notes on normed vector spaces. (You should do non-book problem B1 first.) The due-date for Assignment 1 does not apply to this reading; I do not expect to be using the results of this handout until we get to integration (Chapter VI). If you start reading this handout the first week of the semester, and average two pages a week, you'll be done by the end of the 4th week, which will be in time for application to integration. (However, I recommend that if you start reading an item—a definition, the statement of a proposition, a proof, a remark, etc.—that you finish that item in the same sitting, rather than giving yourself a break just because you've reached the bottom of a page.) Of course, feel free to finish sooner!
The material starting with Theorem 1.7 in these notes is fundamental, and really should be covered in MAA 4211, but I've never been able to squeeze it in during the first semester (except as homework that students have rarely been successful completing).
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