# MAS3711 : Linear Analysis & Topology

• Offered for Year: 2020/21
• Lecturer: Dr Michael Dritschel
• Owning School: Mathematics, Statistics and Physics
• Teaching Location: Newcastle City Campus
##### Semesters
 Semester 1 Credit Value: 10 Semester 2 Credit Value: 10 ECTS Credits: 10

#### Aims

To introduce students to the basic ideas of functional analysis (an important area of research and one in which the School is currently active).

To present the basic ideas of topology essential to an understanding of modern analysis and geometry.

Module Summary
At the end of the last century, mathematicians began to realise that the methods used to solve differential equations are quite like those involved in solving simultaneous equations and they began to investigate this similarity and make it rigorous. Thus, this topic grew out of an attempt to provide a framework 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 place where we have vectors and a notion of size, and operators, which are like matrices. The course develops the general theory, stressing the similarities between vectors and matrices and the new ideas.

Topology is an elegant and abstract subject which arose from disparate sources but is now fundamental in analysis and geometry. One way of viewing topology is to say it answers the question: what are the last features of a subset on n-dimensional Euclidean space to discover when one progressively deforms space? Another approach would be through the question: what do the many limiting procedures in mathematics have in common? It turns out that just three axioms are enough to produce a rich subject which provides the right setting in which to understand both the local aspects of sets and mappings (such as continuity) and the global aspects (such as the overall nature of a set).

#### Outline Of Syllabus

Norms. Cauchy sequences, completeness and Banach spaces. Examples: 3-d vectors, matrices, continuous functions. Hilbert space: the Cauchy-Schwartz inequality. Bounded operators on Banach spaces and Hilbert spaces: operator norm, adjoints, inverses, the spectrum.

Metric spaces and topologies. Open sets, closed sets, neighbourhoods. Interior, closure, boundary. Continuous functions, homeomorphisms. Hausdorff spaces. Compact space, Connected spaces, convergent sequences, completeness.

#### Teaching Methods

Please note that module leaders are reviewing the module teaching and assessment methods for Semester 2 modules, in light of the Covid-19 restrictions. There may also be a few further changes to Semester 1 modules. Final information will be available by the end of August 2020 in for Semester 1 modules and the end of October 2020 for Semester 2 modules.

##### Teaching Activities
Category Activity Number Length Student Hours Comment
Scheduled Learning And Teaching ActivitiesLecture91:009:00Present in Person
Scheduled Learning And Teaching ActivitiesLecture91:009:00Synchronous On Line Material
Structured Guided LearningLecture materials361:0036:00Non Synchronous Activities
Guided Independent StudyAssessment preparation and completion301:0030:00Completion of in course assessments
Structured Guided LearningStructured non-synchronous discussion181:0018:00Non Synchronous Discussion of Lecture Material
Scheduled Learning And Teaching ActivitiesDrop-in/surgery41:004:00Office hour or discussion board activity
Guided Independent StudyIndependent study941:0094:00Lecture preparation, background reading, course review
Total200:00
##### Jointly Taught With
Code Title
MAS8711Linear Analysis & Topology
##### Teaching Rationale And Relationship

Non-synchronous online materials are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on assessed work. Present-in-person and synchronous online sessions are used to help develop the students’ abilities at applying the theory to solving problems and to identify and resolve specific queries raised by students, and to allow students to receive individual feedback on marked work. Students who cannot attend a present-in-person session will be provided with an alternative activity allowing them to access the learning outcomes of that session. In addition, office hours/discussion board activity will provide an opportunity for more direct contact between individual students and the lecturer: a typical student might spend a total of one or two hours over the course of the module, either individually or as part of a group.
Alternatives will be offered to students unable to be present-in-person due to the prevailing C-19 circumstances.
Student’s should consult their individual timetable for up-to-date delivery information.

#### Assessment Methods

Please note that module leaders are reviewing the module teaching and assessment methods for Semester 2 modules, in light of the Covid-19 restrictions. There may also be a few further changes to Semester 1 modules. Final information will be available by the end of August 2020 in for Semester 1 modules and the end of October 2020 for Semester 2 modules.

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

##### Exams
Description Length Semester When Set Percentage Comment
Written Examination1202A8024 hour take home paper
##### Other Assessment
Description Semester When Set Percentage Comment
Written exercise1M8written exercises
Written exercise2M12written exercises
##### Assessment Rationale And Relationship

A substantial formal examination is appropriate for the assessment of the material in this module. The course assessments will 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.