## MA3FA1-Functional Analysis I

Module Provider: Mathematics and Statistics
Number of credits: 10 [5 ECTS credits]
Level:6
Terms in which taught: Spring term module
Pre-requisites: MA1LIN Linear Algebra and MA2CA1 Complex Analysis I
Non-modular pre-requisites:
Co-requisites: MA3TLA Topology and Linear Analysis and MA3MTI Measure Theory and Integration
Modules excluded:
Module version for: 2015/6

Module Convenor: Dr Nikos Katzourakis

Summary module description:
The module continues the study of Banach spaces initiated in MA3TLA.

Aims:
To introduce the ideas and techniques of Functional Analysis and enable students to acquire skills in solving problems in the area. The first part of the module is devoted to the theory of Hilbert spaces, which have a rich geometric structure due to the presence of an inner product. The second part of the module studies the theory of bounded linear operators between Banach spaces, covering the fundamental results of Functional Analysis: the Hahn-Banach Theorem, the Open Mapping Theorem and the Uniform Boundedness Principle.

Assessable learning outcomes:
By the end of the module students are expected to be able to:
• identify and demonstrate understanding of the main definitions concerning Hilbert spaces and bounded linear operators between Banach spaces
• state and prove without the help of notes the main theorems covered in the module
• apply methods and techniques on Hilbert spaces and bounded linear operator between Banach spaces to solve problems in applications

Outline content:
The first part of the module is devoted to the theory of Hilbert spaces. Topics to be covered include: definition and basic properties of Hilbert spaces, examples; orthogonality and orthonormal sets; orthogonal decomposition; equivalent characterizations of countable orthonormal bases; the Riesz-Fischer Theorem; characterization of separability; trigonometric series.
The second part of the module is devoted to the theory of bounded linear operators between normed spaces. Topics to be covered include: definition and basic properties of bounded linear operators; the norm of an operator; the Banach-Steinhaus Theorem (Uniform Boundedness Principle); the Open Mapping Theorem; the dual of a normed space, examples; the Hahn-Banach Theorem, real and complex versions; the Riesz Representation Theorem.

Brief description of teaching and learning methods:
Lectures supported by problem sheets.

Contact hours:
 Autumn Spring Summer Lectures 20 0 Guided independent study 80 Total hours by term 100.00 0.00 Total hours for module 100.00

Summative Assessment Methods:
 Method Percentage Written exam 80 Set exercise 20

Other information on summative assessment:
Two assessed problem sheets and one examination paper.

Formative assessment methods:
Problem sheets.

Penalties for late submission:
The Module Convener will apply the following penalties for work submitted late, in accordance with the University policy.

• where the piece of work is submitted up to one calendar week after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for the piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of five working days;
• where the piece of work is submitted more than five working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

• The University policy statement on penalties for late submission can be found at: http://www.reading.ac.uk/web/FILES/qualitysupport/penaltiesforlatesubmission.pdf
You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Length of examination:
2 hours.

Requirements for a pass:
A mark of 40% overall.

Reassessment arrangements:
One examination paper of 2 hours duration in August/September - the resit module mark will be the higher of the exam mark (100% exam) and the exam mark plus previous coursework marks (80% exam, 20% coursework).

Last updated: 11 March 2015