# Functional Analysis module (MA52006)

Credits

15

Module code

MA52006

This module, aimed at the Level 5 student, specifically aims to develop students' knowledge, skills and understanding of the fundamentals of function spaces (infinite dimensional spaces) and linear operators and functionals defined on function spaces.

Functional analysis plays an important role in many areas of applied mathematics and is essential for the theory of Partial Differential Equations, Numerical Analysis, Probability theory and Theoretical Physics. Functional analysis originated from classical analysis and is formed by the study of infinite dimensional vector spaces and linear functions defined on these spaces. The theory of functional analysis was developed by some of the most famous mathematicians of the 20th century such as Hilbert, Schmidt, Riesz, Banach and von Neumann. Functional analysis can be characterised as a combination of infinite-dimensional linear algebra and classical analysis. Methods of functional analysis will allow us to analyse the properties of function spaces and to characterise solutions of integral and differential equations, arising in modelling of many biological and physical systems.

### Prerequisites

Students taking this module must usually have achieved a pass mark in MA32001, or equivalent.

### Indicative Content

• #### Function Spaces

• Normed spaces and inner product spaces (Banach and Hilbert spaces)
• Lebesgue integral and Lebesgue spaces
• Orthogonality in Hilbert spaces
• Weak derivatives and Sobolev spaces
• Notion of weak convergence in Function Spaces
• #### Linear Operators and Linear Functionals

• Definition of linear operators and linear functionals
• Definition of a norm of a bounded linear operator
• Riesz and Lax-Milgram theorems
• Definition, main properties and examples of compact operators
• Spectral theory for compact operators
• #### Well-posedness results for Integral and Partial Differential Equations

• Application of Lax-Milgram theorem and theory of compact operators to prove the well-posedness results

### Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (60%) and coursework (40%) consisting of homeworks and tests.

### Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.

## Courses

This module is available on following courses: