- Offered for Year: 2022/23
- Module Leader(s): Dr David Seifert
- Owning School: Mathematics, Statistics and Physics
- Teaching Location: Newcastle City Campus
Semesters
|
Semester 1 Credit Value:
|
10
|
|
ECTS Credits:
|
5.0
|
Aims
To introduce students to the basic ideas of functional analysis, an important area of research developed to study integral and differential equations. To introduce students to the notion of convergence and continuous transformations on normed vector spaces.
Module Summary
Around the end of the 19th century, mathematicians began to realise that the methods used to solve differential equations are in some ways similar to those involved in solving simultaneous equations. They soon began to investigate this similarity carefully. The topic of this course grew out of an attempt to provide a rigorous framework within which to explain phenomena in applied mathematics. One needs linear algebra to explain the matrix behaviour, and analysis to explain the calculus. The result is the concept of a Banach space, a structure in which we have vectors and a notion of size, and operators, which in many ways are just like matrices. The course develops the general theory, stressing the similarities between the new ideas and familiar concepts.
Outline Of Syllabus
Norms, inner products. Convergence, Cauchy sequences, completeness, Banach spaces and Hilbert spaces. Examples to include sequence spaces and continuous functions. Bounded linear operators on normed spaces: operator norm, inverses, the spectrum. Dual spaces, adjoint operators.
Teaching Methods
Teaching Activities
| Category |
Activity |
Number |
Length |
Student Hours |
Comment |
|---|
| Scheduled Learning And Teaching Activities | Lecture | 20 | 1:00 | 20:00 | 20 Formal Lectures |
| Scheduled Learning And Teaching Activities | Lecture | 2 | 1:00 | 2:00 | 2 Revision Lectures |
| Scheduled Learning And Teaching Activities | Lecture | 5 | 1:00 | 5:00 | 5 Problem Classes |
| Guided Independent Study | Assessment preparation and completion | 15 | 1:00 | 15:00 | Completion of in course assessments |
| Guided Independent Study | Independent study | 58 | 1:00 | 58:00 | Preparation time for lectures, background reading, coursework review |
| Total | | | | 100:00 | |
Jointly Taught With
| Code |
Title |
|---|
| MAS8702 | Linear analysis |
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 Examination | 120 | 1 | A | 80 | N/A |
Exam Pairings
| Module Code |
Module Title |
Semester |
Comment |
|---|
| MAS8702 | Linear analysis | 1 | N/A |
Other Assessment
| Description |
Semester |
When Set |
Percentage |
Comment |
|---|
| Prob solv exercises | 1 | M | 10 | N/A |
| Prob solv exercises | 1 | M | 10 | N/A |
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.
Note: the exam for MAS8702 is more challenging than the exam for MAS3702
Reading Lists
Timetable
