Metric and Hilbert Spaces (MAST30026)

Undergraduate level 3Points: 12.5On Campus (Parkville)

You’re viewing the 2024 Handbook:
Or view archived Handbooks
You’re currently viewing the 2024 version of this subject

Overview

Availability
Semester 2
FeesLook 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;
  • apply general results on metric and Hilbert spaces to solve problems in other areas of mathematics and physics, including numerical methods and differential equations.
  • analyse bounded linear maps between Hilbert spaces;

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: 21 January 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
MAST10009 Accelerated Mathematics 2
Semester 2 (On Campus - Parkville)
 12.5
MAST20026 Real Analysis
Semester 2 (On Campus - Parkville)
Semester 1 (On Campus - Parkville)
 12.5
MAST20033 Real Analysis: Advanced
Semester 1 (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: 21 January 2025

Assessment

DescriptionTimingPercentage

Two or three written assignments due at regular intervals amounting to a total of up to 50 pages

  • 50 pages
During the teaching period20%

A written examination

  • 3 hours
During the examination period80%

Additional details

This Dual-Delivery subject has On Campus assessment components.

Last updated: 21 January 2025

Dates & times

  • Semester 2
    CoordinatorAlexandru Ghitza
    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 period22 July 2024 to 20 October 2024
    Last self-enrol date 2 August 2024
    Census date 2 September 2024
    Last date to withdraw without fail20 September 2024
    Assessment period ends15 November 2024

    Semester 2 contact information

    aghitza@unimelb.edu.au

Time commitment details

Estimated total time commitment of 170 hours

What do these dates mean

Visit this webpage to find out about these key dates, including how they impact on:

  • Your tuition fees, academic transcript and statements.
  • And for Commonwealth Supported students, your:
       -  Student Learning Entitlement. This applies to all students enrolled in a Commonwealth Supported Place (CSP).
     

    Subjects withdrawn after the census date (including up to the ‘last day to withdraw without fail’) count toward the Student Learning Entitlement.

Additional delivery details

This Dual-Delivery subject has On Campus assessment components.

Last updated: 21 January 2025

Further information

Last updated: 21 January 2025