Econ 519: David Reiley
Due Monday, 14 August 2006

Problem Set #10

The "finite covering property" (Section 29.5) is the most subtle and difficult concept in this chapter. It's most useful for understanding vector spaces other than Rn, such as (infinite-dimensional) spaces of functions (one place we worry about such spaces is in probability theory, where we often want to talk about whether a sequence of functions converges, such as in the Central Limit Theorem). In function spaces, the appropriate norm may not be obvious, so we would like a definition of compactness that doesn't depend on the norm (as "closed and bounded" does). Understanding the "finite covering property" is not essential for your mastery of economics, but it would be useful if you can get familiar with it and use it for practice in understanding proofs. So, if you're struggling, do the last three problems last.

1. Problem 29.1.
2. Problem 29.2.
3. Problem 29.6. (Hint: you'll need one of the theorems from Chapter 12.)
4. Problem 29.8.
5. Problem 29.9.
6. Problem 29.11.
7. Problem 29.12.
8. Problem 29.13.
9. Problem 29.14.
10. Problem 29.16.
11. Problem 29.17. That is, show that the definition in equation (7) is guaranteed to satisfy the three properties of an equivalence relation. (An equivalence relation is a much more general concept than the equivalence of two norms: sets can be equivalent, points can be equivalent, etc.)
12. Problem 29.20.
13. Problem 29.22.
14. Problem 29.23. Don't worry too much about the question "Which of the three was the easiest calculation?" The point is that if you find it easier to show a set is open in norm A, it must be open under norm B, as discussed at the top of page 815.
15. Problem 29.24.
16. Problem 29.27.
17. Problem 29.29.