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# Functional analysis

Modern mathematics places emphasis on classes of functions, rather than special properties of individual functions. Functions are considered as elements of a set and these sets can usually be given the additional structure of a linear and topological space. The interplay of these structures gives the subject its characteristic flavour. Along with the set of functions, this module considers sets of mappings from the set into various other sets, particularly, linear mappings onto the set itself (operators) and into complex numbers (linear functionals).

## What you will study

This MSc in Mathematics module does not consider any of the many applications of the theory, which include approximations, optimisation, classical and quantum mechanics, statistical mechanics, infinite group representations and harmonic analysis, and partial differential equations.

The module is based on the set book Elements of Functional Analysis by I. J. Maddox. We follow that text chapter by chapter, omitting the last one.

### You will learn

Successful study of this module should enhance your skills in understanding complex mathematical texts, working with abstract concepts, constructing solutions to problems logically and communicating mathematical ideas clearly.

## Entry

To study this module you must:

• declare the MSc in Mathematics (or another qualification towards which the module can count) as your qualification intention
• have successfully completed at least one of the intermediate modules M828, M821, M832, M836, M829 or M835.

For this module you are also advised to have studied some or all of the modules Approximation theory (M832), Fractal geometry (M835) and Analytic number theory II (M829), as you must have a sound knowledge of convergence, continuity, complex functions and Cauchy’s theorem, elementary set theory and mappings, topological spaces (particularly metric spaces), compactness, vector spaces and linear maps between them.

All teaching is in English and your proficiency in the English language should be adequate for the level of study you wish to take. We strongly recommend that students have achieved an IELTS (International English Language Testing System) score of at least 7. To assess your English language skills in relation to your proposed studies you can visit the IELTS website.

If you have any doubt about the suitability of the module, please contact our Student Registration & Enquiry Service.

## Qualifications

M826 is an optional module in our:

Some postgraduate qualifications allow study to be chosen from other subject areas. We advise you to refer to the relevant qualification descriptions for information on the circumstances in which this module can count towards these qualifications because from time to time the structure and requirements may change.

## Regulations

As a student of The Open University, you should be aware of the content of the Module Regulations and the Student Regulations which are available on our Essential documents website.

## If you have a disability

You need to be able to read small print and diagrams.

## Study materials

### What's included

Module notes, other printed materials.

## Teaching and assessment

### Support from your tutor

You will have a tutor who will help you with the study material and mark and comment on your written work, and whom you can ask for advice and guidance. Contact our Student Registration & Enquiry Service if you want to know more about study with The Open University before you register.

### Assessment

The assessment details can be found in the facts box above.

You will be expected to submit your tutor-marked assignments (TMAs) online using a special maths eTMA processor, which is used in place of the main eTMA system, unless there are some difficulties which prevent you from doing so. In these circumstances, you must negotiate with your tutor to get their agreement to submit your assignment on paper.

You will, however, be granted the option of submitting on paper if typesetting electronically or merging scanned images of your answers to produce an electronic TMA would take you an unacceptably long time.

## Future availability

The details given here are for the final module start in October 2013.

## Distance learning

The Open University is the world's leading provider of flexible, high quality distance learning. Unlike other universities we are not campus based. You will study in a flexible way that works for you whether you're at home, at work or on the move. As an OU student you'll be supported throughout your studies - your tutor or study adviser will guide and advise you, offer detailed feedback on your assignments, and help with any study issues. Tuition might be in face-to-face groups, via online tutorials, or by phone.