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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
SummaryThis 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: Honours Analysis (MATH10068) AND ( Linear Analysis (MATH10082) OR Linear and Fourier Analysis (MATH10081))
Prohibited Combinations Students MUST NOT also be taking Functional and Real Analysis (MATH11134)
Other requirements Students wishing to take both MATH11135 Functional Analysis and MATH11136 Real Analysis in the same academic session should register for the 20 credit course MATH11134 Functional and Real Analysis.
Information for Visiting Students
High Demand Course? Yes
Course Delivery Information
Academic year 2017/18, Available to all students (SV1) Quota:  None
Course Start Semester 2
Timetable Timetable
Learning and Teaching activities (Further Info) Total Hours: 100 ( Lecture Hours 22, Seminar/Tutorial Hours 5, Summative Assessment Hours 2, Programme Level Learning and Teaching Hours 2, Directed Learning and Independent Learning Hours 69 )
Assessment (Further Info) Written Exam 95 %, Coursework 5 %, Practical Exam 0 %
Additional Information (Assessment) Coursework 5%, Examination 95%
Feedback Not entered
Exam Information
Exam Diet Paper Name Hours & Minutes
Main Exam Diet S2 (April/May)Functional Analysis (MATH11135)2:00
Learning Outcomes
1. Facility with the main, big theorems of functional analysis.
2. Ability to use duality in various contexts and theoretical results from the course in concrete situations.
3. Capacity to work with families of applications appearing in the course, particularly specific calculations needed in the context of Baire Category.
4. Be able to 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 organiserProf A Carbery
Tel: (0131 6)50 5993
Course secretaryMr Martin Delaney
Tel: (0131 6)50 6427
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