 Academic Year:
| 2013/4 |
| Owning Department/School:
| Department of Mathematical Sciences |
| Credits:
| 6 |
| Level:
| Masters UG & PG (FHEQ level 7) |
| Period: |
Semester 2
|
| Assessment:
| EX 100% |
| Supplementary Assessment: |
MA40057 Mandatory Extra Work (where allowed by programme regulations)
|
| Requisites:
| Before taking this unit you must take MA30041 and take MA40043 |
| Description:
| Aims:
To introduce the theory of infinite-dimensional normed vector spaces, the linear mappings between them, and spectral theory.
Learning Outcomes:
By the end of the unit, the students should be able to state and prove the principal theorems relating to Banach spaces, bounded linear operators, compact linear operators, and spectral theory of compact self-adjoint linear operators, and apply these notions and theorems to simple examples.
Skills:
Numeracy T/F, A
Problem Solving T/F, A
Written Communication F (on problem sheets).
Content:
Topics will be chosen from the following: Normed vector spaces and their metric structure. Banach spaces. Examples: Euclidean spaces, function spaces, Hilbert spaces. Riesz Lemma and finite-dimensional subspaces. The space of bounded linear operators. Dual spaces and second duals. Uniform Boundedness Theorem. Open Mapping Theorem. Closed Graph Theorem. Projections onto closed subspaces. Invertible operators form an open set. Power series expansion for (I-T)-1. Compact operators on Banach spaces. Spectrum of an operator - compactness of spectrum. Operators on Hilbert space and their adjoints. Spectral theory of self-adjoint compact operators. Zorn's Lemma. Hahn-Banach Theorem. Canonical embedding, reflexivity. Weak convergence. Riesz representation theorem. Fourier series.
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| Programme availability: |
MA40057 is Optional on the following programmes:
Department of Computer Science
- USCM-AFM14 : MComp (hons) Computer Science and Mathematics (Full-time) - Year 4
- USCM-AKM14 : MComp (hons) Computer Science and Mathematics with Industrial Placement (Full-time with Thick Sandwich Placement) - Year 5
- USCM-AAM14 : MComp (hons) Computer Science and Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 5
Department of Mathematical Sciences
- USMA-AFB15 : BSc (hons) Mathematical Sciences (Full-time) - Year 3
- USMA-AKB16 : BSc (hons) Mathematical Sciences (Full-time with Thick Sandwich Placement) - Year 4
- USMA-AAB16 : BSc (hons) Mathematical Sciences with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
- USMA-AFB13 : BSc (hons) Mathematics (Full-time) - Year 3
- USMA-AKB14 : BSc (hons) Mathematics (Full-time with Thick Sandwich Placement) - Year 4
- USMA-AFB01 : BSc (hons) Mathematics and Statistics (Full-time) - Year 3
- USMA-AKB02 : BSc (hons) Mathematics and Statistics (Full-time with Thick Sandwich Placement) - Year 4
- USMA-AAB02 : BSc (hons) Mathematics and Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
- USMA-AAB14 : BSc (hons) Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
- USMA-AFB05 : BSc (hons) Statistics (Full-time) - Year 3
- USMA-AKB06 : BSc (hons) Statistics (Full-time with Thick Sandwich Placement) - Year 4
- USMA-AAB06 : BSc (hons) Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
- USMA-AFM14 : MMath Mathematics (Full-time) - Year 3
- USMA-AFM14 : MMath Mathematics (Full-time) - Year 4
- USMA-AAM15 : MMath Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
- TSMA-AFM09 : MSc Mathematical Sciences (Full-time)
- TSMA-APM09 : MSc Mathematical Sciences (Part-time)
- TSMA-AFM08 : MSc Modern Applications of Mathematics (Full-time)
- TSMA-AWM14 : MSc Modern Applications of Mathematics (Full-time incorporating placement)
- TSMA-AFL02 : PG Dip Modern Applications of Mathematics (Full-time)
Department of Physics
- USXX-AFM01 : MSci (hons) Mathematics and Physics (Full-time) - Year 4
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