• Offered for Year: 2025/26

• Module Leader(s): Dr Evgenios Kakariadis
• Co-Module Leader: 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 familiarise students with the theory of measure spaces. To understand Lebesgue integration and applications that arise both in pure and applied sciences. 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. To reinforce the ability of students to follow research in Analysis.

Module Summary
Measure theory gives the appropriate language for measuring subsets of a space in a systematic way. The common example is Lebesgue measure on the real line which gives the length of an interval. This idea can be used to further produce a notion of integration. Unlike Riemann integration which is based on a partition of the domain of a function, Lebesgue integration relies on partitions of the range. As such it can tackle, in a sense, more functions than usual. Measure theory is a basic tool for Analysis and Algebra but also has vast applications in Applied Sciences, including Physics, Medicine and Economics. By the end of the course the students will understand Lebesgue integration in Rn and how it can be used as a language to encode a variety of examples through the notion of Hilbert spaces.

Functional analysis 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

Systems of sets and measures. Measure theory on Rn (Lebesgue integration). Comparison with Riemann integration.

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

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

• MAS8757's Reading List

Timetable

• Timetable Website: www.ncl.ac.uk/timetable/
• MAS8757's Timetable