Advanced Analysis 2022

Lecturer:

Demonstrators:

  • Griffin Pinney
  • Lachlan Potter
  • Michael Law
  • Kie Seng Nge

This course mainly concerns analysis on metric spaces. The main reference is the following:
Lecture notes, by John Hutchinson

Each student will engage in a total of 20 hours of lectures, and participate in a total of 10 hours of workshops. Each student will be assigned to one workshop group. The workshops will be run by demonstrators.

The Zoom links of the lectures and the workshops, as well as our contact information, can be found here: Course information

Lecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7 Lecture 8 Lecture 9 Lecture 10

Workshop problems:

The main assessment will be a final exam at the end of the course.

Final Exam paper

Announcements:

  • There will be a quiz during the lecture on 10 May 2022. It covers the materials we discussed in Lectures 1 to 3. It will be a good chance to review the material covered so far.
  • The maximum score for Quiz 1 was 95 out of 100 (3 students), and the median was 65 out of 100. You can now see your score on gradescope. See lecture 6 for the solution of the questions on the quiz.
  • The final exam will cover Chapters 6-13 and Chapter 15 of the lecture notes, with the exceptions of Chapter 9.4, 13.1, 13.2, 13.4, 13.5, 15.3 and 15.8. You'll be allowed to bring a double sided A4 sheet of handwritten notes into the exam to help you remember some of the key definitions / theorems / examples. It will be held on June 7 from 10:00am-12:30pm.
  • Here are some questions that may help you when you revise the material covered in this course:
    • What is a metric? Can you give some examples of metric spaces? How do you check whether a given function is a metric?
    • What is an open set in a metric space? A closed set? How do you check whether a set is open or closed? Can a set be both open and closed at the same time?
    • What is the interior of a set in a metric space? The boundary? The exterior? Why is a set in a metric space the disjoint union of its interior, its exterior and its boundary?
    • How do you find the set of limit points of a given set in a metric space? How do you relate the closure of a set to limit points / boundary points?
    • How do you know whether a sequence converges in a metric space? What is a bounded sequence in a metric space? What is a Cauchy sequence?
    • Must all convergent sequences be bounded? Cauchy? Must all bounded sequences converge? etc.
    • What is a complete metric space? Can you give examples of complete metric spaces? Examples of incomplete metric spaces? How do you check whether a metric space is complete?
    • What is a contraction on a metric space? What is a fixed point of a map on a set? When does a contraction have a fixed point? Look up the contraction mapping principle, and see how one uses that to e.g. construct solutions to ordinary differential equations.
    • How do you know whether a function is continuous on a metric space? Uniformly continuous? What is the difference between a continuous function and a uniformly continuous function? Can you give some examples of uniformly continuous functions on a metric space? And some examples of continuous but not uniformly continuous functions? How can continuity be characterized using preimages of open sets / closed sets?
    • What is a compact set in a metric space? A sequentially compact set? A totally bounded set? There is a theorem that says for a set in a metric space, compactness is equivalent to sequential compactness, which in turn is equivalent to completeness and total boundedness. How do you use this to check whether a set is compact?
    • What is a compact set in the Euclidean space Rn (equipped with the Euclidean metric)? Is it true that a closed and bounded set in Rn is always compact? What if Rn is replaced by a general metric space?
    • When does a sequence of functions converge uniformly on a subset of a metric space? What does it mean to say that a sequence of functions can converge pointwisely on a set without converging uniformly? Why is it good to have a sequence of uniformly convergent functions? How do you know whether the limit of a sequence of functions is continuous / Riemann integrable / differentiable?
    • What does the Arzela-Ascoli theorem say? When does a sequence of functions on a metric space have a uniformly convergent subsequence?
  • The best way of preparing for the final exam is to think through what has been covered. Trying to memorize material from the course will not get you very far. Emphasis of the exam will be on your actual understanding. Ideally, once you have learned the material, you should be able to use it in any subsequent course you will take.
  • The exam will consist of four long questions. Each question will be divided into several parts. The parts may or may not be related to each other; please read the questions and the instructions carefully. You will be asked to give complete, logical arguments to support your answers.