• Offered by Mathematical Sciences Institute
  • ANU College ANU Joint Colleges of Science
  • Classification Advanced
  • Course subject Mathematics
  • Areas of interest Mathematics
  • Academic career PGRD
  • Course convener
    • Prof John Urbas
  • Mode of delivery In Person
  • Co-taught Course
  • Offered in First Semester 2019
    See Future Offerings
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.


Three lectures per week and regular workshops/computer labs.

Requisite and Incompatibility

You will need to contact the Mathematical Sciences Institute to request a permission code to enrol in this course.


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:
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.

6.00 0.12500
Domestic fee paying students
Year Fee
2019 $3840
International fee paying students
Year Fee
2019 $5460
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
2538 25 Feb 2019 04 Mar 2019 31 Mar 2019 31 May 2019 In Person View

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