This course is intended both for mathematics students and for those using mathematics at a high level in theoretical physics, engineering and information technology, and mathematical economics. Selected topics from below will be covered:
Hilbert spaces - orthogonality, bounded linear operators, Riesz representation, adjoints, unitary operators, compact operators, the spectral theorem for compact self-adjoint operators with applications.
Measure theory - abstract measure theory, integration, signed measures, total variation, absolute continuity, the Lebesgue-Radon-Nikodym theorem, Hausdorff measure.
Banach spaces - bounded linear operators, dual spaces, adjoints, the Hahn-Banach theorem, Baire category theorem and its consequences (uniform boundedness principle, closed graph and open mapping theorems), weak convergence, sequential version of Banach-Alaoglu theorem, interpolation.
Applications to harmonic analysis and partial differential equations, such as the Fourier transform, maximal functions, singular integrals, Strichartz estimates.
Note: Graduate students attend joint classes with undergraduates enrolled in MATH3325 but will be assessed separately.
Learning Outcomes
Upon successful completion, students will have the knowledge and skills to:
- Explain the fundamental concepts of functional analysis and their role in modern mathematics and applied contexts
- Demonstrate accurate and efficient use of functional analysis techniques
- Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from functional analysis
- Apply problem-solving using functional analysis techniques applied to diverse situations in physics, engineering and other mathematical contexts
- Use deep knowledge and understanding of advanced functional analysis to formulate responses to complex concrete and abstract problems.
Indicative Assessment
- Regular written assignments Regardless of performance on other assessment items, a minimum score of 35% on the regular written assignments is required to pass the course. (This is known as a 'course hurdle'.) (20) [LO 1,2,3,4,5]
- Quizzes (15) [LO 1,2,3,4]
- Midsemester Exam (20) [LO 1,2,3,4]
- Final Exam (45) [LO 1,2,3,4]
The ANU uses Turnitin to enhance student citation and referencing techniques, and to assess assignment submissions as a component of the University's approach to managing Academic Integrity. While the use of Turnitin is not mandatory, the ANU highly recommends Turnitin is used by both teaching staff and students. For additional information regarding Turnitin please visit the ANU Online website.
Workload
The expected workload will consist of approximately 130 hours throughout the semester including:
• Face-to face component which will consist of 3x1 hours (or 1x2 hours lecture and 1x1 hours) lectures per week for 12 week semester (36 hours) and 10x1 hours of workshops throughout the semester.
• Approximately 84 hours of self-directed study which will include preparation for lectures, workshops, assignments, and exams.
Inherent Requirements
No specific inherent requirements have been identified in this course.
Requisite and Incompatibility
You will need to contact the Mathematical Sciences Institute to request a permission code to enrol in this course.
Prescribed Texts
E. M. Stein & R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces
E. M. Stein & R. Shakarchi, Functional Analysis: Introduction to Further Topics in Analysis
Preliminary Reading
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations
Fees
Tuition fees are for the academic year indicated at the top of the page.
Commonwealth Support (CSP) Students
If you have been offered a Commonwealth supported place, your fees are set by the Australian Government for each course. At ANU 1 EFTSL is 48 units (normally 8 x 6-unit courses). More information about your student contribution amount for each course at Fees.
- Student Contribution Band:
- 1
- Unit value:
- 6 units
If you are a domestic graduate coursework student with a Domestic Tuition Fee (DTF) place or international student you will be required to pay course tuition fees (see below). Course tuition fees are indexed annually. Further information for domestic and international students about tuition and other fees can be found at Fees.
Where there is a unit range displayed for this course, not all unit options below may be available.
| Units | EFTSL |
|---|---|
| 6.00 | 0.12500 |
Course fees
- Domestic fee paying students
| Year | Fee |
|---|---|
| 2025 | $4680 |
- International fee paying students
| Year | Fee |
|---|---|
| 2025 | $6720 |
Offerings, Dates and Class Summary Links
ANU utilises MyTimetable to enable students to view the timetable for their enrolled courses, browse, then self-allocate to small teaching activities / tutorials so they can better plan their time. Find out more on the Timetable webpage.
Class summaries, if available, can be accessed by clicking on the View link for the relevant class number.
Second Semester
| Class number | Class start date | Last day to enrol | Census date | Class end date | Mode Of Delivery | Class Summary |
|---|---|---|---|---|---|---|
| 7609 | 21 Jul 2025 | 28 Jul 2025 | 31 Aug 2025 | 24 Oct 2025 | In Person | N/A |