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:
 EXTH 100%* 
Assessment Detail: 
 Open book Examination with a duration of 24 hours* (EXTH 100%)
*Assessment updated due to Covid19 disruptions 
Supplementary Assessment: 
 Likeforlike 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 selfadjoint, 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: Innerproduct spaces, Hilbert spaces. CauchySchwarz inequality, parallelogram identity. Examples. Orthogonality, GramSchmidt 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, selfadjoint operators. Spectrum of an operator. Spectral theory of selfadjoint, compact operators.
Applications and further topics, which might include: Fourier series, Gauss approximation problem, LaxMilgram theorem.

Programme availability: 
MA40256 is Optional on the following programmes:
Department of Computer Science
 USCMAFM14 : MComp(Hons) Computer Science and Mathematics (Year 4)
 USCMAAM14 : MComp(Hons) Computer Science and Mathematics with Study year abroad (Year 5)
 USCMAKM14 : MComp(Hons) Computer Science and Mathematics with Year long work placement (Year 5)
Department of Economics
 UHESAFB04 : BSc(Hons) Economics and Mathematics (Year 3)
 UHESAAB04 : BSc(Hons) Economics and Mathematics with Study year abroad (Year 4)
 UHESAKB04 : BSc(Hons) Economics and Mathematics with Year long work placement (Year 4)
 UHESACB04 : BSc(Hons) Economics and Mathematics with Combined Placement and Study Abroad (Year 4)
Department of Mathematical Sciences
 RSMAAFM16 : Integrated PhD Statistical Applied Mathematics
 TSMAAFM17 : MRes Statistical Applied Mathematics
 TSMAAFM16 : MSc Statistical Applied Mathematics
 USMAAFB15 : BSc(Hons) Mathematical Sciences (Year 3)
 USMAAAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
 USMAAKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
 USMAAFB13 : BSc(Hons) Mathematics (Year 3)
 USMAAAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
 USMAAKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
 USMAAFB01 : BSc(Hons) Mathematics and Statistics (Year 3)
 USMAAAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 4)
 USMAAKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 4)
 USMAAFB05 : BSc(Hons) Statistics (Year 3)
 USMAAAB06 : BSc(Hons) Statistics with Study year abroad (Year 4)
 USMAAKB06 : BSc(Hons) Statistics with Year long work placement (Year 4)
 USMAAFM14 : MMath(Hons) Mathematics (Year 3)
 USMAAFM14 : MMath(Hons) Mathematics (Year 4)
 USMAAAM15 : MMath(Hons) Mathematics with Study year abroad (Year 4)
 USMAAKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)
 USMAAKM15 : MMath(Hons) Mathematics with Year long work placement (Year 5)
Department of Physics
 USXXAFB03 : BSc(Hons) Mathematics and Physics (Year 3)
 USXXAAB04 : BSc(Hons) Mathematics and Physics with Study year abroad (Year 4)
 USXXAKB04 : BSc(Hons) Mathematics and Physics with Year long work placement (Year 4)
 USXXAFM01 : MSci(Hons) Mathematics and Physics (Year 4)
 USXXAAM01 : MSci(Hons) Mathematics and Physics with Study year abroad (Year 5)
 USXXAKM01 : MSci(Hons) Mathematics and Physics with Year long work placement (Year 5)
