Metric and Hilbert Spaces

This subject has the following teaching availabilities in 2010:

One of

• 620-297 Group Theory and Linear Algebra
• 620-222 Linear and Abstract Algebra (prior to 2009)

and one of

• 620-295 Real Analysis with Applications
• 620-158 Accelerated Mathematics 2

Students who have completed either of the following may not enrol in this subject for credit.

• 620-311 Metric Spaces (prior to 2010)
• 620-312 Linear Analysis (prior to 2010)

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.

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.

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%).

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

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

• Bachelor of Commerce
• Bachelor of Environments
• Bachelor of Music

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.

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.