| Unit name | Analysis 2 |
|---|---|
| Unit code | MATH20200 |
| Credit points | 20 |
| Level of study | I/5 |
| Teaching block(s) |
Teaching Block 1 (weeks 1 - 12) |
| Unit director | Dr. Netrusov |
| Open unit status | Not open |
| Pre-requisites |
MATH11002, MATH11003 and MATH11521 |
| Co-requisites |
None |
| School/department | School of Mathematics |
| Faculty | Faculty of Science |
This unit generalises some theorems about convergence and continuity from Level 1 analysis, and develops a theory of convergence in Rn and more generally in metric spaces. Topics will include basic topology (open, closed, compact, connected sets), continuity of functions, completeness, the contraction mapping theorem, compactness and topological spaces.
Aims:
To introduce metric spaces and to extend some theorems about convergence and continuity in the case of sequences of real numbers and real-valued functions of one real variable.
Relation to other units
This unit is a member of a sequence of analysis units at levels 2, 3, 4. It is a prerequisite for Measure Theory & Integration and for Functional Analysis.
Syllabus
Metric spaces (definition, examples, open sets, closed sets, interior, closure, limit points, equivalent metrics, product metrics).
Completeness (limits, continuity, Cauchy sequence, complete sets, isometries, completion of a metric space, contraction mapping theorem, existence and uniqueness of ordinary differential equations).
Compactness (definition, examples, continuous functions, uniform continuity, Heine-Borel theorem, criteria for compactness).
Connectedness (definition, examples, Rn, components, continuous functions, path connectedness).
Uniform convergence
At the end of the course the student should know and understand the definitions and theorems (and their proofs) and should be able to use the ideas of the course in unseen situations.
Transferable Skills
Assimilation of abstract ideas and reasoning in an abstract context. Setting out a sustained argument in a form comprehensible to others.
A standard lecture course of 33 lectures with 8 - 10 problem classes.
The final assessment mark for Metric Spaces is calculated from a 2½-hour written examination in April. The paper consists of FIVE questions. A candidate's FOUR best answers will be used for assessment. Calculators are NOT permitted.
