Handbook

MAST30026 Metric and Hilbert Spaces

Credit Points: 12.5
Level: 3 (Undergraduate)
Dates & Locations:

This subject has the following teaching availabilities in 2017:

Semester 2, Parkville - Taught on campus.Show/hide details
Pre-teaching Period Start not applicable
Teaching Period 24-Jul-2017 to 22-Oct-2017
Assessment Period End 17-Nov-2017
Last date to Self-Enrol 04-Aug-2017
Census Date 31-Aug-2017
Last date to Withdraw without fail 22-Sep-2017


Timetable can be viewed here.
For information about these dates, click here.
Time Commitment: Contact Hours: 3 x one hour lectures per week, 1 x one hour practice class per week
Total Time Commitment:

Estimated total time commitment of 170 hours

Prerequisites:
Subject
Study Period Commencement:
Credit Points:

and one of

Subject
Study Period Commencement:
Credit Points:
Semester 1, Semester 2
12.50
Corequisites: None
Recommended Background Knowledge: None
Non Allowed Subjects: None
Core Participation Requirements:

For the purposes of considering request for Reasonable Adjustments under the Disability Standards for Education (Cwth 2005), and Student Support and Engagement Policy, academic requirements for this subject are articulated in the Subject Overview, Learning Outcomes, Assessment and Generic Skills sections of this entry.

It is University policy to take all reasonable steps to minimise the impact of disability upon academic study, and reasonable adjustments will be made to enhance a student's participation in the University's programs. Students who feel their disability may impact on meeting the requirements of this subject are encouraged to discuss this matter with a Faculty Student Adviser and Student Equity and Disability Support: http://services.unimelb.edu.au/disability

Coordinator

Prof Arun Ram

Contact

Email: yaoban@unimelb.edu.au

Subject Overview:

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.

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.
Assessment:

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%).

Prescribed Texts: None
Recommended Texts:

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

Breadth Options:

This subject potentially can be taken as a breadth subject component for the following courses:

You should visit learn more about breadth subjects and read the breadth requirements for your degree, and should discuss your choice with your student adviser, before deciding on your subjects.

Fees Information: Subject EFTSL, Level, Discipline & Census Date
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.
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.

Related Majors/Minors/Specialisations: Mathematical Physics
Pure Mathematics
Pure Mathematics
Pure Mathematics
Pure Mathematics
Pure Mathematics (specialisation of Mathematics and Statistics major)
Science-credited subjects - new generation B-SCI and B-ENG.
Selective subjects for B-BMED

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