Unit information: Functional Analysis 34 in 2013/14

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

Description including Unit Aims

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.

Intended Learning Outcomes

By the end of the unit, students will

• understand basic concepts and results in functional analysis;
• be able to solve routine problems;
• have developed skills in applying the techniques of the course to unseen situations;
• have acquired independent self-study skills through guided reading;
• have acquired presentation-giving skills.

Transferable Skills:

Deductive thinking; problem-solving; mathematical exposition; presentation skills.

Teaching Information

Lectures (30), recommended problems, guided reading and presentation.

Assessment Information

Formative assessment will consist in a number of marked homeworks.

The final assessment mark for Functional Analysis 34 will be made up as follows:

• 10% from a 20-minute presentation, based on independent guided reading and assessed by the unit director.

• 90% from a 2½-hour examination in May/June consisting of FIVE questions. (It will be identical to the Level 3 Functional Analysis examination.) A candidate's best FOUR answers will be used for assessment. Calculators are NOT permitted in this examination.

Reading and References

The course will follow portions of the text Kreyszig, E., Introductory Functional Analysis with Applications, John Wiley & Sons (1989).