Last updated Sun Mar 7 10:15 EST 2021
Due-date: Friday 3/12/21
You are required to do all of the problems and reading below (except for anything explicitly labeled "optional"). 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 are to 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: Tao exercises:
Of these, hand in only 2.2/ 10; 2.4/ 1, 2, 7, 8
- 2.1/ 4
- 2.2/ 5, 6, 7, 10, 11
- 2.3/ 6 (minus the last sentence)
- 2.4/ 1, 2, 7, 8, 9
- 3.2/ 1
- 3.3/ 8
Notes on some of these problems:
- Problem 2.2.6: "Let \( {\bf R}^m\) and \({\bf R}^n\) be Euclidean spaces" means: "Treat \( {\bf R}^m\) as the metric space \( ( {\bf R}^m, d_{\rm Euc} ) = ({\bf R}^m,d_{\ell^2}) \), and treat \({\bf R}^n\) analogously."
- Problems 2.2/ 7, 10, 11; 2.3/ 6: Tao forgot to specify metrics on \({\bf R}^n\) (\(n=k\) in 2.2.7; \(n=2\) in 2.2.10–11 and 2.3.6). Whenever this happens, unless I say otherwise, take the metric to be either \(d_{\ell^2}, d_{\ell^1},\) or \(d_{\ell^\infty}\), but say which one you're using. By results proven in class and/or in homework about equivalent metrics and continuity, the choice among these three metrics will not affect the truth of what you're being asked to show. Usually, when mathematicians speak of \({\bf R}^n\) as a metric space without remembering to specify a metric, they're implicitly thinking of the Euclidean metric.
- Exercise 2.4.9: In the hint, treat the reference to Exercise 2.4.6 as a reference to non-book problem B4. I didn't assign the former since it's a special case of the latter.
- B: Click for non-book problems. There is a lot you should learn by doing these problems. Problem B9 and B13 involve important applications of compactness; B13–B15 have you prove some fundamental facts about finite-dimensional vector spaces; B17 (for which B16 is a prerequisite) applies several ideas we've covered, and shows you that the infinite-dimensional analog of the Heine-Borel Theorem is false; B18 gives you a use for uniform continuity; B19 is a non-obvious application of uniform continuity and compactness.
Problems B16–B20 belong in the current assignment by virtue of subject matter, but will not be due on Mar. 12; they will be repeated as part of the next assignment. I've left them in the current assignment to give you more time with them; the due-date for Assignment 4 may be as early as Mar. 19. I'm postponing their due-dates just because of the length of the current assignment, and because B20 is quite challenging. (Problem B19(b) would also have been quite challenging had I not structured the proof for you in a hint, but with the hint it shouldn't be too hard.)Of the non-book problems, hand in only B1, B4, B9bc, B12, B13, B15 (you may assume B14 in your write-up of B15).
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