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MA32041: Advanced analysis

[Page last updated: 22 May 2025]

Academic Year: 2025/26
Owning Department/School: Department of Mathematical Sciences
Credits: 10 [equivalent to 20 CATS credits]
Notional Study Hours: 200
Level: Honours (FHEQ level 6)
Period:
Semester 1
Assessment Summary: EXCB 100%
Assessment Detail:
  • Examination (EXCB 100%)
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Requisites: Before taking this module you must take MA22033 OR take MA20219
In taking this module you cannot take MA30252
Learning Outcomes: By the end of the unit, students should be able to:
  • demonstrate a good knowledge and understanding of metric spaces, Banach spaces, Hilbert spaces, and the mappings between them;
  • state and prove the principal theorems relating to completeness, compactness, and dense sets in metric spaces and to bounded linear operators and dual spaces;
  • apply the theory to simple examples.



Synopsis: This unit will introduce you to ideas that arise from a more abstract perspective. You will explore a range of tools leading to the techniques and methods of modern mathematical analysis.

Content: Revision of metric spaces, normed spaces, and inner product spaces. Banach spaces and Hilbert spaces. Cauchy-Schwarz inequality, parallelogram identity. Convergence and continuity. Examples. Completion of metric, normed, and inner product spaces. Dense and nowhere dense sets, separable spaces, Baire category theorem. Equivalence of compactness and sequential compactness in metric spaces, relatively compact sets, Arzela-Ascoli theorem. Uniform approximation of continuous functions, Polynomial and trigonometric polynomial approximation. The space of bounded linear operators. Dual spaces. Riesz representation theorem. Weak and weak
* convergence.

Course availability:

MA32041 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)

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.