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Programme & Unit Catalogues

## MA40043: Real & abstract analysis

Owning Department/School: Department of Mathematical Sciences
Credits: 6
Level: Masters UG & PG (FHEQ level 7)
Period: Semester 1
Assessment: EX 100%
Supplementary Assessment: MA40043 Mandatory Extra Work (where allowed by programme regulations)
Requisites: Before taking this unit you must take MA20216 and take MA20218 and while taking this unit you must take MA30041
Description: 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 unit in functional analysis.

Learning Outcomes:
By the end of the unit, students should be able to state and prove the principal theorems relating to uniform continuity and uniform convergence for real functions on metric spaces, compactness in spaces of continuous functions, and elementary Hilbert space theory, and to apply these notions and the 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: Uniform continuity and uniform limits of continuous functions on [0,1]. Abstract Stone-Weierstrass Theorem. Uniform approximation of continuous functions. Polynomial and trigonometric polynomial approximation, separability of C[0,1]. Total Boundedness. Diagonalisation. Ascoli-Arzelà Theorem. Complete metric spaces. Baire Category Theorem. Nowhere differentiable function. Completion of a metric space. Inner-product spaces. Hilbert spaces. Cauchy-Schwarz inequality, parallelogram identity. Examples. Orthogonality, Gram-Schmidt process. Bessel's inequality, Pythagoras' Theorem. Projections and subspaces. Orthogonal complements. Complete orthonormal sets in separable Hilbert spaces.
Programme availability:

#### MA40043 is Optional on the following programmes:

Department of Computer Science
• USCM-AFM14 : MComp (hons) Computer Science and Mathematics (Full-time) - Year 4
• USCM-AKM14 : MComp (hons) Computer Science and Mathematics with Industrial Placement (Full-time with Thick Sandwich Placement) - Year 5
• USCM-AAM14 : MComp (hons) Computer Science and Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 5
Department of Mathematical Sciences
• USMA-AFB15 : BSc (hons) Mathematical Sciences (Full-time) - Year 3
• USMA-AKB16 : BSc (hons) Mathematical Sciences (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB16 : BSc (hons) Mathematical Sciences with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFB13 : BSc (hons) Mathematics (Full-time) - Year 3
• USMA-AKB14 : BSc (hons) Mathematics (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AFB01 : BSc (hons) Mathematics and Statistics (Full-time) - Year 3
• USMA-AKB02 : BSc (hons) Mathematics and Statistics (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB02 : BSc (hons) Mathematics and Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AAB14 : BSc (hons) Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFB05 : BSc (hons) Statistics (Full-time) - Year 3
• USMA-AKB06 : BSc (hons) Statistics (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB06 : BSc (hons) Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFM14 : MMath Mathematics (Full-time) - Year 3
• USMA-AFM14 : MMath Mathematics (Full-time) - Year 4
• USMA-AAM15 : MMath Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• TSMA-AFM09 : MSc Mathematical Sciences (Full-time) - Year 1
• TSMA-APM09 : MSc Mathematical Sciences (Part-time) - Year 1
• TSMA-APM09 : MSc Mathematical Sciences (Part-time) - Year 2
• TSMA-AFM08 : MSc Modern Applications of Mathematics (Full-time) - Year 1
• TSMA-AFL02 : PG Dip Modern Applications of Mathematics (Full-time) - Year 1
Department of Physics
• USXX-AFM01 : MSci (hons) Mathematics and Physics (Full-time) - Year 4

Notes:
* This unit catalogue is applicable for the 2012/13 academic year only. Students continuing their studies into 2013/14 and beyond should not assume that this unit will be available in future years in the format displayed here for 2012/13.
* Programmes and units are subject to change at any time, 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.