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Module

MAS8702 : Linear analysis

  • 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 ActivitiesLecture201:0020:0020 Formal Lectures
Scheduled Learning And Teaching ActivitiesLecture21:002:002 Revision Lectures
Scheduled Learning And Teaching ActivitiesLecture51:005:005 Problem Classes
Guided Independent StudyAssessment preparation and completion151:0015:00Completion of in course assessments
Guided Independent StudyIndependent study581:0058:00Preparation time for lectures, background reading, coursework review
Total100:00
Jointly Taught With
Code Title
MAS3702Linear 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 Examination1201A80N/A
Exam Pairings
Module Code Module Title Semester Comment
MAS3702Linear analysis1N/A
Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises1M10N/A
Prob solv exercises1M10N/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