Autumn semester

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)

Linear Algebra 
Real Analysis 2

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.

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]) , L¹, and L².

   

• 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.

The contents of the lectures will include:

·        Metric and normed spaces, their definitions and basic examples, including Euclidean space, discrete metric, and the L¹ and L²-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. Cauchy-Schwarz inequality and parallelogram law.

·        Orthogonal systems, Bessel’s inequality.

·        Fourier analysis and applications (such as the wave equation).

• Compact sets, dense sets and closures.
• Complete orthogonal systems and Parseval’s equality.

Teaching is by lectures, tutorials, discussion sessions and suggestions for study in the course texts. Learning takes place through lectures, tutorials, discussion sessions, exercises and study of the course texts and background reading.

3 hours of lectures/tutorials/example classes per week for 10 weeks.

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).

31 July 2007