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)

MS113 Linear Algebra, MS202 Introduction to Metric Spaces

The objectives are to introduce the students to the concept of a Hilbert space, which generalises Euclidean space to infinite dimensions, to introduce analysis and operators in a Hilbert space, and to indicate the range of applications where such concepts are needed.

At the end of the module the student should have

• a firm understanding of the concept of a function space, the basics of Hilbert spaces, the basic properties of operators acting on these spaces, as well as the basics of unbounded operators on a Hilbert space.
• a grasp of the importance of the concept of function spaces in applications, particularly in the analysis of differential and integral equations.

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).

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