MAS8220 : Topology and Functional Analysis

  • Module Leader(s): Dr Zinaida Lykova
  • Owning School: Mathematics & Statistics
Semesters
Semester 2 Credit Value: 20
ECTS Credits: 10.0

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.

Teaching Methods

Teaching Activities
Category Activity Number Length Student Hours Comment
Scheduled Learning And Teaching ActivitiesLecture21:002:00Revision lectures
Guided Independent StudyAssessment preparation and completion123:0023:00Revision for unseen Exam
Guided Independent StudyAssessment preparation and completion12:152:15Unseen Exam
Scheduled Learning And Teaching ActivitiesLecture461:0046:00Formal lectures and tutorials
Guided Independent StudyAssessment preparation and completion86:0048:00Written assignments
Scheduled Learning And Teaching ActivitiesDrop-in/surgery240:000:00Office Hours in a staff office
Guided Independent StudyIndependent study81:008:00Assignment review
Guided Independent StudyIndependent study170:4570:45Studying, practising and gaining understanding of course material
Total200:00
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. Tutorials are used to discuss the course material, identify and resolve specific queries raised by students and to allow students to receive individual feedback on marked work. Office hours provide an opportunity for more direct contact between individual students and the lecturer.

Assessment Methods

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

Exams
Description Length Semester When Set Percentage Comment
Written Examination1352A90unseen
Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises2M10N/A
Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. Written assignments (approximately 8 pieces of work of approximately equal weight) 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; this is thus formative as well as summative assessment.

Reading Lists

Timetable

Disclaimer: The University will use all reasonable endeavours to deliver modules in accordance with the descriptions set out in this catalogue. Every effort has been made to ensure the accuracy of the information, however, the University reserves the right to introduce changes to the information given including the addition, withdrawal or restructuring of modules if it considers such action to be necessary.