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MATH4371: Functional Analysis and Applications IV

Please ensure you check the module availability box for each module outline, as not all modules will run in each academic year. Each module description relates to the year indicated in the module availability box, and this may change from year to year, due to, for example: changing staff expertise, disciplinary developments, the requirements of external bodies and partners, and student feedback. Current modules are subject to change in light of the ongoing disruption caused by Covid-19.

Type Open
Level 4
Credits 20
Availability Available in 2023/24
Module Cap None.
Location Durham
Department Mathematical Sciences

Prerequisites

  • Analysis III (MATH3011)

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 Michealmas 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 examination3 hours100none

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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