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 | 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.
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Information for Visiting Students
Pre-requisites | None |
High Demand Course? |
Yes |
Course Delivery Information
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Academic year 2015/16, Available to all students (SV1)
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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 )
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Assessment (Further Info) |
Written Exam
95 %,
Coursework
5 %,
Practical Exam
0 %
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Additional Information (Assessment) |
Coursework 5%, Examination 95% |
Feedback |
Not entered |
Exam Information |
Exam Diet |
Paper Name |
Hours & Minutes |
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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.
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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 | Dr Martin Dindos
Tel:
Email: |
Course secretary | Mr Thomas Robinson
Tel: (0131 6)50 4885
Email: |
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© Copyright 2015 The University of Edinburgh - 21 October 2015 12:28 pm
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