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DRPS : Course Catalogue : School of Mathematics : Mathematics

Undergraduate Course: Functional Analysis (MATH11135)

Course Outline
SchoolSchool of Mathematics CollegeCollege of Science and Engineering
Credit level (Normal year taken)SCQF Level 11 (Year 5 Undergraduate) AvailabilityAvailable to all students
SCQF Credits10 ECTS Credits5
SummaryNB. This course is delivered *biennially* with the next instance being in 2023 - 24. It is anticipated that it would then be delivered every other session thereafter.

This course will cover the foundations of functional analysis in the context of normed linear spaces The Big Theorems (Hahn-Banach, Baire Category, Uniform Boundedness, Open Mapping and Closed Graph) will be presented and several applications will be analysed. The important notion of duality will be developed in Banach and Hilbert spaces and an introduction to spectral theory for compact operators will be given.

Course description This course covers major theorems of Functional Analysis that have applications in Harmonic and Fourier Analysis, Ordinary and Partial Differential Equations. This course is a natural follow on of the course Linear Analysis; while the main focus of the Linear Analysis is on Hilbert spaces with its rich geometrical structures this course will work with normed linear spaces. Hilbert space is a special case of a normed linear space. Despite working in this more general framework many results encountered in Linear Algebra will be re-introduced in this course in more general form. For example the spectral theorem will be presented for all compact operators.
Review of linear spaces and their norms.
The Hahn-Banach, Baire Category, Uniform Boundedness Principle, Open Mapping and Closed Graph theorems.
Duality in Banach and Hilbert spaces.
Spectral theory for compact operators on Banach spaces. Fredholm alternative.
Weak topologies, Banach-Alaoglu and the Arzela-Ascoli theorem.
Entry Requirements (not applicable to Visiting Students)
Pre-requisites Students MUST have passed: Linear Analysis (MATH10082) AND Metric Spaces (MATH10101)
Prohibited Combinations Other requirements A pass in Honours Analysis (MATH10068) in 2020-21 or earlier is an acceptable substitute for a pass in Metric Spaces (MATH10101)
Information for Visiting Students
Pre-requisitesVisiting students are advised to check that they have studied the material covered in the syllabus of each prerequisite course before enrolling
High Demand Course? Yes
Course Delivery Information
Not being delivered
Learning Outcomes
On completion of this course, the student will be able to:
  1. Demonstrate facility with the main, big theorems of functional analysis.
  2. Use duality in various contexts and theoretical results from the course in concrete situations.
  3. Work with families of applications appearing in the course, particularly specific calculations needed in the context of Baire Category.
  4. Produce examples and counterexamples illustrating the mathematical concepts presented in the course.
  5. Understand the statements and proofs of important theorems and be able to explain the key steps in proofs, sometimes with variation.
Reading List

1. Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis. Universitext, Springer.

2. Elements of Functional Analysis, by Robert Zimmer, University of
Chicago Lecture Series.
Additional Information
Graduate Attributes and Skills Not entered
Course organiserDr Robert Bickerton
Tel: (0131 6)50 5284
Course secretaryMr Martin Delaney
Tel: (0131 6)50 6427
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