# MAS8714 : Topics in Analysis & Functional Analysis

• Offered for Year: 2021/22
• Module Leader(s): Dr Zinaida Lykova
• Owning School: Mathematics, Statistics and Physics
• Teaching Location: Newcastle City Campus
##### Semesters
 Semester 1 Credit Value: 10 Semester 2 Credit Value: 20 ECTS Credits: 15

#### Aims

To equip students with a range of tools and methods for operators on infinite dimensional Hilbert spaces. To understand Lebesgue integration and applications that arise both in pure and applied sciences. To reinforce the ability of students to follow research in Analysis.

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

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.

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

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

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.

#### Teaching Methods

##### Teaching Activities
Category Activity Number Length Student Hours Comment
Scheduled Learning And Teaching ActivitiesLecture501:0050:00Formal Lectures 20 on line 30 present in person
Scheduled Learning And Teaching ActivitiesLecture31:003:00Revision Lectures – Present in Person
Scheduled Learning And Teaching ActivitiesLecture101:0010:00Problem Classes - Online
Guided Independent StudyAssessment preparation and completion501:0050:00Completion of in course assessments
Guided Independent StudyIndependent study1871:00187:00Preparation time for lectures, background reading, coursework review
Total300:00
##### Teaching Rationale And Relationship

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
Description Length Semester When Set Percentage Comment
Written Examination1502A80N/A
##### Other Assessment
Description Semester When Set Percentage Comment
Written exercise1M4written exercise 1
Written exercise1M4written exercise 2
Written exercise2M4written exercise 3
Written exercise2M4written exercise 4
Written exercise2M4written exercise 5
##### Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. 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.

In the event of on-campus examinations not being possible, an on-line alternative assessment will be used for written examination 1.