This is the final problem sheet for this unit! On the other hand, many of you will probably be working on various exercises and working through old exams all of the way up until the exam. If there is ever any work that you would like me to look over and give feedback on, please do get in touch over email.
Section B.11 Problem Sheet 11
Due 15 December at 3pm, either in class or in the pigeon hole in 4W.
- Exercise 6.1.4 (PS11) Baire for linear subspaces (from Section 6.1 only need Definition 6.2 and, for the last part, the statement of Corollary 6.6).
Exercise 5.2.1 gives you some practice checking if things are algebras or lattices, and thinking about how this relates to density. Exercise 5.2.5 similarly gets you thinking about separating points. We’ve talked about (the unassigned) Exercise 5.2.3 in class as a model application of Corollary 5.9. The similarly unassigned Exercise 5.1.2 gives an application of Theorem 5.5, but is relatively tricky — for next year (or during a problems class or Revision Week) I would hope to split it into more parts and add some more helpful hints.
Exercise 6.1.1 gets you thinking about this new definition of ‘nowhere dense’ that we’re going to be working with in Chapter 6, and would have made sense at the end of Chapter 1. Finally, Exercise 6.1.4 is a nice exercise which gets you thinking about closures and interiors of linear subspaces, eventually leading up to an important application of Corollary 6.6.
Please feel free to email me, or drop by office hours (Tuesdays 1:15–2:05 in 4W 1.12), with any questions about these problems whatsoever. I am also more than happy to meet one-on-one or in a small group.
