5M: Advanced Functional Analysis (SMSTC) MATHS5040
- Academic Session: 2022-23
- School: School of Mathematics and Statistics
- Credits: 20
- Level: Level 5 (SCQF level 11)
- Typically Offered: Semester 2
- Available to Visiting Students: No
- Available to Erasmus Students: No
This course gives a high level introduction to general functional analysis techniques.
Two hours of lectures a week (delivered by live video link across the Scottish Mathematical Sciences Training Council (SMSTC) network).
Ten hours of tutorial time spread over the semester.
Requirements of Entry
Mandatory Entry Requirements
4H Functional Analysis
Recommended Entry Requirements
100% Final Exam
In accordance with the University's Code of Assessment reassessments are normally set for all courses which do not contribute to the honours classifications. For non honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students, and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions are listed below in this box.
Main Assessment In: April/May
Are reassessment opportunities available for all summative assessments? No
Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below.
This course gives a high level introduction to general functional analysis techniques. The course will study completeness in normed spaces and inner product spaces. The Baire category theorem will be established and used to establish the uniform boundedness principle. The weak and weak*-topologies arising from duality between Banach spaces will be introduced and the Hahn-Banach theorem established. The notion of compactness of an operator will be introduced and the spectral theorem for compact self-adjoint operators will be established. This leads to the von Neumann-Schatten classes (Sp), which are natural noncommutative analogues of Lp spaces. The abstract notion of a Banach algebra will be introduced and spectal theory will be introduced in this context. Abelian Banach algebras will be studied via the Gelfand transform. The course ends with a brief introduction to C* algebras.
Intended Learning Outcomes of Course
By the end of the course students will be able to:
Demonstrate knowledge and the ability to apply central definitions and facts from the theory of functional analysis and use these to solve problems of a theoretical, logical or numerical nature. Specific
■ The structure and geometry of Banach and Hilbert spaces and operators between them;
■ completeness and the Baire category theorem with applications to existence results, such as the existence of a continuous but nowhere differentiable real function, and to the open
mapping and closed graph theorems;
■ convexity in normed spaces and extension results including the Hahn-Banach theorem; weak and weak*topologies on normed and Banach spaces and appropriate tools, such as nets, from general topology to handle calculations in non-metrisable topololgies; the Krein-Milman theorem and it's applications to functional analysis (such as the Banach-Stone theorem) and various
fixed point theorems;
■ compact operators, particularly on Hilbert spaces, approximation results for compact operators and the spectral theorem for compact self-adjoint operators;
■ the von Neumann-Schatten classes;
■ abstract Banach algebras and spectral theory, with applications to harmonic analysis such as Weiner's theorem.
Minimum Requirement for Award of Credits
Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.