Module Catalogue 2024/25

Pre Requisite Comment

N/A

Co Requisite Comment

N/A

Aims

To deepen the students’ understanding of Functional Analysis and Topology, and to show how
the interplay between topology, analysis and algebra can be exploited. Students will gain a
knowledge of that part of topology relevant to functional analysis, algebras of linear
transformations on Banach and Hilbert spaces, and Banach algebras.
Module summary
This subject constitutes a synthesis of some of the main trends in analysis over the past century. One
studies functions not individually, but as a collection which admits natural operations of addition and
multiplication and has geometric structure. An algebra is a vector space with an associative
multiplication. There is an abundance of natural examples, many of them having the structure of a
Banach space. Examples are the spaces of n by n matrices and the continuous functions on the
interval [0,1], with suitable norms. Putting together algebras and norms one is led to the idea of a
Banach algebra. A rich and elegant theory of such objects was developed over the second half of the
twentieth century. Several members of staff have research interests close to this area.

Outline Of Syllabus

Further topics in topology, bounded linear operators, the Hahn-Banach theorem, the open mapping
theorem, weak and weak-* topologies, introduction to Banach algebras, the group of units and
spectrum, the Gelfand-Mazur theorem, commutative Banach algebras, characters and maximal
ideals, the Gelfand topology and Gelfand representation theorem, examples and applications.

Intended Knowledge Outcomes

The students will have a broad understanding of topology and functional analysis.

Intended Skill Outcomes

The students will be able to demonstrate skill in proving theorems about the topological and algebraic
structures of some of the principal objects occurring in functional analysis, as well as their
applications.
Students will develop skills across the cognitive domain (Bloom’s taxonomy, 2001 revised edition):
remember, understand, apply, analyse, evaluate and create

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking. Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.

General Notes

N/A