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MA40256: Analysis in Hilbert spaces

[Page last updated: 05 August 2021]

Academic Year: 2021/2
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 100%
Assessment Detail:
  • Examination (EX 100%)
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Requisites: Before taking this module you must take MA30252
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)

Notes:

  • This unit catalogue is applicable for the 2021/22 academic year only. Students continuing their studies into 2022/23 and beyond should not assume that this unit will be available in future years in the format displayed here for 2021/22.
  • Programmes 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.