• Offered by Mathematical Sciences Institute
• ANU College ANU Joint Colleges of Science
• Course subject Mathematics
• Areas of interest Mathematics
• Course convener
• Prof John Urbas
• Mode of delivery In Person
• Co-taught Course
• Offered in First Semester 2020
Analysis 2: Lebesgue Integration and Hilbert Spaces (MATH6212)

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 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.

## Learning Outcomes

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

1. Explain the fundamental concepts of advanced analysis such as topology and Lebeque integration 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. Use deep knowledge and understanding of advanced analysis, such as topology and Lebesgue integration, to formulate responses to complex concrete and abstract problems.

## 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.

To be determined

## 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.

## Fees

Tuition fees are for the academic year indicated at the top of the page.

If you are a domestic graduate coursework or international student you will be required to pay tuition fees. Tuition fees are indexed annually. Further information for domestic and international students about tuition and other fees can be found at Fees.

Student Contribution Band:
2
Unit value:
6 units

If you are an undergraduate student and 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). You can find your student contribution amount for each course 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
2020 \$4050
International fee paying students
Year Fee
2020 \$5760
Note: Please note that fee information is for current year only.

## Offerings, Dates and Class Summary Links

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
3586 24 Feb 2020 02 Mar 2020 31 Mar 2020 29 May 2020 In Person View