MA32053: Functional analysis: Hilbert spaces
[Page last updated: 22 May 2025]
| Academic Year: | 2025/26 |
| Owning Department/School: | Department of Mathematical Sciences |
| Credits: | 5 [equivalent to 10 CATS credits] |
| Notional Study Hours: | 100 |
| Level: | Honours (FHEQ level 6) |
| Period: |
- Semester 2
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| Assessment Summary: | EXCB 100% |
| Assessment Detail: |
- Closed-book written examination (EXCB 100%)
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| Supplementary Assessment: |
- Like-for-like reassessment (where allowed by programme regulations)
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| Requisites: |
Before or while taking this module you must take MA32041 AND take MA32061
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| Learning Outcomes: |
By the end of the unit, you should be able to:
- state and prove the principal theorems relating to Hilbert space theory and spectral theory of self-adjoint, compact linear operators;
- apply these notions and theorems to simple examples and applications.
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| Synopsis: | Building on ideas from Advanced Analysis, this unit offers insight into a theory that is central to mathematical analysis and underpins methods and techniques in areas as diverse as geometry, scientific computing, probability theory, or quantum physics.
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| Content: | Revision of Hilbert spaces. Examples. Orthogonality, Gram-Schmidt process. Bessel's inequality. Orthogonal complements. Complete orthonormal sets in separable Hilbert spaces. Projection 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.
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| Course availability: |
MA32053 is Optional on the following courses:
Department of Mathematical Sciences
- 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-AFB30 : 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-AFB32 : 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-AFM30 : MMath(Hons) Mathematics (Year 3)
- USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)
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Notes: - This unit catalogue is applicable for the 2025/26 academic year only. Students continuing their studies into 2026/27 and beyond should not assume that this unit will be available in future years in the format displayed here for 2025/26.
- 好色tv and units are subject to change in accordance with normal University procedures.
- Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.
- Find out more about these and other important University terms and conditions here.
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