• Offered by Mathematical Sciences Institute
• ANU College ANU Joint Colleges of Science
• Course subject Mathematics
• Areas of interest Mathematics
• Academic career UGRD
• Course convener
• Dr Joseph Feneuil
• Mode of delivery In Person
• Co-taught Course
• Offered in Second Semester 2023
Advanced Functional Analysis, Spectral theory and Applications (MATH3325)

This course is intended both for mathematics students continuing to honours work and for other students using mathematics at a high level in theoretical physics, engineering and information technology, and mathematical economics.

Topics to be covered include:

Hilbert spaces - bounded linear operators, compact operators, the spectral theorem for compact self-adjoint operators; Fourier transform, applications to partial differential equations and the central limit theorem.

Measure theory - abstract measure theory, integration,  Fubini-Tonelli theorem, Radon-Nikodym theorem, Hausdorff measure, fractals.

Banach spaces and linear operators - basic properties, Baire category theorem and its consequences (uniform boundedness principle, closed graph and open mapping theorems), Hahn-Banach theorem and dual spaces, sequential version of Banach-Alaoglu theorem, dual spaces of L^p spaces and spaces of continuous functions. Applications to Fourier series, fractals.

Note: This is an HPC. It emphasises mathematical rigour and proof and continues the development of modern analysis from an abstract viewpoint.

## Learning Outcomes

Upon successful completion, students will have the knowledge and skills to:

On satisfying the requirements of this course, students will have the knowledge and skills to:

1. Explain the fundamental concepts of functional analysis and their role in modern mathematics and applied contexts
2. Demonstrate accurate and efficient use of functional analysis techniques
3. Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from functional analysis
4. Apply problem-solving using functional analysis techniques applied to diverse situations in physics, engineering and other mathematical contexts

## Indicative Assessment

Assessment will be based on:

• Four assignments (40% total; LO 1-4)
• Essay paper (15%; LO 1-4)
• Attendance and participation in lectures and wokshops (5%; LO 1-4)
• Take home exam (40%; LO 1-4)

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.

Three lectures per week, workshops by arrangement.

## Requisite and Incompatibility

To enrol in this course you must have successfully completed MATH3320 with a mark of 60 or above. Incompatible with MATH6214.

## 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
2023 \$4320
International fee paying students
Year Fee
2023 \$6180
Note: Please note that fee information is for current year only.

## 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.

The list of offerings for future years is indicative only.
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
5943 24 Jul 2023 31 Jul 2023 31 Aug 2023 27 Oct 2023 In Person N/A

Responsible Officer: Registrar, Student Administration / Page Contact: Website Administrator / Frequently Asked Questions