|Semester 2 Credit Value:||20|
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.
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.
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.
|Guided Independent Study||Assessment preparation and completion||8||6:00||48:00||Written assignments|
|Scheduled Learning And Teaching Activities||Lecture||2||1:00||2:00||Revision lectures|
|Guided Independent Study||Assessment preparation and completion||1||23:00||23:00||Revision for unseen Exam|
|Guided Independent Study||Assessment preparation and completion||1||2:15||2:15||Unseen Exam|
|Scheduled Learning And Teaching Activities||Lecture||46||1:00||46:00||Formal lectures and tutorials|
|Scheduled Learning And Teaching Activities||Drop-in/surgery||24||0:00||0:00||Office Hours in a staff office|
|Guided Independent Study||Independent study||8||1:00||8:00||Assignment review|
|Guided Independent Study||Independent study||1||70:45||70:45||Studying, practising and gaining understanding of course material|
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.
The format of resits will be determined by the Board of Examiners
|Prob solv exercises||2||M||10||N/A|
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.
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.