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Metric and Hilbert Spaces (MAST30026)

Undergraduate level 3Points: 12.5On Campus (Parkville)

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Year of offer2019
Subject levelUndergraduate Level 3
Subject codeMAST30026
Semester 2
FeesSubject EFTSL, Level, Discipline & Census Date

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.

Eligibility and requirements


Code Name Teaching period Credit Points
MAST20022 Group Theory and Linear Algebra
Semester 2

and one of

Code Name Teaching period Credit Points
MAST20026 Real Analysis
Semester 1
Semester 2
MAST10009 Accelerated Mathematics 2
Semester 2



Non-allowed subjects


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



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%).

Dates & times

  • Semester 2
    Principal coordinatorDaniel Murfet
    Mode of deliveryOn Campus — Parkville
    Contact hours3 x one hour lectures per week, 1 x one hour practice class per week
    Total time commitment170 hours
    Teaching period29 July 2019 to 27 October 2019
    Last self-enrol date 9 August 2019
    Census date31 August 2019
    Last date to withdraw without fail27 September 2019
    Assessment period ends22 November 2019

    Semester 2 contact information

Time commitment details

Estimated total time commitment of 170 hours

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

    This subject is available for science credit to students enrolled in the BSc (both pre-2008 and new degrees), BASc or a combined BSc course.

  • Breadth options

    This subject is available as breadth in the following courses:

  • 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.

    Entry requirements including prerequisites may apply. Please refer to the CAP applications page for further information.

  • Available to Study Abroad and/or Study Exchange Students

    This subject is available to students studying at the University from eligible overseas institutions on exchange and study abroad. Students are required to satisfy any listed requirements, such as pre- and co-requisites, for enrolment in the subject.

Last updated: 2 August 2019