Course overview
Solving equations is a crucial aspect of working in mathematics, physics, engineering, and many other fields. These equations might be straightforward algebraic statements, or complicated systems of differential equations, but there are some fundamental questions common to all of these settings: does a solution exist? If so, is it unique? And if we know of the existence of some specific solution, how do we determine it explicitly or as accurately as possible? This course develops the foundations required to rigorously establish the existence of solutions to various equations, thereby laying the basis for the study of such solutions. Through an understanding of the foundations of analysis, we obtain insight critical in numerous areas of application, such areas ranging across physics, engineering, economics and finance.
Topics covered are: sets, functions, metric spaces and normed linear spaces, compactness, connectedness, and completeness. Banach fixed point theorem and applications, uniform continuity and convergence. General topological spaces, generating topologies, topological invariants, quotient spaces. Introduction to Hilbert spaces and bounded operators on Hilbert spaces.
Course learning outcomes
- Demonstrate an understanding of the concepts of metric spaces and topological spaces, and their role in mathematics
- Demonstrate familiarity with a range of examples of these structures
- Prove basic results about completeness, compactness, connectedness and convergence within these structures
- Use the Banach fixed point theorem to demonstrate the existence and uniqueness of solutions to differential equations
- Demonstrate an understanding of the concepts of Hilbert spaces and Banach spaces, and their role in mathematics
- Demonstrate familiarity with a range of examples of these structures
- Prove basic results about Hilbert spaces and Banach spaces and operators between such spaces
- Apply the theory in the course to solve a variety of problems at an appropriate level of difficulty
- Demonstrate skills in communicating mathematics orally and in writing