MA30252: Advanced real analysis
[Page last updated: 03 August 2022]
Academic Year:  2022/23 
Owning Department/School:  Department of Mathematical Sciences 
Credits:  6 [equivalent to 12 CATS credits] 
Notional Study Hours:  120 
Level:  Honours (FHEQ level 6) 
Period: 
 Semester 1

Assessment Summary:  EX 100% 
Assessment Detail:  
Supplementary Assessment: 
 Likeforlike reassessment (where allowed by programme regulations)

Requisites: 
Before taking this module you must take MA20219

Learning Outcomes:
 By the end of the unit, students should be able to state and prove the principal theorems relating to completeness, compactness, and dense sets in metric and normed spaces, and to apply these notions and theorems to simple examples.

Aims:  To introduce and study abstract spaces and general ideas in analysis, to apply them to examples, and to lay the foundations for the MA4 analysis units.

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

Content:  Metric spaces and normed spaces, convergence and continuity. Examples.
Completion of a metric space. Dense and nowhere dense sets, separable spaces, Baire category theorem. Equivalence of compactness and sequential compactness in metric spaces, relatively compact sets, ArzelaAscoli theorem. Uniform approximation of continuous functions, Polynomial and trigonometric polynomial approximation.
Further topics, which might include: the abstract StoneWeierstrass theorem, existence of nowhere differentiable functions, connectedness and path connectedness.

Programme availability: 
MA30252 is Compulsory on the following programmes:
Department of Physics
 USXXAFM01 : MSci(Hons) Mathematics and Physics (Year 3)
 USXXAAM01 : MSci(Hons) Mathematics and Physics with Study year abroad (Year 4)
 USXXAKM01 : MSci(Hons) Mathematics and Physics with Year long work placement (Year 4)
MA30252 is Optional on the following programmes:
Department of Computer Science
 USCMAFB20 : BSc(Hons) Computer Science and Mathematics (Year 3)
 USCMAAB20 : BSc(Hons) Computer Science and Mathematics with Study year abroad (Year 4)
 USCMAKB20 : BSc(Hons) Computer Science and Mathematics with Year long work placement (Year 4)
 USCMAFM14 : MComp(Hons) Computer Science and Mathematics (Year 3)
 USCMAAM14 : MComp(Hons) Computer Science and Mathematics with Study year abroad (Year 3)
 USCMAKM14 : MComp(Hons) Computer Science and Mathematics with Year long work placement (Year 3)
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
 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)

Notes:  This unit catalogue is applicable for the 2022/23 academic year only. Students continuing their studies into 2023/24 and beyond should not assume that this unit will be available in future years in the format displayed here for 2022/23.
 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 prerequisite rules.
 Find out more about these and other important University terms and conditions here.
