 Academic Year: | 2019/0 |
| Owning Department/School: | Department of Mathematical Sciences |
| Credits: | 6 [equivalent to 12 CATS credits] |
| Notional Study Hours: | 120 |
| Level: | Masters UG & PG (FHEQ level 7) |
| Period: |
- Semester 2
|
| Assessment Summary:
| EX-TH 100%* |
| Assessment Detail: |
- Open book Examination with a duration of 24 hours* (EX-TH 100%)
*Assessment updated due to Covid-19 disruptions |
| Supplementary Assessment: |
- Like-for-like reassessment (where allowed by programme regulations)
|
| Requisites: |
Before taking this module you must take MA30252 OR take MA40043
|
| Description:
| Aims:
To introduce and study the theory of Hilbert spaces, the mappings between them, and spectral theory.
Learning Outcomes:
By the end of the unit, students should be able to state and prove the principal theorems relating to Hilbert space theory and spectral theory of self-adjoint, compact linear operators, and to apply these notions and theorems to simple examples and applications.
Skills:
Numeracy T/F, A
Problem Solving T/F, A
Written Communication F (on problem sheets).
Content:
Inner-product spaces, Hilbert spaces. Cauchy-Schwarz inequality, parallelogram identity. Examples. Orthogonality, Gram-Schmidt process. Bessel's inequality.
Orthogonal complements. Complete orthonormal sets in separable Hilbert spaces.
Projection theorem. Bounded linear operators, dual spaces. Riesz representation theorem. Compact operators. Adjoint of an operator, self-adjoint operators. Spectrum of an operator. Spectral theory of self-adjoint, compact operators.
Applications and further topics, which might include: Fourier series, Gauss approximation problem, Lax-Milgram theorem.
|
| Programme availability: |
MA40256 is Optional on the following programmes:
Department of Computer Science
- USCM-AFM14 : MComp(Hons) Computer Science and Mathematics (Year 4)
- USCM-AAM14 : MComp(Hons) Computer Science and Mathematics with Study year abroad (Year 5)
- USCM-AKM14 : MComp(Hons) Computer Science and Mathematics with Year long work placement (Year 5)
Department of Economics
- UHES-AFB04 : BSc(Hons) Economics and Mathematics (Year 3)
- UHES-AAB04 : BSc(Hons) Economics and Mathematics with Study year abroad (Year 4)
- UHES-AKB04 : BSc(Hons) Economics and Mathematics with Year long work placement (Year 4)
- UHES-ACB04 : BSc(Hons) Economics and Mathematics with Combined Placement and Study Abroad (Year 4)
Department of Mathematical Sciences
- RSMA-AFM16 : Integrated PhD Statistical Applied Mathematics
- TSMA-AFM17 : MRes Statistical Applied Mathematics
- TSMA-AFM16 : MSc Statistical Applied Mathematics
- USMA-AFB15 : BSc(Hons) Mathematical Sciences (Year 3)
- USMA-AAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
- USMA-AKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
- USMA-AFB13 : BSc(Hons) Mathematics (Year 3)
- USMA-AAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
- USMA-AKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
- USMA-AFB01 : BSc(Hons) Mathematics and Statistics (Year 3)
- USMA-AAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 4)
- USMA-AKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 4)
- USMA-AFB05 : BSc(Hons) Statistics (Year 3)
- USMA-AAB06 : BSc(Hons) Statistics with Study year abroad (Year 4)
- USMA-AKB06 : BSc(Hons) Statistics with Year long work placement (Year 4)
- USMA-AFM14 : MMath(Hons) Mathematics (Year 3)
- USMA-AFM14 : MMath(Hons) Mathematics (Year 4)
- USMA-AAM15 : MMath(Hons) Mathematics with Study year abroad (Year 4)
- USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)
- USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 5)
Department of Physics
- USXX-AFB03 : BSc(Hons) Mathematics and Physics (Year 3)
- USXX-AAB04 : BSc(Hons) Mathematics and Physics with Study year abroad (Year 4)
- USXX-AKB04 : BSc(Hons) Mathematics and Physics with Year long work placement (Year 4)
- USXX-AFM01 : MSci(Hons) Mathematics and Physics (Year 4)
- USXX-AAM01 : MSci(Hons) Mathematics and Physics with Study year abroad (Year 5)
- USXX-AKM01 : MSci(Hons) Mathematics and Physics with Year long work placement (Year 5)
|
