• Offered by Mathematical Sciences Institute
• ANU College ANU Joint Colleges of Science
• Course subject Mathematics
• Areas of interest Mathematics
• Course convener
• EmPr John Urbas
• Mode of delivery In Person
• Co-taught Course
• Offered in First Semester 2025
Advanced Analysis 2: Lebesgue Integration and Hilbert Spaces (MATH3320)

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 will normally include topics 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.

This is an Honours Pathway Course. It emphasises mathematical rigour and proof and develops modern analysis from an abstract viewpoint.

## Learning Outcomes

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

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

## Indicative Assessment

1. 5 assignments (30) [LO 1,2,3,4]
2. Mid semester exam (20-30%) (30) [LO 1,2,3,4]
3. Final exam (40-50%) (40) [LO 1,2,3,4]
4. Precise weights to be determined in consultation with the class at first lecture. (null) [LO null]

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.

The expected workload will consist of approximately 130-140 hours throughout the semester including:

• Face-to face component which will consist of 12 x 3 hours lectures per semester (36 hours) and 10 hours of workshops throughout the semester.

• Approximately 80-90 hours of self-study which will include preparation for lectures, workshops, assignments and exams.

## Inherent Requirements

To be determined.

## Requisite and Incompatibility

To enrol in this course you must have successfully completed MATH2320 with a mark of 60 and above.

## Prescribed Texts

No prescribed text.

## 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
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.

### First Semester

Class number Class start date Last day to enrol Census date Class end date Mode Of Delivery Class Summary
3559 17 Feb 2025 24 Feb 2025 31 Mar 2025 23 May 2025 In Person N/A