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MATH42920: Functional Analysis and Applications

It is possible that changes to modules or programmes might need to be made during the academic year, in response to the impact of Covid-19 and/or any further changes in public health advice.

Type Open
Level 4
Credits 20
Availability Available in 2024/2025
Module Cap None.
Location Durham
Department Mathematical Sciences

Prerequisites

  • Prior knowledge of Analysis

Corequisites

  • None

Excluded Combinations of Modules

  • None

Aims

  • To introduce key concepts in Functional Analysis and to explore its applications in fields such as Spectral Theory and/or Partial Differential Equations (PDEs).

Content

  • Spaces and operators: Banach and Hilbert spaces; linear operators and dual spaces; strong and weak convergence.
  • Cornerstones of Functional Analysis: Hahn-Banach theorem; Baire category theorem and uniform boundedness principle; open mapping theorem and closed graph theorem.
  • Applications - a selection of the following: Spectral theory; Hilbert space methods for PDEs; calculus of variations and optimal transport.

Learning Outcomes

Subject-specific Knowledge:

  • By the end of the module students will:
  • Be able to solve novel and/or complex problems in the field of Functional Analysis.
  • Have an understanding of specialised and complex theoretical mathematics in the field of Functional Analysis.
  • Have mastered a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: Spectral theorem for compact self-adjoint operators; Sobolev spaces and regularity of solutions of PDEs; Monge-Kantorovich problems and gradient flows.

Subject-specific Skills:

  • Students will have developed advanced technical and scholastic skills in the area of Functional Analysis.

Key Skills:

  • Students will have highly specialised skills in the following areas: problem solving, abstract reasoning.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

  • Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
  • Problem classes show how to solve example problems in an ideal way, revealing also the thought processes behind such solutions.
  • Formative assessments provide feedback to guide students in the correct development of their knowledge and skills in preparation for the summative assessment.
  • The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

ActivityNumberFrequencyDurationTotalMonitored
Lectures422 per week in Michaelmas and Epiphany; 2 in Easter1 hour42 
Problems Classes8Fortnightly in Michaelmas and Epiphany1 hour8 
Preparation and Reading150 
Total200 

Summative Assessment

Component: ExaminationComponent Weighting: 100%
ElementLength / DurationElement WeightingResit Opportunity
End of year written examination3100 

Formative Assessment

Eight written or electronic assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

More information

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