## MA3FA1-Functional Analysis I

Module Provider: Mathematics and Statistics
Number of credits: 10 [5 ECTS credits]
Level:6
Terms in which taught: Autumn term module
Pre-requisites: MA1LA Linear Algebra and MA2CA1 Complex Analysis I or MA2RCA Real and Complex Analysis
Non-modular pre-requisites:
Co-requisites: MA3TLA Topology and Linear Analysis and MA3MTI Measure Theory and Integration
Modules excluded: MA4FA1 Functional Analysis I
Module version for: 2017/8

Module Convenor: Dr Jani Virtanen

Summary module description:

Functional analysis is a vast area of modern mathematics that in its simplest form studies infinite-dimensional linear (vector) spaces equipped with a given topology. The two main directions are the study of the geometry of the linear space and of the linear operators acting on the space. Our focus is on the former, while MA3/4OT (Operator Theory) is concerned with the latter. The types of linear spaces we study are Banach spaces and Hilbert spaces, which are the two most fundamental structures with great importance in other parts of mathematical analysis and its applications.

Aims:

To introduce the basic theory of Hilbert spaces, discuss their geometry, which generalizes the notion of the finite-dimensional Euclidean spaces, and demonstrate the effect of the dimension of the space in its study. To recall the concept of Banach spaces, introduced in MA3TLA, study their further properties, and provide examples of useful Banach spaces that are not Hilbert spaces. To discuss applications of functional analysis in other parts of mathematics and mathematical physics. To prepare students for further (MSc and Phd) studies in the area of analysis and its applications, and for a career in industry as a highly skilled research mathematician that can deal with a variety of (physical) applications.

Assessable learning outcomes:

By the end of the module students are expected to be able to:

• demonstrate understanding of the geometry of Hilbert spaces;

• demonstrate understanding of the difference between Hilbert and Banach spaces, and the role that the inner product of Hilbert space plays in it;

• prove the main theorems and use them to solve problems in other areas of mathematics.

To understand the role that functional analysis plays in other branches of mathematics, mathematical physics and their applications.

Outline content:

Hilbert spaces, examples of Hilbert spaces, orthogonality, the Riesz representation theorem, orthonormal sets, isomorphic Hilbert spaces. Banach spaces, examples of Banach spaces, bounded linear operators and functionals, the Hahn-Banach theorem, the dual space, the open mapping theorem, the closed graph theorem, the principle of uniform boundedness. Applications.

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 100

Other information on summative assessment:
One examination paper.

Formative assessment methods:
Problem sheets.

Penalties for late submission:
The Module Convenor 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.