Module Code: MAT3004 
Module Title: INTRODUCTION TO FUNCTION SPACES 

Module Provider: Mathematics

Short Name: MS310

Level: HE3

Module Coordinator: ZELIK S Dr (Maths)

Number of credits: 15

Number of ECTS credits: 7.5




Module Availability 
Autumn 


Assessment Pattern 
Unit(s) of Assessment

Weighting Towards Module Mark( %)

Coursework (closed book class tests and exercises)

25

Written examination (2 hours, unseen)

75

Qualifying Condition(s) A weighted aggregate mark of 40% is required to pass the module.



Module Overview 
The objective is to introduce the subject of (infinite dimensional) spaces consisting of functions and show how they are structured by metric, norm or inner product. Cauchy sequences and convergent sequences and completeness are presented and the Contraction Mapping Theorem is discussed and applied to derive the Implicit Function Theorem. The relation between orthogonal bases and Fourier analysis is made clear and applied to practical problems. 


Prerequisites/Corequisites 
MAT1016 Linear Algebra MAT2004 Real Analysis 2 


Module Aims 



Learning Outcomes 
At the end of the module the student should:
 Have a basic understanding of the abstract concept of a metric and normed space and apply them to e.g. Euclidean space, C([0,1]) , L^{1}, and L^{2}.
 Be able to determine whether simple sequences of functions converge pointwise, uniform and/or in norm and appreciate that convergence depends on the choice of norm.
 Be able to apply the Contraction Mapping Theorem and Implicit Function Theorem in practical situations.
 Have a firm understanding of inner product spaces and the role of orthogonality in applications; particularly Fourier Theory.



Module Content 
The contents of the lectures will include:
· Metric and normed spaces, their definitions and basic examples, including Euclidean space, discrete metric, and the L^{1} and L^{2}norm.
· Open and closed sets, Cauchy and convergent sequences, completeness.
· Pointwise versus uniform convergence and uniform limits of continuous functions.
· Fixed points and the Contraction Mapping Theorem; applications to e.g. (Newton) iteration, the Implicit Function Theorem and existence of solutions of ODEs.
· Inner product spaces, their definition and basic examples. CauchySchwarz inequality and parallelogram law.
· Orthogonal systems, Bessel’s inequality. Fourier analysis and applications (such as the wave equation). 


Methods of Teaching/Learning 
Teaching is by lectures and example classes. Learning takes place through lectures, exercises (example sheets), preparation for tests and background reading. 


Selected Texts/Journals 
K. Saxe, Beginning Functional Analysis, Springer, (2001).
E. Kreyszig, Introductory Functional Analysis with Applications, Wiley (1978).
V. Bryant, Metric Spaces: Iteration and Application,
Cambridge
University Press (1985).
J.E. Marsden, Elementary Classical Analysis, W.H. Freeman & Co. (1974).



Last Updated 
27 July 2009 


