PMH3 Functional Analysis
General Information
This page relates to the Pure Mathematics Honours course "Functional Analysis".
Lecturer for this course: Alexander Fish.
For general information on honours in the School of Mathematics and Statistics, refer to the relevant honours handbook.
Lecturer Contact Information
My office is Carslaw room 712 and I will often be available there to answer question. You can also email me with this link to ask questions or make an appointment.
Class Times
The lecture times for the course are Monday and Tuesday 10am-11am in Carslaw 830.
Textbook
Most of the course is close to parts of the lecture notes by A/Prof Daniel Daners ``Introduction to Functional Analysis". I also warmly recommend the new monograph by M. Einsiedler and T. Ward ``Functional Analysis, Spectral Theory, and Applications".
The textbook contains many exercises, some of which will be assigned as homework.
Exercise Sheets
- March 7 - Normed vector spaces, completions and Banach spaces
- March 14 - Linear operators and the operator norm
- March 26 - Hilbert Spaces
- April 17 - Uniform boundness and the Open Mapping Theorem
- May 8 - Dual spaces and the Hahn-Banach Theorem
- May 15 - Spectral Theory
Course Outline
Functional Analysis is one of the major areas of modern mathematics. Roughly speaking it is an infinite-dimensional generalisation of Linear Algebra -- it is studying various properties of linear continuous transformations on normed infinite-dimensional spaces. Functional Analysis plays a fundamental role in such areas as Differential Equations/PDE, Representation Theory, and Probability.
We will cover the following topics:
- Normed vector spaces, completions and Banach spaces
- Linear Operators and Operator norm
- Hilbert spaces and the Stone-Weierstrass Theorem
- Uniform boundedness and the Open Mapping Theorem
- Dual spaces and the Hahn-Banach Theorem
- Spectral theory of compact self-adjoint operators
Each of the above 6 topics has an associated Exercise Sheet, available on the course webpage. We will not have formal tutorials but you should work through the exercises on these sheets.
Background Knowledge
The formal prerequisites are MATH3961 Metric Spaces (Advanced) and MATH3969 Measure Theory and Fourier Analysis (Advanced). You will also need to recall some basic linear algebra in general vector spaces, from MATH2961 Linear Algebra and Vector Calculus (Advanced).
Assessment
30% assignments, 70% exam.
Three assignments each worth 10%, due at the end of weeks 4, 8 and 12, and one closed-book exam worth 70%. Assignment questions will be selected from the Exercise Sheets, and the exam may also contain questions from the Exercise Sheets. These assignments are to be submitted through Turnitin; see the question sheets for further instructions. As usual, you are encouraged to work on exercises with other students, but in a small enough group that everyone can make a contribution, and you should write up your assignment solutions independently.
- Assignment 1 will be due before midnight on Wednesday March 28 (Week 4).
- Assignment 2 will be due before midnight on Wednesday May 2 (Week 8).
- Assignment 3 will be due before midnight on Wednesday May 30 (Week 12).
The final exam, worth 70%, will be 2 hours plus 10 minutes reading time, closed-book but with a `formula sheet' provided listing some of the main definitions and formulas from the course. Last year's exam paper will be posted and will be discussed on the Week 13 at a specially organised session. This year's exam will have essentially the same format.