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MA40256: Analysis in Hilbert spaces

[Page last updated: 22 May 2025]

Academic Year: 2025/26
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
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.


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.

Skills: Numeracy T/F, A
Problem Solving T/F, A
Written Communication F (on problem sheets).

Aims: To introduce and study the theory of Hilbert spaces, the mappings between them, and spectral theory.

Course availability:

MA40256 is Optional on the following courses:

Department of Mathematical Sciences
  • 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 5)
Department of Physics
  • 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 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.
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