[an error occurred while processing this directive]
MT5815 ADVANCED FUNCTIONAL ANALYSIS
This course consists of the existing MT4515 module with additional advanced material leading to a project.
Aims
The module will familiarise students with the basic notions of functional analysis, that is analysis on (typically infinite dimensional) normed spaces and Hilbert space. The course will cover normed spaces, convergence and completeness, Hilbert spaces, bounded linear operators, spectral theory, the Hahn-Banach theorem, and product Banach spaces and quotient Banach spaces
Objectives
By the end of the course students are expected to
1. understand the concept of a norm on a vector space and work with norms in specific and general instances;
2. appreciate the notion and consequences of completeness;
3. be able to work with operators on normed spaces ;
4. understand the statement of the spectral theorem and be able to apply it to simple situations.
5. understand the statement of the Hahn-Banach theorum and be able to apply it to simple situations
Syllabus
Normed spaces.
Examples of normed spaces : and .
Bounded linear operator on norm spaces.
The operator norm.
Completeness.
Banach spaces.
Inner product spaces and Hilbert spaces.
The spectrum of a bounded operator.
Compact operators.
The spectral theorem for compact self-abjoint operators.
Hahn-Banach's theorem. The axiom of Choice. The role of the Axiom of Choice in modern analysis. Applications of Hahn-Banach's theorem including Banach limits.
The principle of uniform boundedness and applications
Construction of new Banach spaces from old ones including product spaces and quotient spaces.
Textbooks
Introduction to Hilbert Space: N Young; Cambridge University Press. Students are strongly encouraged to purchase this textbook.
Prerequisites
MT2002
Availability
Academic year 2002/3 in semester 2: Wednesdays, Fridays and odd Mondays at 12
Lecturer
Dr B O Stratmann
Click here to see the University Course Catalogue entry.
Revised: JOC (May 2002)
Found a problem with the site?
Click here and
let us know.
