Last updated Wed Jan 16 16:56 EST 2019
Due-date: Wednesday, 1/23/19 (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 have indicated below--> 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, 34–38. (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, 32, 34ac (i.e. the first and third sequences in the problem), 38 (modified as indicated below).
- Reminder: for sequences of functions, Rosenlicht's "converges", absent the modifier "uniformly", is my "converges pointwise" (see p. 83).
- If you do not find #21 harder than the other proof-problems in the list above, you are probably doing something wrong.
- 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.
- In #38, assume only that the domain \(X\) of the functions in your sequence is a nonempty set; it is irrelevant whether \(X\) is a metric space. (Recall that for any set \(X\) and metric space \(Y\), a function \(f:X\to Y\) is bounded if the range of \(f\) is a bounded subset of \(Y\).)
- B. Click here for non-book problems. Of these, hand in only B1b, B2a, B3. In doing parts of B2 and B3, you are allowed to assume the results of earlier problem-parts on this page, whether or not you were successful in proving them.
Since posting the "finalized" nonbook problems on or about Jan. 14, I've found and fixed several typos, so you may need to refresh your browser to get the more finalized version.
- 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 this reading 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 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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