Metric and Hilbert Spaces (MAST30026)
Undergraduate level 3Points: 12.5On Campus (Parkville)
Overview
| Availability | Semester 2 |
|---|---|
| Fees | Look up fees |
This subject provides a basis for further studies in modern analysis, geometry, topology, differential equations and quantum mechanics.It introduces the idea of a metric space with a general distance function, and the resulting concepts of convergence, continuity, completeness, compactness and connectedness. The subject also introduces Hilbert spaces: infinite dimensional vector spaces (typically function spaces) equipped with an inner product that allows geometric ideas to be used to study these spaces and linear maps between them.
Topics include: metric and normed spaces, limits of sequences, open and closed sets, continuity, topological properties, compactness, connectedness; Cauchy sequences, completeness, contraction mapping theorem; Hilbert spaces, orthonormal systems, bounded linear operators and functionals, applications.
Intended learning outcomes
On completion of this subject, students should understand:
- the definition and fundamental properties of metric spaces, including the ideas of convergence, continuity, completeness, compactness and connectedness;
- the definition and fundamental properties of Hilbert spaces, and bounded linear maps between them;
- how basic concepts of geometry and linear algebra can be generalised to infinite dimensional spaces;
and should be able to:
- prove simple results about metric spaces and Hilbert spaces;
- analyse bounded linear maps between Hilbert spaces;
- apply general results on metric and Hilbert spaces to solve problems in other areas of mathematics and physics, including numerical methods and differential equations.
Generic skills
In addition to learning specific skills that will assist students in their future careers in science, they will have the opportunity to develop generic skills that will assist them in any future career path. These include:
- problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
- analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
- collaborative skills: the ability to work in a team;
- time-management skills: the ability to meet regular deadlines while balancing competing commitments.
Last updated: 9 April 2025
Eligibility and requirements
Prerequisites
| Code | Name | Teaching period | Credit Points |
|---|---|---|---|
| MAST20022 | Group Theory and Linear Algebra | Semester 2 (On Campus - Parkville) |
12.5 |
and one of
| Code | Name | Teaching period | Credit Points |
|---|---|---|---|
| MAST20026 | Real Analysis |
Semester 2 (On Campus - Parkville)
Semester 1 (On Campus - Parkville)
|
12.5 |
| MAST10009 | Accelerated Mathematics 2 | Semester 2 (On Campus - Parkville) |
12.5 |
Corequisites
None
Non-allowed subjects
None
Inherent requirements (core participation requirements)
The University of Melbourne is committed to providing students with reasonable adjustments to assessment and participation under the Disability Standards for Education (2005), and the Assessment and Results Policy (MPF1326). Students are expected to meet the core participation requirements for their course. These can be viewed under Entry and Participation Requirements for the course outlines in the Handbook.
Further details on how to seek academic adjustments can be found on the Student Equity and Disability Support website: http://services.unimelb.edu.au/student-equity/home
Last updated: 9 April 2025
Assessment
Additional details
Two or three written assignments due at regular intervals during semester amounting to a total of up to 50 pages (20%), and a 3-hour written examination in the examination period (80%).
Last updated: 9 April 2025
Dates & times
- Semester 2
Principal coordinator Arun Ram Mode of delivery On Campus (Parkville) Contact hours 3 x one hour lectures per week, 1 x one hour practice class per week Total time commitment 170 hours Teaching period 24 July 2017 to 22 October 2017 Last self-enrol date 4 August 2017 Census date 31 August 2017 Last date to withdraw without fail 22 September 2017 Assessment period ends 17 November 2017 Semester 2 contact information
Email: aram@unimelb.edu.au
Time commitment details
Estimated total time commitment of 170 hours
Last updated: 9 April 2025
Further information
- Texts
Prescribed texts
Recommended texts and other resources
J. J. Koliha, Metrics, Norms and Integrals: An introduction to Contemporary Analysis, World Scientific, 2008
L. Debnath and P. Mikusinski, Introduction to Hilbert Spaces with Applications, 2nd Ed, Academic Press, 1999
E. Kreyszig, Introductory Functional Analysis with Applications, Wiley, 1989
- Subject notes
- Related Handbook entries
This subject contributes to the following:
Type Name Informal specialisation Pure Mathematics Informal specialisation Pure Mathematics Major Mathematical Physics Informal specialisation Science-credited subjects - new generation B-SCI and B-ENG. Informal specialisation Selective subjects for B-BMED Informal specialisation Pure Mathematics Major Pure Mathematics - Breadth options
- Available through the Community Access Program
About the Community Access Program (CAP)
This subject is available through the Community Access Program (also called Single Subject Studies) which allows you to enrol in single subjects offered by the University of Melbourne, without the commitment required to complete a whole degree.
Please note Single Subject Studies via Community Access Program is not available to student visa holders or applicants
Entry requirements including prerequisites may apply. Please refer to the CAP applications page for further information.
Last updated: 9 April 2025