This module develops and consolidates the student’s knowledge of elementary real analysis, with a view to providing a sound foundation for more advanced studies in pure and applied mathematics. The concept of limits of sequences is extended to sequences of functions and the student is introduced to elementary topological notions, basic properties of metric spaces, Banach’s Fixed-Point Theorem and some of its applications.
Sequences and Series of Functions
Sequences of functions: pointwise and uniform convergence, applications of uniform convergence.
Definition, elementary properties and examples including: Euclidean Metric, Taxicab metric (application in pattern recognition), Sup metric,
Discrete metric, Hamming metric (applications in Coding Theory). Open, closed and bounded sets; accumulation points, closure, interior and
boundary of a set. Sequences, continuous functions, Cauchy sequences and completeness.
Banach's Fixed Point Theorem
Statement and proof of the theorem. Applications to differential and linear equations.
2 hours of lectures and 1 hour tutorial session per week.
The lectures will provide theoretical material which will be underpinned by many examples to demonstrate the use of this material. The tutorial sessions will provide students with supervised practice time using appropriate exercises.
|Module Content & Assessment