Undergraduate Course: Functional Analysis (MATH11135)
Course Outline
School  School of Mathematics 
College  College of Science and Engineering 
Credit level (Normal year taken)  SCQF Level 11 (Year 5 Undergraduate) 
Availability  Available to all students 
SCQF Credits  10 
ECTS Credits  5 
Summary  NB. This course is delivered *biennially* with the next instance being in 2019  20. 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 (HahnBanach, 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.
In academic year 202223 and later, MATH10101 Metric Spaces is a prerequisite for this course. 
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 reintroduced 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 HahnBanach, 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, BanachAlaoglu and the ArzelaAscoli theorem.

Entry Requirements (not applicable to Visiting Students)
Prerequisites 
Students MUST have passed:
Honours Analysis (MATH10068) AND
Linear Analysis (MATH10082)

Corequisites  
Prohibited Combinations  
Other requirements  In academic year 202223 and later, MATH10101 Metric Spaces is a prerequisite for this course. 
Information for Visiting Students
Prerequisites  Visiting 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:
 Demonstrate facility with the main, big theorems of functional analysis.
 Use duality in various contexts and theoretical results from the course in concrete situations.
 Work with families of applications appearing in the course, particularly specific calculations needed in the context of Baire Category.
 Produce examples and counterexamples illustrating the mathematical concepts presented in the course.
 Understand the statements and proofs of important theorems and be able to explain the key steps in proofs, sometimes with variation.

Reading List
Recommended:
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 
Keywords  FunAn 
Contacts
Course organiser  Mr Silouanos Brazitikos
Tel: (0131 6)50 5060
Email: Silouanos.Brazitikos@ed.ac.uk 
Course secretary  Mr Martin Delaney
Tel: (0131 6)50 6427
Email: Martin.Delaney@ed.ac.uk 

