• Offered by Department of Mathematics
• ANU College ANU Joint Colleges of Science
Specialist
• Course subject Mathematics
• Areas of interest Mathematics
• Course convener
• Dr Bai-Ling Wang
• Mode of delivery In Person
• Offered in First Semester 2014
Analysis 2: Topology, 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 include:

• Topological Spaces
• Continuity
• Homeomorphisms
• Convergence
• Hausdorff spaces
• Compactness
• Connectedness
• Path connectedness
• 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 as Banach spaces
• Maximal Functions
• Vitali covers, Lebesgue differentiation, and density results
• Hilbert Spaces
• Elementary properties such as Cauchy Schwartz inequality and polarization
• Orthogonal complements
• Linear operators
• Riesz duality
• 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:

On satisfying the requirements of this course, 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. Apply problem-solving using advanced analysis techniques applied to diverse situations in physics, engineering and other mathematical contexts.

## Indicative Assessment

Assessment will be based on Assignments and Final Exam after class discussion (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.

## Requisite and Incompatibility

You will need to contact the Department of Mathematics to request a permission code to enrol in this course.

## 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. Students continuing in their current program of study will have their tuition fees indexed annually from the year in which you commenced your program. 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
1994-2003 \$1650
2004 \$2160
2005 \$2520
2006 \$2520
2007 \$2520
2008 \$2916
2009 \$2916
2010 \$2916
2011 \$2946
2012 \$2946
2013 \$2946
2014 \$2946
International fee paying students
Year Fee
1994-2003 \$3606
2004 \$3618
2005 \$3618
2006 \$3618
2007 \$3618
2008 \$3618
2009 \$3618
2010 \$3750
2011 \$3756
2012 \$3756
2013 \$3756
2014 \$3762
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
3332 17 Feb 2014 07 Mar 2014 31 Mar 2014 30 May 2014 In Person N/A