Analysis module (MA32001)

Credits

15

Module code

MA32001

This module provides an in-depth study of Analysis aimed at Level 3 or 4 students in which the concepts are defined precisely and the results are proved rigorously. This module is mandatory for Level 3 students taking a BSc or MMath in Mathematics. This module may be taken in combination with other Level 3 or 4 modules by Level 4 students on Mathematics combined degrees other than those taking the MSci in Mathematics and Physics. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

Prerequisites

Students taking this module must have achieved a pass mark in each of the modules MA21001 and MA22001, or equivalents.

Indicative Content

• Normed and Metric Spaces

Supremum, Completeness Axiom. Definitions and properties of normed and metric spaces, convergence of sequences, continuity, closed sets (in terms of limit points)

• Connectedness and Completeness

Connected sets: definition in metric spaces; relation to the concept of continuity. Cauchy sequences, completeness and relation to closed sets, Banach's contraction mapping theorem

• Compact Sets

General definition with open sets of the notion of compactness, its sequences characterisations on metric spaces, and its connection with closed subsets. "Closed and bounded" characterisation of the compact sets in Rn (i.e., the Heine-Borel Theorem), connection with limit points, Weierstrass Theorem. Connection between continuity and compactness. The concept of uniform continuity and its connection with compactness. Urysohn's Lemma [1 lecture]. Partition of Unity on Rn

• Convergence and Equicontinuity

Uniform convergence of sequences of functions. The concept of equicontinuity of a family of functions and Arzela-Ascoli Theorem.

• Series

Ratio test, comparison test, Weierstrass M-test. Power series, Taylor series (mention of Taylor's theorem).

Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).

Credit Rating

This module is a Scottish Higher Education Level 3 or SCQF level 9 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.