This course is intended both for continuing mathematics students and for other students using mathematics at a high level in theoretical physics, engineering and information technology, and mathematical economics.
Topics to be covered will be selected from the following, with some additions and variations each year:
- Measure and Integration - Lebesgue outer measure, measurable sets and integration, Lebesgue integral and basic properties, convergence theorems, connection with Riemann integration, Fubini's theorem, approximation theorems for measurable sets, Lusin's theorem, Egorov's theorem, Lp spaces, general measure theory, Radon-Nikodym theorem
- Hilbert Spaces - elementary properties such as Cauchy Schwartz inequality and polarization, nearest point, orthogonal complements, linear operators, Riesz duality, adjoint operator, basic properties or unitary, self adjoint and normal operators, review and discussion of these operators in the complex and real setting, applications to L2 spaces and integral operators, projection operators, orthonormal sets, Bessel's inequality, Fourier expansion, Parseval's equality, applications to Fourier series.
Note: Graduate students attend joint classes with undergraduates but will be assessed separately.
Learning Outcomes
Upon successful completion, students will have the knowledge and skills to:
- Explain the fundamental concepts of advanced analysis such as topology and Lebeque integration and their role in modern mathematics and applied contexts
- Demonstrate accurate and efficient use of advanced analysis techniques
- Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from advanced analysis
- Apply problem-solving using advanced analysis techniques applied to diverse situations in physics, engineering and other mathematical contexts.
- Analyse complex topics related to measure theory and advanced analysis and communicate results.
Indicative Assessment
- Regular written assignments (30) [LO 1,2,3,4,5]
- Mid-semester exams (typically two) (30) [LO 1,2,3,4]
- Final exam (30) [LO 1,2,3,4]
- Written or oral report (10) [LO 1,2,3,4,5]
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 as follows:
• Face-to face component which may consist of 12 x 3 hours lectures per semester (36 hours) and 10 hours of workshops throughout the semester.
• Approximately 84 hours of self-directed study which will include preparation for lectures, workshops, and assessment tasks.
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
No prescribed text.
Preliminary Reading
Real Analysis by Stein and Shakarchi
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 |
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.
First Semester
| Class number | Class start date | Last day to enrol | Census date | Class end date | Mode Of Delivery | Class Summary |
|---|---|---|---|---|---|---|
| 2556 | 23 Feb 2026 | 02 Mar 2026 | 31 Mar 2026 | 29 May 2026 | In Person | N/A |