• Offered for Year: 2024/25

• Module Leader(s): Dr Zinaida Lykova

• Owning School: Mathematics, Statistics and Physics
• Teaching Location: Newcastle City Campus

Semesters

Your programme is made up of credits, the total differs on programme to programme.

Aims

To deepen the students’ understanding of Functional Analysis, and to show how the interplay between topology, analysis and algebra can be exploited. Students will gain a knowledge of functional analysis, algebras of linear operators 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

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.

Teaching Methods

Teaching Activities

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 Methods

The format of resits will be determined by the Board of Examiners

Exams

Other Assessment

Formative Assessments

Formative Assessment is an assessment which develops your skills in being assessed, allows for you to receive feedback, and prepares you for being assessed. However, it does not count to your final mark.

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.

Reading Lists

• MAS8756's Reading List

Timetable

• Timetable Website: www.ncl.ac.uk/timetable/
• MAS8756's Timetable