Unit name | Functional Analysis 34 |
---|---|
Unit code | MATHM6202 |
Credit points | 20 |
Level of study | M/7 |
Teaching block(s) |
Teaching Block 2 (weeks 13 - 24) |
Unit director | Dr. McGillivray |
Open unit status | Not open |
Pre-requisites |
none |
Co-requisites |
none |
School/department | School of Mathematics |
Faculty | Faculty of Science |
This unit sets out to explore some core notions in functional analysis. Functional analysis originated partly in the study of integral equations. It forms the basis of the theory of operators acting in infinite dimensional spaces. It has found broad applicability in diverse areas of mathematics (for example, spectral theory). Students will be introduced to the theory of Banach and Hilbert spaces. This will be followed by an exposition of four fundamental theorems relating to Banach spaces (Hahn-Banach theorem, uniform boundedness theorem, open mapping theorem, closed graph theorem).
Aims
The unit aims to provide students with a firm grounding in the theory and techniques of functional analysis and to offer students ample opportunity to build on their problem-solving ability in this area. It also aims to equip students with independent self-study and presentation-giving skills.
Syllabus
Banach spaces: bounded linear operators; bounded linear functionals; dual space
Hilbert spaces: orthogonal complement; total orthonormal set; representation of functionals on Hilbert spaces; Hilbert adjoint operator; self-adjoint, unitary and normal operators
Fundamental theorems for normed and Banach spaces: Zorn's Lemma; Hahn-Banach Theorem; Category Theorem; Uniform Boundedness Theorem; strong and weak convergence; convergence of sequences of operators; Open Mapping Theorem; Closed Graph Theorem
Relation to Other Units
This is a level M version of Functional Analysis 3, and students may not take both units. See Assessment Methods for the differences.
By the end of the unit, students will
Transferable Skills:
Deductive thinking; problem-solving; mathematical exposition; presentation skills.
Lectures (30), recommended problems, guided reading and presentation.
Formative assessment will consist in a number of marked homeworks.
The final assessment mark for Functional Analysis 34 will be made up as follows:
The course will follow portions of the text Kreyszig, E., Introductory Functional Analysis with Applications, John Wiley & Sons (1989).