Section H.5 The 2022 exam
This exam was written by me, but unlike this year’s exam it was an open book exam. All of these questions would have also been fair game for this year’s exam except for Q4(b). Before attempting that problem, read the conclusion of Exercise 5.1.2, which was not assigned this year but was assigned in the year of this exam.
A small notational thing to be aware of is that in previous years we wrote \(L^2(a,b)\) rather than \(L^2([a,b])\text{.}\)
Also, on an in person exam you should expect to be asked to state a few definitions and results from lectures. For instance, the first parts of the first two questions might be changed to the following.
Exercises Some ‘closed book’ modifications to 2022 exam
1. Question 1.
(a)
Let \((X,d)\) be a metric space and let \(S \subseteq X\text{.}\)
(i)
Define what it means for \(S\) to be open.
(ii)
Define what it means for \(S\) to be dense.
(iii)
Define what it means for \(S\) to be relatively compact.
2. Question 2.
(a)
Let \((X,d)\) be a metric space.
(i)
Define what it means to be an open cover of \((X,d)\text{.}\)
(ii)
Define what it means for \((X,d)\) to be compact.
