Metric and Hilbert Spaces
Subject MAST30026 (2010)
Note: This is an archived Handbook entry from 2010.
Credit Points:  12.50  

Level:  3 (Undergraduate)  
Dates & Locations:  This subject has the following teaching availabilities in 2010: Semester 2, Parkville  Taught on campus.
Lectures and practice classes. 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 120 hours  
Prerequisites: 
One of
and one of  
Corequisites:  None  
Recommended Background Knowledge:  None  
Non Allowed Subjects: 
Students who have completed either of the following may not enrol in this subject for credit.
 
Core Participation Requirements:  It is University policy to take all reasonable steps to minimise the impact of disability upon academic study and reasonable steps will be made to enhance a student's participation in the University's programs. Students who feel their disability may impact upon their active and safe participation in a subject are encouraged to discuss this with the relevant subject coordinator and the Disability Liaison Unit. 
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. 

Objectives: 
On completion of this subject, students should understand:
and should be able to:

Assessment: 
Two or three written assignments due at regular intervals during semester amounting to a total of up to 50 pages (20%), and a 3hour 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:

Notes:  This subject is available for science credit to students enrolled in the BSc (both pre2008 and new degrees), BASc or a combined BSc course. 
Related Course(s): 
Bachelor of Science 
Related Majors/Minors/Specialisations: 
Mathematical Physics Mathematics and Statistics (Mathematical Physics specialisation) Mathematics and Statistics (Pure Mathematics specialisation) Pure Mathematics 
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