**First semester, 2017-2018**

This module presents the concept of a metric space, which is a set with a notion of distance defined on it; this includes subsets of real or complex Euclidean space, or, more generally, vector spaces with inner products (scalar products) defined on them. Metric spaces are fundamental objects in modern analysis, which allow one to talk about notions such as convergence and continuity in a much more general framework. The theory of metric spaces will be applied to find approximate solutions of equations and differential equations. Finally, more advanced topological notions such as connectedness and compactness are introduced.

**Syllabus:**

1. Definition and fundamental properties of a metric space. Open sets, closed sets, closure and interior. Convergence of sequences. Continuity of mappings.

2. Real inner-product spaces, orthonormal sequences, perpendicular distance to a subspace, applications in approximation theory.

3. Cauchy sequences, completeness of **R** with the standard metric; uniform convergence and completeness of C[a,b] with the uniform metric.

4. The contraction mapping theorem, with applications in the solution of equations and differential equations.

5. Connectedness and path-connectedness.
Introduction to compactness and sequential compactness, including subsets of
**R**^{n}.

**Booklist:**

Victor Bryant, Metric spaces: iteration and application, Cambridge, 1985.

M. Ó. Searcóid, Metric Spaces, Springer Undergraduate Mathematics Series, 2006.

D. Kreider, An introduction to linear analysis, Addison-Wesley, 1966.

Information from module catalogue

Last updated 1st September 2017