Academic Year:
 2013/4 
Owning Department/School:
 Department of Mathematical Sciences 
Credits:
 6 
Level:
 Masters UG & PG (FHEQ level 7) 
Period: 
Semester 2

Assessment:
 EX 100% 
Supplementary Assessment: 
MA40057 Mandatory Extra Work (where allowed by programme regulations)

Requisites:
 Before taking this unit you must take MA30041 and take MA40043 
Description:
 Aims: To introduce the theory of infinitedimensional normed vector spaces, the linear mappings between them, and spectral theory.
Learning Outcomes: By the end of the unit, the students should be able to state and prove the principal theorems relating to Banach spaces, bounded linear operators, compact linear operators, and spectral theory of compact selfadjoint linear operators, and apply these notions and theorems to simple examples.
Skills: Numeracy T/F, A
Problem Solving T/F, A
Written Communication F (on problem sheets).
Content: Topics will be chosen from the following: Normed vector spaces and their metric structure. Banach spaces. Examples: Euclidean spaces, function spaces, Hilbert spaces. Riesz Lemma and finitedimensional subspaces. The space of bounded linear operators. Dual spaces and second duals. Uniform Boundedness Theorem. Open Mapping Theorem. Closed Graph Theorem. Projections onto closed subspaces. Invertible operators form an open set. Power series expansion for (IT)^{1}. Compact operators on Banach spaces. Spectrum of an operator  compactness of spectrum. Operators on Hilbert space and their adjoints. Spectral theory of selfadjoint compact operators. Zorn's Lemma. HahnBanach Theorem. Canonical embedding, reflexivity. Weak convergence. Riesz representation theorem. Fourier series.

Programme availability: 
MA40057 is Optional on the following programmes:
Department of Computer Science
 USCMAFM14 : MComp (hons) Computer Science and Mathematics (Fulltime)  Year 4
 USCMAKM14 : MComp (hons) Computer Science and Mathematics with Industrial Placement (Fulltime with Thick Sandwich Placement)  Year 5
 USCMAAM14 : MComp (hons) Computer Science and Mathematics with Study Year Abroad (Fulltime with Study Year Abroad)  Year 5
Department of Mathematical Sciences
 USMAAFB15 : BSc (hons) Mathematical Sciences (Fulltime)  Year 3
 USMAAKB16 : BSc (hons) Mathematical Sciences (Fulltime with Thick Sandwich Placement)  Year 4
 USMAAAB16 : BSc (hons) Mathematical Sciences with Study Year Abroad (Fulltime with Study Year Abroad)  Year 4
 USMAAFB13 : BSc (hons) Mathematics (Fulltime)  Year 3
 USMAAKB14 : BSc (hons) Mathematics (Fulltime with Thick Sandwich Placement)  Year 4
 USMAAFB01 : BSc (hons) Mathematics and Statistics (Fulltime)  Year 3
 USMAAKB02 : BSc (hons) Mathematics and Statistics (Fulltime with Thick Sandwich Placement)  Year 4
 USMAAAB02 : BSc (hons) Mathematics and Statistics with Study Year Abroad (Fulltime with Study Year Abroad)  Year 4
 USMAAAB14 : BSc (hons) Mathematics with Study Year Abroad (Fulltime with Study Year Abroad)  Year 4
 USMAAFB05 : BSc (hons) Statistics (Fulltime)  Year 3
 USMAAKB06 : BSc (hons) Statistics (Fulltime with Thick Sandwich Placement)  Year 4
 USMAAAB06 : BSc (hons) Statistics with Study Year Abroad (Fulltime with Study Year Abroad)  Year 4
 USMAAFM14 : MMath Mathematics (Fulltime)  Year 3
 USMAAFM14 : MMath Mathematics (Fulltime)  Year 4
 USMAAAM15 : MMath Mathematics with Study Year Abroad (Fulltime with Study Year Abroad)  Year 4
 TSMAAFM09 : MSc Mathematical Sciences (Fulltime)
 TSMAAPM09 : MSc Mathematical Sciences (Parttime)
 TSMAAFM08 : MSc Modern Applications of Mathematics (Fulltime)
 TSMAAWM14 : MSc Modern Applications of Mathematics (Fulltime incorporating placement)
 TSMAAFL02 : PG Dip Modern Applications of Mathematics (Fulltime)
Department of Physics
 USXXAFM01 : MSci (hons) Mathematics and Physics (Fulltime)  Year 4
